Real-Life Math (2 vols) - K. Lerner, B. Lerner (Thomson Gale, 2006), Notas de estudo de Matemática

Baixe Real-Life Math (2 vols) - K. Lerner, B. Lerner (Thomson Gale, 2006) e outras Notas de estudo em PDF para Matemática, somente na Docsity! Real- Life Math Real-Life Math K. Lee Lerner and Brenda Wilmoth Lerner, Editors Project Editor Kimberley A. McGrath Editorial Luann Brennan, Meggin M. Condino, Madeline Harris, Paul Lewon, Elizabeth Manar Editorial Support Services Andrea Lopeman Indexing Factiva, a Dow Jones & Reuters Company Rights and Acquisitions Margaret Abendroth, Timothy Sisler Imaging and Multimedia Lezlie Light, Denay Wilding Product Design Pamela Galbreath, Tracey Rowens Composition Evi Seoud, Mary Beth Trimper Manufacturing Wendy Blurton, Dorothy Maki © 2006 Thomson Gale, a part of the Thomson Corporation. Thomson and Star Logo are trademarks and Gale and UXL are registered trademarks used herein under license. For more information, contact: Thomson Gale 27500 Drake Rd. Farmington Hills, MI 48331-3535 Or you can visit our Internet site at http://www.gale.com ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, record- ing, taping, Web distribution, or information storage retrieval systems—without the written permission of the publisher. For permission to use material from this product, submit your request via Web at http://www.gale-edit.com/permissions, or you may download our Permissions Request form and submit your request by fax or mail to: Permissions Thomson Gale 27500 Drake Rd. Farmington Hills, MI 48331-3535 Permissions Hotline: 248-699-8006 or 800-877-4253, ext. 8006 Fax: 248-699-8074 or 800-762-4058 While every effort has been made to ensure the reliability of the information presented in this publication, Thomson Gale does not guar- antee the accuracy of the data contained herein. Thomson Gale accepts no payment for listing; and inclusion in the publication of any organization, agency, institution, publication, service, or individual does not imply endorse- ment of the editors or publisher. Errors brought to the attention of the publisher and verified to the satisfaction of the publisher will be corrected in future editions. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Real-life math / K. Lee Lerner and Brenda Wilmoth Lerner, editors. p. cm. Includes bibliographical references and index. ISBN 0-7876-9422-3 (set : hardcover: alk. paper)— ISBN 0-7876-9423-1 (v. 1)—ISBN 0-7876-9424-X (v. 2) 1. Mathematics—Encyclopedias. I. Lerner, K. Lee. II. Lerner, Brenda Wilmoth. QA5.R36 2006 510’.3—dc22 2005013141 This title is also available as an e-book, ISBN 1414404999 (e-book set). ISBN: 0-7876-9422-3 (set); 0-7876-9423-1 (v1); 0-7876-9424-X (v2) Contact your Gale sales representative for ordering information. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 R E A L - L I F E M A T H v Table of Contents Entries (With Areas of Discussion) . . . . . . . vii Introduction . . . . . . . . . . . . . . xix List of Advisors and Contributors . . . . . . . xxi Entries . . . . . . . . . . . . . . . 1 Volume 1: A–L Addition . . . . . . . . . . . . . . . 1 Algebra . . . . . . . . . . . . . . . 9 Algorithms . . . . . . . . . . . . . . 26 Architectural Math . . . . . . . . . . . . 33 Area . . . . . . . . . . . . . . . . 45 Average. . . . . . . . . . . . . . . . 51 Base . . . . . . . . . . . . . . . . 59 Business Math . . . . . . . . . . . . . 62 Calculator Math . . . . . . . . . . . . 69 Calculus . . . . . . . . . . . . . . . 80 Calendars . . . . . . . . . . . . . . . 97 Cartography . . . . . . . . . . . . . . 100 Charts . . . . . . . . . . . . . . . . 107 Computers and Mathematics . . . . . . . . 114 Conversions . . . . . . . . . . . . . . 122 Coordinate Systems . . . . . . . . . . . 131 Decimals . . . . . . . . . . . . . . . 138 Demographics . . . . . . . . . . . . . 141 Discrete Mathematics . . . . . . . . . . 144 Division . . . . . . . . . . . . . . . 149 Domain and Range . . . . . . . . . . . 156 Elliptic Functions . . . . . . . . . . . . 159 Estimation . . . . . . . . . . . . . . 161 Exponents . . . . . . . . . . . . . . 167 Factoring . . . . . . . . . . . . . . . 180 Financial Calculations, Personal . . . . . . . 184 Fractals . . . . . . . . . . . . . . . 198 Fractions . . . . . . . . . . . . . . . 203 Functions . . . . . . . . . . . . . . . 210 Game Math . . . . . . . . . . . . . . 215 Game Theory . . . . . . . . . . . . . 225 Geometry. . . . . . . . . . . . . . . 232 Graphing . . . . . . . . . . . . . . . 248 Imaging . . . . . . . . . . . . . . . 262 Information Theory . . . . . . . . . . . 269 Inverse . . . . . . . . . . . . . . . . 278 Iteration . . . . . . . . . . . . . . . 284 Linear Mathematics . . . . . . . . . . . 287 Logarithms . . . . . . . . . . . . . . 294 Logic . . . . . . . . . . . . . . . . 300 T a b l e o f C o n t e n t s vi R E A L - L I F E M A T H Volume 2: M–Z Matrices and Determinants . . . . . . . . . 303 Measurement . . . . . . . . . . . . . 307 Medical Mathematics . . . . . . . . . . . 314 Modeling . . . . . . . . . . . . . . . 328 Multiplication . . . . . . . . . . . . . 335 Music and Mathematics . . . . . . . . . . 343 Nature and Numbers . . . . . . . . . . . 353 Negative Numbers . . . . . . . . . . . . 356 Number Theory . . . . . . . . . . . . 360 Odds . . . . . . . . . . . . . . . . 365 Percentages . . . . . . . . . . . . . . 372 Perimeter . . . . . . . . . . . . . . . 385 Perspective . . . . . . . . . . . . . . 389 Photography Math . . . . . . . . . . . 398 Plots and Diagrams . . . . . . . . . . . 404 Powers . . . . . . . . . . . . . . . 416 Prime Numbers . . . . . . . . . . . . . 420 Probability . . . . . . . . . . . . . . 423 Proportion . . . . . . . . . . . . . . 430 Quadratic, Cubic, and Quartic Equations . . . . 438 Ratio . . . . . . . . . . . . . . . . 441 Rounding . . . . . . . . . . . . . . 449 Rubric . . . . . . . . . . . . . . . . 453 Sampling . . . . . . . . . . . . . . . 457 Scale . . . . . . . . . . . . . . . . 465 Scientific Math . . . . . . . . . . . . . 473 Scientific Notation . . . . . . . . . . . 484 Sequences, Sets, and Series . . . . . . . . . 491 Sports Math . . . . . . . . . . . . . . 495 Square and Cube Roots . . . . . . . . . . 511 Statistics . . . . . . . . . . . . . . . 516 Subtraction . . . . . . . . . . . . . . 529 Symmetry . . . . . . . . . . . . . . 537 Tables . . . . . . . . . . . . . . . . 543 Topology . . . . . . . . . . . . . . . 553 Trigonometry . . . . . . . . . . . . . 557 Vectors . . . . . . . . . . . . . . . 568 Volume . . . . . . . . . . . . . . . 575 Word Problems . . . . . . . . . . . . . 583 Zero-sum Games . . . . . . . . . . . . 595 Glossary . . . . . . . . . . . . . . . 599 Field of Application Index . . . . . . . . . 605 General Index . . . . . . . . . . . . . 609 E n t r i e s ( W i t h A r e a s o f D i s c u s s i o n ) R E A L - L I F E M A T H ix Charts Bar Charts . . . . . . . . . . . . . . 109 Basic Charts . . . . . . . . . . . . . . 107 Choosing the Right Type of Chart For the Data . . 112 Clustered Column Charts . . . . . . . . . 110 Column and Bar Charts . . . . . . . . . . 109 Line Charts . . . . . . . . . . . . . . 107 Pie Charts . . . . . . . . . . . . . . 110 Stacked Column Charts . . . . . . . . . . 110 Using the Computer to Create Charts . . . . . 112 X-Y Scatter Graphs . . . . . . . . . . . 109 Computers and Mathematics Algorithms . . . . . . . . . . . . . . 115 Binary System . . . . . . . . . . . . . 114 Bits . . . . . . . . . . . . . . . . 116 Bytes . . . . . . . . . . . . . . . . 116 Compression. . . . . . . . . . . . . . 118 Data Transmission . . . . . . . . . . . . 119 Encryption . . . . . . . . . . . . . . 120 IP Address . . . . . . . . . . . . . . 117 Pixels, Screen Size, and Resolution . . . . . . 117 Subnet Mask. . . . . . . . . . . . . . 118 Text Code . . . . . . . . . . . . . . 116 Conversions Absolute Systems . . . . . . . . . . . . 127 Arbitrary Systems . . . . . . . . . . . . 128 Cooking or Baking Temperatures . . . . . . . 127 Derived Units . . . . . . . . . . . . . 124 English System . . . . . . . . . . . . . 123 International System of Units (SI) . . . . . . 123 Metric Units . . . . . . . . . . . . . . 123 Units Based On Physical or “Natural” Phenomena . . . . . . . . . . . . 124 Weather Forecasting . . . . . . . . . . . 126 Coordinate Systems 3-D Systems On Ordinance Survey Maps . . . . 136 Cartesian Coordinate Plane . . . . . . . . . 132 Changing Between Coordinate Systems . . . . . 132 Choosing the Best Coordinate System . . . . . 132 Commercial Aviation . . . . . . . . . . . 135 Coordinate Systems Used in Board Games . . . . 134 Coordinate Systems Used for Computer Animation . . . . . . . . . . . . . 134 Dimensions of a Coordinate System . . . . . . 131 Longitude and John Harrison . . . . . . . . 135 Modern Navigation and GPS . . . . . . . . 135 Paper Maps of the World . . . . . . . . . 134 Polar Coordinates . . . . . . . . . . . . 133 Radar Systems and Polar Coordinates . . . . . 136 Vectors . . . . . . . . . . . . . . . 132 Decimals Grade Point Average Calculations . . . . . . . 139 Measurement Systems . . . . . . . . . . 139 Science . . . . . . . . . . . . . . . 139 Demographics Census . . . . . . . . . . . . . . . 142 Election Analysis . . . . . . . . . . . . 141 Geographic Information System Technology . . . 143 Discrete Mathematics Algorithms . . . . . . . . . . . . . 145 Boolean Algebra . . . . . . . . . . . . 145 Combinatorial Chemistry . . . . . . . . . 147 Combinatorics . . . . . . . . . . . . . 145 Computer Design . . . . . . . . . . . . 146 Counting Jaguars Using Probability Theory . . . 147 Cryptography . . . . . . . . . . . . . 146 Finding New Drugs with Graph Theory . . . . 147 Graphs . . . . . . . . . . . . . . . 146 Logic, Sets, and Functions . . . . . . . . . 144 Looking Inside the Body With Matrices . . . . . 147 Matrix Algebra . . . . . . . . . . . . . 146 Number Theory . . . . . . . . . . . . 145 Probability Theory . . . . . . . . . . . . 145 Searching the Web . . . . . . . . . . . . 146 Shopping Online and Prime Numbers . . . . . 147 Division Averages . . . . . . . . . . . . . . . 152 Division and Comparison . . . . . . . . . 151 Division and Distribution . . . . . . . . . 150 Division, Other Uses . . . . . . . . . . . 153 Practical Uses of Division For Students . . . . . 153 Domain and Range Astronomers . . . . . . . . . . . . . 157 Calculating Odds and Outcomes . . . . . . . 157 E n t r i e s ( W i t h A r e a s o f D i s c u s s i o n ) x R E A L - L I F E M A T H Computer Control and Coordination . . . . . 157 Computer Science . . . . . . . . . . . . 158 Engineering . . . . . . . . . . . . . . 157 Graphs, Charts, Maps . . . . . . . . . . . 158 Physics . . . . . . . . . . . . . . . 157 Elliptic Functions The Age of the Universe . . . . . . . . . . 160 Conformal Maps . . . . . . . . . . . . 159 E-Money . . . . . . . . . . . . . . . 160 Estimation Buying a Used Car . . . . . . . . . . . . 162 Carbon Dating . . . . . . . . . . . . . 165 Digital Imaging . . . . . . . . . . . . . 164 Gumball Contest . . . . . . . . . . . . 163 Hubble Space Telescope . . . . . . . . . . 165 Population Sampling . . . . . . . . . . . 164 Software Development . . . . . . . . . . 166 Exponents Bases and Exponents . . . . . . . . . . . 167 Body Proportions and Growth (Why Elephants Don’t Have Skinny Legs) . . . . . . . . . . . . 179 Credit Card Meltdown . . . . . . . . . . 178 Expanding Universe . . . . . . . . . . . 178 Exponential Functions . . . . . . . . . . 168 Exponential Growth . . . . . . . . . . . 171 Exponents and Evolution . . . . . . . . . 174 Integer Exponents . . . . . . . . . . . . 167 Interest and Inflation . . . . . . . . . . . 177 Non-Integer Exponents . . . . . . . . . . 168 Radioactive Dating . . . . . . . . . . . 177 Radioactive Decay . . . . . . . . . . . . 175 Rotting Leftovers . . . . . . . . . . . . 173 Scientific Notation . . . . . . . . . . . . 171 Factoring Codes and Code Breaking . . . . . . . . . 182 Distribution . . . . . . . . . . . . . . 182 Geometry and Approximation of Size . . . . . 182 Identification of Patterns and Behaviors . . . . . . . . . . . 181 Reducing Equations . . . . . . . . . . . 181 Skill Transfer . . . . . . . . . . . . . 182 Financial Calculations, Personal Balancing a Checkbook . . . . . . . . . . 189 Budgets . . . . . . . . . . . . . . . 188 Buying Music . . . . . . . . . . . . . 184 Calculating a Tip . . . . . . . . . . . . 194 Car Purchasing and Payments . . . . . . . . 187 Choosing a Wireless Plan . . . . . . . . . 187 Credit Cards . . . . . . . . . . . . . . 185 Currency Exchange . . . . . . . . . . . 195 Investing . . . . . . . . . . . . . . . 190 Retirement Investing . . . . . . . . . . . 192 Social Security System . . . . . . . . . . 190 Understanding Income Taxes . . . . . . . . 189 Fractals Astronomy . . . . . . . . . . . . . . 202 Building Fractals . . . . . . . . . . . . 199 Cell Phone and Radio Antenna . . . . . . . 202 Computer Science . . . . . . . . . . . . 202 Fractals and Nature . . . . . . . . . . . 200 Modeling Hurricanes and Tornadoes . . . . . . 201 Nonliving Systems . . . . . . . . . . . . 201 Similarity . . . . . . . . . . . . . . . 199 Fractions Algebra . . . . . . . . . . . . . . . 205 Cooking and Baking . . . . . . . . . . . 206 Fractions and Decimals . . . . . . . . . . 204 Fractions and Percentages . . . . . . . . . 204 Fractions and Voting . . . . . . . . . . . 208 Music . . . . . . . . . . . . . . . . 206 Overtime Pay . . . . . . . . . . . . . 208 Radioactive Waste . . . . . . . . . . . . 206 Rules For Handling Fractions . . . . . . . . 204 Simple Probabilities . . . . . . . . . . . 207 Tools and Construction . . . . . . . . . . 208 Types of Fractions . . . . . . . . . . . . 203 What Is a Fraction? . . . . . . . . . . . 203 Functions Body Mass Index . . . . . . . . . . . . 214 Finite-Element Models . . . . . . . . . . 212 Functions, Described . . . . . . . . . . . 210 Functions and Relations . . . . . . . . . . 210 Guilloché Patterns . . . . . . . . . . . 211 Making Airplanes Fly . . . . . . . . . . . 211 E n t r i e s ( W i t h A r e a s o f D i s c u s s i o n ) R E A L - L I F E M A T H xi The Million-Dollar Hypothesis . . . . . . . 212 Nuclear Waste . . . . . . . . . . . . . 213 Synths and Drums . . . . . . . . . . . . 213 Game Math Basic Board Games. . . . . . . . . . . . 220 Card Games . . . . . . . . . . . . . . 218 Magic Squares . . . . . . . . . . . . . 221 Math Puzzles . . . . . . . . . . . . . 223 Other Casino Games . . . . . . . . . . . 219 Game Theory Artificial Intelligence . . . . . . . . . . . 230 Decision Theory . . . . . . . . . . . . 228 eBay and the Online Auction World . . . . . . 230 Economics . . . . . . . . . . . . . . 229 Economics and Game Theory . . . . . . . . 228 Evolution and Animal Behavior . . . . . . . 229 General Equilibrium . . . . . . . . . . . 229 Infectious Disease Therapy . . . . . . . . . 230 Nash Equilibrium . . . . . . . . . . . . 229 Geometry Architecture . . . . . . . . . . . . . . 237 Fireworks . . . . . . . . . . . . . . . 241 Fourth Dimension . . . . . . . . . . . . 245 Global Positioning . . . . . . . . . . . . 239 Honeycombs . . . . . . . . . . . . . 239 Manipulating Sound . . . . . . . . . . . 241 Pothole Covers . . . . . . . . . . . . . 236 Robotic Surgery. . . . . . . . . . . . . 245 Rubik’s Cube . . . . . . . . . . . . . 243 Shooting an Arrow . . . . . . . . . . . 244 Solar Systems . . . . . . . . . . . . . 242 Stealth Technology . . . . . . . . . . . . 244 Graphing Aerodynamics and Hydrodynamics . . . . . . 259 Area Graphs . . . . . . . . . . . . . . 252 Bar Graphs . . . . . . . . . . . . . . 249 Biomedical Research . . . . . . . . . . . 258 Bubble Graphs . . . . . . . . . . . . . 257 Computer Network Design . . . . . . . . . 259 Finding Oil . . . . . . . . . . . . . . 258 Gantt Graphs . . . . . . . . . . . . . 254 Global Warming . . . . . . . . . . . . 257 GPS Surveying . . . . . . . . . . . . . 258 Line Graphs . . . . . . . . . . . . . . 251 Physical Fitness . . . . . . . . . . . . . 259 Picture Graphs . . . . . . . . . . . . . 254 Pie Graphs . . . . . . . . . . . . . . 252 Radar Graphs . . . . . . . . . . . . . 253 X-Y Graphs . . . . . . . . . . . . . . 254 Imaging Altering Images . . . . . . . . . . . . . 263 Analyzing Images . . . . . . . . . . . . 263 Art . . . . . . . . . . . . . . . . . 267 Compression . . . . . . . . . . . . . 264 Creating Images . . . . . . . . . . . . 263 Dance . . . . . . . . . . . . . . . . 266 Forensic Digital Imaging . . . . . . . . . . 266 Meat and Potatoes . . . . . . . . . . . . 266 Medical Imaging . . . . . . . . . . . . 264 Optics . . . . . . . . . . . . . . . . 264 Recognizing Faces: a Controversial Biometrics Application . . . . . . . . . 264 Steganography and Digital Watermarks . . . . . 266 Information Theory Communications . . . . . . . . . . . . 273 Error Correction . . . . . . . . . . . . 275 Information and Meaning . . . . . . . . . 273 Information Theory in Biology and Genetics . . . 274 Quantum Computing . . . . . . . . . . 276 Unequally Likely Messages . . . . . . . . . 271 Inverse Anti-Sound . . . . . . . . . . . . . . 282 The Brain and the Inverted Image On the Eye . . . . . . . . . . . . . 281 Cryptography . . . . . . . . . . . . . 280 Definition of an Inverse . . . . . . . . . . 278 Fluid Mechanics and Nonlinear Design . . . . . 281 Inverse Functions . . . . . . . . . . . . 279 The Multiplicative Inverse . . . . . . . . . 278 Negatives Used in Photography . . . . . . . 281 Operations Where the Inverse Does Not Exist . . . . . . . . . . . . . 279 Operations With More Than One Inverse . . . . . . . . . . . . . . 279 Stealth Submarine Communications . . . . . . 282 Stereo . . . . . . . . . . . . . . . . 282 E n t r i e s ( W i t h A r e a s o f D i s c u s s i o n ) xiv R E A L - L I F E M A T H Photography Math The Camera . . . . . . . . . . . . . . 398 Depth of Field . . . . . . . . . . . . . 400 Digital Image Processing . . . . . . . . . . 403 Digital Photography . . . . . . . . . . . 401 Film Speed . . . . . . . . . . . . . . 398 Lens Aperture . . . . . . . . . . . . . 400 Lens Focal Length . . . . . . . . . . . . 399 Photomicrography . . . . . . . . . . . . 403 Reciprocity . . . . . . . . . . . . . . 401 Shutter Speed . . . . . . . . . . . . . 399 Sports and Wildlife Photography . . . . . . . 402 Plots and Diagrams Area Chart . . . . . . . . . . . . . . 406 Bar Graphs . . . . . . . . . . . . . . 406 Body Diagram . . . . . . . . . . . . . 414 Box Plot . . . . . . . . . . . . . . . 405 Circuit Diagram. . . . . . . . . . . . . 414 Diagrams . . . . . . . . . . . . . . . 404 Fishbone Diagram . . . . . . . . . . . . 406 Flow Chart . . . . . . . . . . . . . . 411 Gantt Charts . . . . . . . . . . . . . 413 Line Graph . . . . . . . . . . . . . . 408 Maps . . . . . . . . . . . . . . . . 413 Organization Charts . . . . . . . . . . . 413 Other Diagrams . . . . . . . . . . . . 414 Pie Graph. . . . . . . . . . . . . . . 406 Polar Chart . . . . . . . . . . . . . . 406 Properties of Graphs . . . . . . . . . . . 404 Scatter Graph . . . . . . . . . . . . . 405 Stem and Leaf Plots . . . . . . . . . . . 405 Street Signs . . . . . . . . . . . . . . 414 Three-Dimensional Graph . . . . . . . . . 407 Tree Diagram . . . . . . . . . . . . . 412 Triangular Graph . . . . . . . . . . . . 407 Weather Maps . . . . . . . . . . . . . 414 Powers Acids, Bases, and pH Level . . . . . . . . . 418 Areas of Polygons and Volumes of Solid Figures . . . . . . . . . . . 417 Astronomy and Brightness of Stars . . . . . . 418 Computer Science and Binary Logic . . . . . . . . . . . . . . 417 Earthquakes and the Richter Scale . . . . . . 417 The Powers of Nanotechnology . . . . . . . 418 Prime Numbers Biological Applications of Prime Numbers . . . . 421 Probability Gambling and Probability Myths . . . . . . . 425 Probability in Business and Industry . . . . . . 427 Probability, Other Uses . . . . . . . . . . 428 Probability in Sports and Entertainment . . . . 426 Security . . . . . . . . . . . . . . . 424 Proportion Architecture . . . . . . . . . . . . . . 432 Art, Sculpture, and Design . . . . . . . . . 432 Chemistry . . . . . . . . . . . . . . 435 Diets . . . . . . . . . . . . . . . . 436 Direct Proportion . . . . . . . . . . . . 431 Engineering Design . . . . . . . . . . . 435 Ergonomics . . . . . . . . . . . . . . 434 Inverse Proportion . . . . . . . . . . . . 431 Maps . . . . . . . . . . . . . . . . 434 Medicine . . . . . . . . . . . . . . . 434 Musical Instruments . . . . . . . . . . . 435 Proportion in Nature . . . . . . . . . . . 436 Solving Ratios With Cross Products . . . . . . 430 Stock Market . . . . . . . . . . . . . 436 Quadratic, Cubic, and Quartic Equations Acceleration . . . . . . . . . . . . . . 439 Area and Volume . . . . . . . . . . . . 439 Car Tires . . . . . . . . . . . . . . . 439 Guiding Weapons . . . . . . . . . . . . 440 Hospital Size . . . . . . . . . . . . . 440 Just in Time Manufacturing . . . . . . . . 440 Ratio Age of Earth . . . . . . . . . . . . . . 446 Automobile Performance . . . . . . . . . 445 Cleaning Water . . . . . . . . . . . . . 446 Cooking . . . . . . . . . . . . . . . 446 Cost of Gas . . . . . . . . . . . . . . 443 Determination of the Origination of the Moon . . 447 Genetic Traits . . . . . . . . . . . . . 443 Healthy Living . . . . . . . . . . . . . 446 Length of a Trip . . . . . . . . . . . . 443 Music . . . . . . . . . . . . . . . . 445 E n t r i e s ( W i t h A r e a s o f D i s c u s s i o n ) R E A L - L I F E M A T H xv Optimizing Livestock Production . . . . . . . 447 Sports . . . . . . . . . . . . . . . . 445 Stem Cell Research. . . . . . . . . . . . 446 Student-Teacher Ratio . . . . . . . . . . 445 Rounding Accounting . . . . . . . . . . . . . . 451 Bulk Purchases . . . . . . . . . . . . . 450 Decimals . . . . . . . . . . . . . . . 450 Energy Consumption . . . . . . . . . . . 451 Length and Weight . . . . . . . . . . . 450 Lunar Cycles . . . . . . . . . . . . . 451 Mileage . . . . . . . . . . . . . . . 452 Pi . . . . . . . . . . . . . . . . . 450 Population . . . . . . . . . . . . . . 451 Precision . . . . . . . . . . . . . . . 452 Time . . . . . . . . . . . . . . . . 452 Weight Determination . . . . . . . . . . 451 Whole Numbers. . . . . . . . . . . . . 449 Rubric Analytic Rubrics and Holistic Rubrics . . . . . 455 General Rubrics and Task-Specific Rubrics . . . . . . . . . . . . . . 455 Scoring Rubrics . . . . . . . . . . . . 453 Sampling Agriculture . . . . . . . . . . . . . . 459 Archeology . . . . . . . . . . . . . . 463 Astronomy . . . . . . . . . . . . . . 462 Demographic Surveys . . . . . . . . . . 462 Drug Manufacturing . . . . . . . . . . . 460 Environmental Studies . . . . . . . . . . 462 Market Assessment . . . . . . . . . . . 463 Marketing . . . . . . . . . . . . . . 463 Non-Probability Sampling . . . . . . . . . 458 Plant Analysis . . . . . . . . . . . . . 460 Probability Sampling . . . . . . . . . . . 457 Scientific Research . . . . . . . . . . . . 460 Soil Sampling . . . . . . . . . . . . . 460 Weather Forecasts . . . . . . . . . . . . 461 Scale Architecture . . . . . . . . . . . . . . 468 Atmospheric Pressure Using Barometer . . . . . 469 The Calendar . . . . . . . . . . . . . 469 Expanse of Scale From the Sub-Atomic to the Universe . . . . . . . . . . . 471 Interval Scale . . . . . . . . . . . . . 466 Linear Scale . . . . . . . . . . . . . . 465 Logarithmic Scale . . . . . . . . . . . . 465 Map Scale . . . . . . . . . . . . . . 467 Measuring Wind Strength . . . . . . . . . 469 The Metric System of Measurement . . . . . . 472 Music . . . . . . . . . . . . . . . . 471 Nominal Scale . . . . . . . . . . . . . 467 Ordinal Scale . . . . . . . . . . . . . 467 Ratio Scale . . . . . . . . . . . . . . 466 The Richter Scale . . . . . . . . . . . . 470 Sampling . . . . . . . . . . . . . . . 472 Technology and Imaging . . . . . . . . . 469 Toys . . . . . . . . . . . . . . . . 471 Weighing Scale . . . . . . . . . . . . . 468 Scientific Math Aviation and Flights . . . . . . . . . . . 478 Bridging Chasms . . . . . . . . . . . . 478 Discrete Math . . . . . . . . . . . . . 474 Earthquakes and Logarithms . . . . . . . . 482 Equations and Graphs . . . . . . . . . . 476 Estimating Data Used for Assessing Weather . . . 477 Functions and Measurements . . . . . . . . 473 Genetics . . . . . . . . . . . . . . . 483 Logarithms . . . . . . . . . . . . . . 475 Matrices and Arrays . . . . . . . . . . . 475 Medical Imaging . . . . . . . . . . . . 480 Rocket Launch . . . . . . . . . . . . . 480 Ships . . . . . . . . . . . . . . . . 482 Simple Carpentry . . . . . . . . . . . . 479 Trigonometry and the Pythagorean Theorem . . . 474 Weather Prediction . . . . . . . . . . . 476 Wind Chill in Cold Weather . . . . . . . . 476 Scientific Notation Absolute Dating . . . . . . . . . . . . 489 Chemistry . . . . . . . . . . . . . . 486 Computer Science . . . . . . . . . . . . 487 Cosmology . . . . . . . . . . . . . . 487 Earth Science . . . . . . . . . . . . . 489 Electrical Circuits . . . . . . . . . . . . 486 Electronics . . . . . . . . . . . . . . 489 Engineering . . . . . . . . . . . . . . 487 Environmental Science . . . . . . . . . . 488 Forensic Science . . . . . . . . . . . . 488 Geologic Time Scale and Geology . . . . . . . 488 E n t r i e s ( W i t h A r e a s o f D i s c u s s i o n ) xvi R E A L - L I F E M A T H Light Years, the Speed of Light, and Astronomy . . . . . . . . . . . 486 Medicine . . . . . . . . . . . . . . . 488 Nanotechnology . . . . . . . . . . . . 490 Proteins and Biology . . . . . . . . . . . 490 Sequences, Sets, and Series Genetics . . . . . . . . . . . . . . . 493 Operating On Sets . . . . . . . . . . . . 492 Ordering Things . . . . . . . . . . . . 493 Sequences . . . . . . . . . . . . . . 491 Series . . . . . . . . . . . . . . . . 492 Sets . . . . . . . . . . . . . . . . . 491 Using Sequences . . . . . . . . . . . . 493 Sports Math Baseball . . . . . . . . . . . . . . . 498 Basketball . . . . . . . . . . . . . . 499 Cycling—Gear Ratios and How They Work . . . . . . . . . . . . . . 505 Football—How Far was the Pass Thrown? . . . . 507 Football Tactics—Math as a Decision-Making Tool . . . . . . . . . 501 Golf Technology . . . . . . . . . . . . 506 Math and the Science of Sport . . . . . . . . 504 Math and Sports Wagering . . . . . . . . . 508 Math to Understand Sports Performance . . . . 497 Mathematics and the Judging of Sports . . . . . 504 Money in Sport—Capology 101 . . . . . . . 507 North American Football . . . . . . . . . 499 Pascal’s Triangle and Predicting a Coin Toss . . . 500 Predicting the Future: Calling the Coin Toss . . . 500 Ratings Percentage Index (RPI) . . . . . . . 503 Rules Math . . . . . . . . . . . . . . 496 Soccer—Free Kicks and the Trajectory of the Ball . . . . . . . . . . . . . 506 Understanding the Sports Media Expert . . . . 502 Square and Cube Roots Architecture . . . . . . . . . . . . . . 513 Global Economics . . . . . . . . . . . . 515 Hiopasus’s Fatal Discovery . . . . . . . . . 513 Names and Conventions . . . . . . . . . . 512 Navigation . . . . . . . . . . . . . . 514 Pythagorean Theorem . . . . . . . . . . 513 Sports . . . . . . . . . . . . . . . . 514 Stock Markets . . . . . . . . . . . . . 515 Statistics Analysis of Variance . . . . . . . . . . . 522 Average Values . . . . . . . . . . . . . 519 Confidence Intervals . . . . . . . . . . . 522 Correlation and Curve Fitting . . . . . . . . 521 Cumulative Frequencies and Quantiles . . . . . 521 Geostatistics . . . . . . . . . . . . . . 525 Measures of Dispersion . . . . . . . . . . 520 Minimum, Maximum, and Range . . . . . . . 518 Populations and Samples . . . . . . . . . 516 Probability . . . . . . . . . . . . . . 517 Public Opinion Polls . . . . . . . . . . . 527 Quality Assurance . . . . . . . . . . . . 526 Statistical Hypothesis Testing . . . . . . . . 522 Using Statistics to Deceive . . . . . . . . . 523 Subtraction Subtraction in Entertainment and Recreation . . . . . . . . . . . 533 Subtraction in Financial Calculations . . . . . 531 Subtraction in Politics and Industry . . . . . . 535 Tax Deductions . . . . . . . . . . . . . 532 Symmetry Architecture . . . . . . . . . . . . . . 541 Exploring Symmetries . . . . . . . . . . 539 Fractal Symmetries . . . . . . . . . . . 541 Imperfect Symmetries . . . . . . . . . . 542 Symmetries in Nature . . . . . . . . . . 542 Tables Converting Measurements . . . . . . . . . 545 Daily Use . . . . . . . . . . . . . . . 549 Educational Tables . . . . . . . . . . . . 545 Finance . . . . . . . . . . . . . . . 546 Health . . . . . . . . . . . . . . . . 548 Math Skills . . . . . . . . . . . . . . 544 Travel . . . . . . . . . . . . . . . . 549 Topology Computer Networking . . . . . . . . . . 555 I.Q. Tests . . . . . . . . . . . . . . . 555 Möbius Strip . . . . . . . . . . . . . 555 Visual Analysis . . . . . . . . . . . . . 554 Visual Representation . . . . . . . . . . . 555 I n t r o d u c t i o n xx R E A L - L I F E M A T H practicing engineers and scientists who use math on a daily basis. However, RLM is not intended to be a book about real-life applications as used by mathematicians and scientists but rather, wherever possible, to illustrate and discuss applications within the experience—and that are understandable and interesting—to younger readers. RLM is intended to maximize readability and accessi- bility by minimizing the use of equations, example prob- lems, proofs, etc. Accordingly, RLM is not a math textbook, nor is it designed to fully explain the mathematics involved in each concept. Rather, RLM is intended to compliment the mathematics curriculum by serving a general reader for maths by remaining focused on fundamental math concepts as opposed to the history of math, biographies of mathematicians, or simply interesting applications. To be sure, there are inherent difficulties in presenting mathe- matical concepts without the use of mathematical nota- tion, but the authors and editors of RLM sought to use descriptions and concepts instead of mathematical nota- tion, problems, and proofs whenever possible. To the extent that RLM meets these challenges it becomes a valuable resource to students and teachers of mathematics. The editors modestly hope that Real-Life Math serves to help students appreciate the scope of the importance and influence of math on everyday life. RLM will achieve its highest purposes if it intrigues and inspires students to continue their studies in maths and so advance their understanding of the both the utility and elegance of mathematics. “[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word.” Galilei, Galileo (1564–1642) K. Lee Lerner and Brenda Wilmoth Lerner, Editors R E A L - L I F E M A T H xxi List of Advisors and Contributors In compiling this edition, we have been fortunate in being able to rely upon the expertise and contributions of the following scholars who served as contributing advisors or authors for Real-Life Math, and to them we would like to express our sincere appreciation for their efforts: William Arthur Atkins Mr. Atkins holds a BS in physics and math- ematics as well as an MBA. He lives at writes in Perkin, Illinois. Juli M. Berwald, PhD In addition to her graduate degree in ocean sciences, Dr. Berwald holds a BA in mathe- matics from Amherst College, Amherst, Massachusetts. She currently lives and writes in Chicago, Illinois. Bennett Brooks Mr. Brooks is a PhD graduate student in mathematics. He holds a BS in mathematics, with departmental honors, from University of Redlands, Redlands, California, and currently works as a writer based in Beaumont, California. Rory Clarke, PhD Dr. Clark is a British physicist conducting research in the area of high-energy physics at the University of Bucharest, Romania. He holds a PhD in high energy particle physics from the University of Birmingham, an MSc in theoretical physics from Imperial College, and a BSc degree in physics from the University of London. Raymond C. Cole Mr. Cole is an investment banking financial analyst who lives in New York. He holds an MBA from the Baruch Zicklin School of Business and a BS in business administration from Fordham University. Bryan Thomas Davies Mr. Davies holds a Bachelor of Laws (LLB) from the University of Western Ontario and has served as a criminal prosecutor in the Ontario Ministry of the Attorney General. In addition to his legal experience, Mr. Davies is a nationally certified basketball coach. John F. Engle Mr. Engle is a medical student at Tulane University Medical School in New Orleans, Louisiana. L i s t o f A d v i s o r s a n d C o n t r i b u t o r s xxii R E A L - L I F E M A T H William J. Engle Mr. Engle is a retired petroleum engineer who lives in Slidell, Louisiana. Paul Fellows Dr. Fellows is a physicist and mathematician who lives in London, England. Renata A. Ficek Ms. Ficek is a graduate mathematics student at the University of Queensland, Australia. Larry Gilman, PhD Dr. Gilman holds a PhD in electrical engi- neering from Dartmouth College and an MA in English literature from Northwestern University. He lives in Sharon, Vermont. Amit Gupta Mr. Gupta holds an MS in information systems and is managing director of Agarwal Management Consultants P. Ltd., in Ahmedabad, India. William C. Haneberg, PhD Dr. Haneberg is a professional geologist and writer based in Seattle, Washington. Bryan D. Hoyle, PhD Dr. Hoyle is a microbiologist and science writer who lives in Halifax, Nova Scotia, Canada. Kenneth T. LaPensee, PhD In addition to professional research in epidemiology, Dr. LaPensee directs Skylands Healthcare Consulting located in Hampton, New Jersey. Holly F. McBain Ms. McBain is a science and math writer who lives near New Braunfels, Texas. Mark H. Phillips, PhD Dr. Phillips serves as an assistant professor of management at Abilene Christian University, located in Abilene, Texas. Nephele Tempest Ms. Tempest is a writer based in Los Angeles, California. David Tulloch Mr. Tulloch holds a BSc in physics and an MS in the history of science. In addition to research and writing he serves as a radio broadcaster in Ngaio, Wellington, New Zealand. James A. Yates Mr. Yates holds a MMath degree from Oxford University and is a teacher of maths in Skegnes, England. ACKNOWLEDGMENTS The editors would like to extend special thanks to Connie Clyde for her assistance in copyediting. The editors also wish to especially acknowledge Dr. Larry Gilman for his articles on calculus and exponents as well as his skilled corrections of the entire text. The editors are profoundly grateful to their assistant editors and proofreaders, includ- ing Lynn Nettles and Bill Engle, who read and corrected articles under the additional pressures created by evacua- tions mandated by Hurricane Katrina. The final editing of this book was interrupted as Katrina damaged the Gulf Coast homes and offices of several authors, assistant edi- tors, and the editors of RLM just as the book was being pre- pared for press. Quite literally, many pages were read and corrected by light produced by emergency generators— and in some cases, pages were corrected from evacuation shelters. The editors are forever grateful for the patience and kind assistance of many fellow scholars and colleagues during this time. The editors gratefully acknowledge the assistance of many at Thompson Gale for their help in preparing Real-Life Math. The editors wish to specifically thank Ms. Meggin Condino for her help and keen insights while launching this project. The deepest thanks are also offered to Gale Senior Editor Kim McGrath for her tireless, skilled, good-natured, and intelligent guidance. A d d i t i o n R E A L - L I F E M A T H 3 found deep inside every microprocessor (as well as in a simple programming language which bears his name in honor of his pioneering work). Modern computers offer user-friendly graphic interfaces and require little or no math or programming knowledge on the part of the aver- age user. But at the lowest functional level, even a cutting edge processor relies on simple operations performed in its arithmetic logic unit, or ALU. When this basic pro- cessing unit receives an instruction, that instruction has typically been broken down into a series of simple processes which are then completed one at a time. Ironi- cally, though the ALU is the mathematical heart of a modern computer, a typical ALU performs only four functions, the same add, subtract, multiply, and divide found on the earliest electronic calculators of the 1970s. By performing these simple operations millions of times each second, and leveraging this power through modern operating systems and applications software, even a process as simple as addition can produce startling results. Real-life Applications SPORTS AND F I TNESS ADD I T ION Many aspects of popular sports require the use of addition. For example, some of the best-known records tracked in most sports are found by simply adding one success to another. Records for the most homeruns, the most 3-point shots made, the most touchdown passes completed, and the most major golf tournaments won in a career are nothing more than the result of lengthy addi- tion problems stretched out over an entire career. On the business side of sports are other addition applications, including such routine tasks as calculating the number of fans at a ballgame or the number of hotdogs sold, both of which are found by simply adding one more person or sausage to the running total. Many sports competitions are scored on the basis of elapsed time, which is found by simply adding fractions of a second to a total until the event ends, at which time the smallest total is determined to be the winning score. In the case of motor sports, racers compete for the chance to start the actual race near the front of the field, and these qualifying attempts are often separated by mere hundredths or even thousandths of a second. Track events such as the decathlon, which requires participants to attempt ten separate events including sprints, jumps, vaults, and throwing events over the course of two gruel- ing days, are scored by adding the tallies from each sepa- rate event to determine a final score. In the same way, track team scores are found by adding the scores from each individual event, relay, and field event to determine a total score. Although the sport of bowling is scored using only addition, this popular game has one of the more unusual scoring systems in modern sports. Bowlers compete in games consisting of ten frames, each of which includes up to two attempts to knock down all ten bowling pins. Depending on a bowler’s performance in one frame, he may be able to add some shots twice, significantly raising his total score. For example, a bowler who knocks down all ten pins in a single roll is awarded a strike, worth ten plus the total of the next two balls bowled in the follow- ing frames, while a bowler who knocks down all ten pins in two rolls is scored a spare and receives ten plus the next one ball rolled. Without this scoring system, the maxi- mum bowling score would be earned by bowling ten, ten- point strikes in a row for a perfect game total of 100. But with bowling’s bonus scoring system, each of the ten frames is potentially worth thirty points to a bowler who bowls a strike followed by two more strikes, creating a maximum possible game score of 300. While many programs exist to help people lose weight, none is more basic, or less liked, than the straight- forward process of counting calories. Calorie counting is based on a simple, immutable principle of physics: if a human body consumes more calories than it burns, it will store the excess calories as fat, and will become heavier. For this reason, most weight loss plans address, at least to some degree, the number of calories being consumed. A calorie is a measure of energy, and 3,500 calories are required to produce one pound of body weight. Using simple addition, it becomes clear that eating an extra 500 calories per day will add up to 3,500 calories, or one pound gained, per week. While this use of addition allows one to calculate the waistline impact of an additional dessert or several soft drinks, a similar process defines the amount of exercise required to lose this same amount of weight. For exam- ple, over the course of a week, a man might engage in a variety of physical activities, including an hour of vigor- ous tennis, an hour of slow jogging, one hour of swim- ming, and one hour officiating a basketball game. Each of these activities burns calories at a different rate. Using a chart of calorie burn rates, we determine that tennis burns 563 calories per hour, jogging burns 493 calories per hour, swimming burns 704 calories per hour, and officiating a basketball game burns 512. Adding these val- ues up we find that the man has exercised enough to burn a total of 2,272 calories over the course of the week. Depending on how many calories he consumes, this may be adequate to maintain his weight. However if he is A d d i t i o n 4 R E A L - L I F E M A T H consuming an extra 3,500 calories per week, he will need to burn an additional 1,228 calories to avoid storing these extra calories as fat. Over the course of a year, this excess of 1,228 calories will eventually add up to a net gain of more than 63,000 calories, or a weight gain of more than 18 pounds. While healthy activities help prolong life, the same result can be achieved by reducing unhealthy activities. Cigarette smoking is one of the more common behaviors believed to reduce life expectancy. While most smokers believe they would be healthier if they quit, and cigarette companies openly admit the dangers of their product, placing a health value or cost on a single cigarette can be difficult. A recent study published in the British Medical Journal tried to estimate the actual cost, in terms of reduced life expectancy, of each cigarette smoked. While this calculation is admittedly crude, the study concluded that each cigarette smoked reduces average life-span by eleven minutes, meaning that a smoker who puffs through all 20 cigarettes in a typical pack can simply add up the minutes to find that he has reduced his life expectancy by 220 minutes, or almost four hours. Simple addition also tells him that his pack-a-day habit is costing him 110 hours of life for each month he continues, or about four and one-half days of life lost for each month of smoking. When added up over a lifetime, the study concluded that smokers typically die more than six years earlier than non-smokers, a result of adding up the seem- ingly small effects of each individual cigarette. F INANC IAL ADD I T ION One of the more common uses of addition is in the popular pastime of shopping. Most adults understand that the price listed on an item’s price-tag is not always the full amount they will pay. For example, most states charge sales tax, meaning that a shopper with $20.00 to spend will need to add some set percentage to his item total in order to be sure he stays under budget and doesn’t come up short at the checkout counter. Many people esti- mate this add-on unconsciously, and in most cases, the amount added is relatively small. In the case of buying a car, however, various add-ons can quickly raise the total bill, as well as the monthly pay- ments. While paying 7% sales tax on a $3.00 purchase adds only twenty-one cents to the total, paying this same flat rate on a $30,000 automobile adds $2,100 to the bill. In addition, a car purchased at a dealership will invariably include a lengthy list of additional items such as docu- mentation fees, title fees, and delivery charges, which must be added to the sticker price to determine the actual cost to the buyer. As of 1999, Americans spent almost 40 cents of every food dollar at the 300,000 fast food restaurants in the country. Because they are often in a hurry to order, many customers choose one of the so-called value meals offered at most outlets. But in some cases, simple addition demonstrates that the actual savings gained by ordering a value meal is only a few cents. By adding the separate costs of the individual items in the meal, the customer can compare this total to learn just how much he is sav- ing. He can also use this simple addition to make other choices, such as substituting a smaller order of French fries for the enormous order usually included or choosing a small soda or water in place of a large drink. Because most customers order habitually, few actually know the value of what they are receiving in their value meals, and many could save money by buying à la carte (piece by piece). Deciding whether to fly or to drive is often based on cost, such as when a family of six elects to drive to their vacation destination rather than purchasing six airline tickets. In other cases, such as when a couple in Los Ange- les visits relatives in Connecticut over spring break, the choice is motivated by sheer distance. But in some situa- tions, the question is less clear, and some simple addition may reveal that the seemingly obvious choice is not actu- ally superior. Consider a student living in rural Okla- homa who wishes to visit his family in St. Louis. This student knows from experience that driving home will take him eight hours, so he is enthusiastic about cutting that time significantly by flying. But as he begins adding up the individual parts of the travel equation, he realizes the difference is not as large as he initially thought. The actual flight time from Tulsa to St. Louis is just over one hour, but the only flight with seats available stops in Kansas City, where he will have to layover for two hours, making his total trip time from Tulsa to St. Louis more than three hours. Added to this travel time is the one hour trip from his home to the Tulsa airport, the one hour early he is required to check in, the half hour he will spend in St. Louis collecting his baggage and walking to the car, and the hour he will spend driving in St. Louis traffic to his family’s home. Assuming no weather delays occur and all his flight arrive on time, the student can expect to spend close to seven hours on his trip, a net sav- ings of one hour over his expected driving time. Simple addition can help this student decide whether the price of the plane ticket is worth the one hour of time saved. In the still-developing world of online commerce, many web pages use an ancient method of gauging popularity: counting attendance. At the bottom of many web pages is a web counter, sometimes informing the A d d i t i o n R E A L - L I F E M A T H 5 visitor, “You are guest number . . .”. While computer gurus still hotly debate the accuracy of such counts, they are a common feature on websites, providing a simple assess- ment of how many guests visited a particular site. In some cases, simple addition is used to make a political point. Because the United States government finances much of its operations using borrowed money, concerns are frequently raised about the rapidly rising level of the national debt. In 1989, New York businessman Seymour Durst decided to draw attention to the spiraling level of public debt by erecting a National Debt Clock one block from Times Square. This illuminated billboard pro- vided a continuously updated total of the national debt, as well as a sub-heading detailing each family’s individual share of the total. During most of the clock’s lifetime, the national debt climbed so quickly that the last digits on the counter were simply a blur. The clock ran continuously from 1989 until the year 2000, when federal budget sur- pluses began to reduce the $5.5 trillion debt, and the clock was turned off. But two years later, with federal borrowing on the rise once again, Durst’s son restarted the clock, which displayed a national debt of over $6 trillion. By early 2005, the national debt was approach- ing $8 trillion. POKER , PROBAB IL I TY, AND OTHER USES OF ADD I T ION While predicting the future remains difficult even for professionals such as economists and meteorologists, addition provides a method to make educated guesses about which events are more or less likely to occur. Prob- ability is the process of determining how likely an event is to transpire, given the total number of possible outcomes. A simple illustration involves the roll of a single die; the probability of rolling the value three is found by adding up all the possible outcomes, which in this case would be 1, 2, 3, 4, 5, or 6 for a total of six possible outcomes. By adding up all the possibilities, we are able to determine that the chance of rolling a three is one chance in six, meaning that over many rolls of the die, the value three would come up about 1/6 of the time. While this type of calculation is hardly useful for a process with only six possible outcomes, more complex systems lend them- selves well to probabilistic analysis. Poker is a card game with an almost infinite number of variations in rules and procedures. But whichever set of rules is in play, the basic objective is simple: to take and discard cards such that a superior hand is created. Probability theory provides sev- eral insights into how poker strategy can be applied. Consider a poker player who has three Jacks and is still to be dealt her final card. What chance does she have of receiving the last Jack? Probability theory will first add up the total number of cards still in the dealer’s stack, which for this example is 40. Assuming the final Jack has not been dealt to another player and is actually in the stack, her chance of being dealt the card she wants is 1 in 40. Other situations require more complex calculations, but are based on the same process. For example, a player with two pair might wonder what his chance is of draw- ing a card to match either pair, producing a hand known as a full house. Since a card matching either pair would produce the full house, and since there are four cards in the stack which would produce this outcome, the odds of drawing one of the needed cards is now better than in the previous example. Once again assuming that 40 cards remain in the dealer’s stack and that the four possible cards are all still available to be dealt, the odds now improve to 4 in 40, or 1 in 10. Experienced poker players have a solid grasp of the likelihood of completing any given hand, allowing them to wager accordingly. Probability theory is frequently used to answer ques- tions regarding death, specifically how likely one is to die due to a specific cause. Numerous studies have examined how and why humans die, with sometimes surprising findings. One study, published by the National Safety Council, compiled data collected by the National Center for Health Statistics and the U.S. Census Bureau to pre- dict how likely an American is to die from one of several specific causes including accidents or injury. These statis- tics from 2001 offer some insight into how Americans sometimes die, as well as some reassurance regarding unlikely methods of meeting one’s end. Not surprisingly, many people die each year in trans- portation-related accidents, but some methods of trans- portation are much safer than others. For example, the lifetime odds of dying in an automobile accident are 1 in 247, while the odds of dying in a bus are far lower, around 1 in 99,000. In comparison, other types of accidents are actually far less likely; for instance, the odds of being killed in a fireworks-related accident are only 1 in 615,000, and the odds of dying due to dog bites is 1 in 147,000. Some types of accidents seem unlikely, but are actually far more probable than these. For example, more than 300 people die each year by drowning in the bath- tub, making the lifetimes odds of this seemingly unlikely demise a surprising 1 in 11,000. Yet the odds of choking to death on something other than food are higher by a factor of ten, at 1 in 1,200, and about the same as the odds of dying in a structural fire (1 in 1,400) or being poisoned (1 in 1,300). Unfortunately, these odds are roughly equiv- alent to the lifetime chance of dying due to medical or surgical errors or complications, which is calculated at 1 in 1,200. A d d i t i o n 8 R E A L - L I F E M A T H Where to Learn More Books Orkin, Mike. What are the Odds? Chance in Everyday Life. New York: W.H. Freedman and Company, 2000. Seiter, Charles. Everyday Math for Dummies. Indianapolis: Wiley Publishing, 1995. Walker, Roger S. Understanding Computer Science. Indianapolis: Howare W. Sams & Co., 1984. Periodicals Lin, B-H., E. Frazao, and J. Guthrie. “Away-From-Home Foods Increasingly Important to Quality of American Diet,” Agricultural Information Bulletin, U.S. Department of Agri- culture and U.S. Department of Health and Human Services. (1999). Shaw, Mary, Richard Mitchell, and Danny Dorling. “Time for a smoke? One cigarette reduces your life by 11 minutes.” British Medical Journal. (2000): 320 (53). Web sites Aetna Intellihealth. “Can We Predict Height?” February 11, 2003. http://www.intelihealth.com/IH/ihtIH/WSIHW000/353 20/35323/360788.html?ddmtHMSContent#bottom (March 15, 2005). Arabic 2000. “The Arabic Alphabet.” http://www.arabic2000 .com/arabic/alphabet.html (March 15, 2005). Brillig.com. “U.S. National Debt Clock.” http://www.brillig. com/debtclock (March 15, 2005). ComputerWorld. “Inside a Microprocessor.” http://www .computerworld.com/hardwaretopics/hardware/story/0,10 801,64489,00.html (March 14, 2005). DECA: The Decathlon Association. “The Decathlon Rules.” http://www.decathlonusa.org/rules.html (March 15, 2005). Gambling Gates. “The truth behind the limits: What maximum and minimum bets are about.” http://www.gambling gates.com/Tips/maximum_bets8448.html (March 15, 2005). Intel Research. “Moore’s Law.” http://www.intel.com/research/ silicon/mooreslaw.htm (March 15, 2005). Mathematics Magazine. “Chisenbop Tutorial.” http:// www. mathematicsmagazine.com/5-2003/Chisenbop_ 5_2003.htm (March 13, 2005). National Safety Council. “What are the odds of dying?” http://www.nsc.org/lrs/statinfo/odds.htm (March 15, 2005). Nutristrategy. “Calories Burned During Exercise.” http://www .nutristrategy.com/activitylist.htm (March 15, 2005). Sigma Educational Supply. “History of the Abacus.” http:// www.citivu.com/usa/sigmaed/ (March 14, 2005). Tallahassee Democrat (AP). “National ‘Debt Clock’ Restarted.” July 11, 2002. http://www.tallahassee.com/mld/tallahassee/ business/3643411.htm (March 15, 2005). The Great Idea Finder. “Fascinating Facts About the Invention of the Abacus by Chinese in 3000 BC.” Inventions. http://www.ideafinder.com/history/inventions/abacus .htm (March 14, 2005). The Math Forum at Drexel. “How Are Roman Numerals Used today?” (March 15, 2005). University of Maryland Physics Department. “A2-61: Exponential Increase - Chessboard and Rice.” http://www.physics .umd.edu/lecdem/services/demos/demosa2/a2-61.htm (March 15, 2005). R E A L - L I F E M A T H 9 Algebra Overview Algebra is the study of mathematical procedures that combine basic arithmetic with a wide range of symbols in order to express quantitative concepts. Arithmetic refers to the study of the basic mathematical operations per- formed on numbers, including addition, subtraction, multiplication, and division, and is widely viewed as a separate field of mathematics because it must be taught to students before they can progress to higher studies; but arithmetic is basically algebra without the symbols and advanced operations. In this sense, algebra is often referred to as a generalization of arithmetic, which can be applied to more sophisticated ideas than numbers alone. From adding up the price of groceries and balancing a checkbook, to preparing medicines or launching humans into space, algebra enables almost any idea to be written in standard mathematical notation that can be utilized by people around the world. No matter how advanced the mathematics involved, algebraic rules and notations pro- vide the instructions that dictate how to handle the vari- ous combinations of numbers and symbols. Fundamental Mathematical Concepts and Terms Algebraic symbols can be classified into symbols for representing quantities (usually numbers and letters); symbols for representing operations (such as addition, subtraction, multiplication, division, exponents, and roots); symbols representing equality and inequality (equal to, approximately equal to, less than, greater than, less than or equal to, greater than or equal to, and not equal to); and symbols for separating and organizing terms, and determining the order of operations (typically parentheses and brackets). Multiplication in an algebraic expression is often represented by a dot when written out by hand (e.g., 4 • 5
20), or an asterisk when using a computer or graphing calculator (e.g., 4*5
20). Adjacent sets of parentheses also signify multiplication, as in (4)(5)
20. A number or variable attached to the outside of a set of parentheses also signifies multiplication. That is, 60  t
(60)(t)
(60)t
60(t). These notations are used instead of 4  5 and 60  t in order to avoid confusion between the multiplication sign and the commonly used variable x. The symbol for multiplication is often omitted from an equation altogether: aside from the notation, 60t is identical to 60  t. When two numbers are multiplied together, there must always be some sort of symbol to A l g e b r a 10 R E A L - L I F E M A T H indicate the multiplication in order to avoid confusion. For example, (2)(3)
2  3  23. Repeated multiplication can be simplified using expo- nential notation. If the letter n is used to represent a generic number, then n  n
n2 (n squared), n  n  n
n3 (n cubed or n to the 3rd power), and so on. For example, if n
3, then n3
33
3  3  3
27. In general, the value of n multiplied by itself y times can be expressed as ny, read n to the power of y. Performing operations in the proper order is essen- tial to finding the correct solution to an equation. In gen- eral, the proper order of operations is as follows: 1. Using the following guidelines, always perform oper- ations moving from left to right; 2. Perform operations within parentheses or brackets first; 3. Next, evaluate exponents; 4. Then perform multiplication and division operations; 5. Finally, perform addition and subtraction operations. In algebraic equations, numbers are typically referred to as constants because their values do not change. Letters are most often used as variables, which represent either unknown values or placeholders that can be replaced with any value from a range of numbers. For example, if a car is traveling at a speed of 60 miles per hour, then the distance that the car has traveled can be represented as d
60t, where d represents the distance in miles that the car has traveled and t represents the num- ber of hours that the car has been moving. The variable t can be replaced with any nonnegative value (zero and the positive numbers); as time progresses, t increases, and as would be expected, the distance d increases. An expression that involves variables, numbers, and operations is called a variable expression, or algebraic expression. For example, x2  3x is a variable expression. An equation, like x2  3x
18, is created when a variable expression is set equal to a number, variable, or another variable expression. An algebraic inequality is expressed Mathematician Dr. Tasha Inniss corrects a factorization shown on blackboard. Can you spot the error? AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. A l g e b r a R E A L - L I F E M A T H 13 number zero outside of India, introducing it directly to Arabs, and later to Europe when his writings were translated to Latin and other European languages. Khawarizmi’s original book on arithmetic was lost, leav- ing only translations. Another of Khawarizmi’s books, Kitab al-Jabr w’al- Muqabala, sparked the analysis of algebra as a well- organized form of mathematics. The title of the book has been interpreted in various ways, including “Rules of Reintegration and Reduction” and “The book of sum- mary concerning calculating by transposition and reduc- tion.” The word algebra is derived from the term al-jabr in the title of the book, which can be taken to mean “reunion of broken parts,” “reduction,” “connection,” or “completion.” The rest of the title loosely translates to “to set equal to” or “to balance.” The title of the book relates to the fundamental procedures involved in solving alge- braic problems, such as shifting terms from one side of an equation to the other and combining like terms. The methods described in Khawarizmi’s books have been built upon ever since; and the word algebra evolved for centuries before it was spelled and used as it is today. At the beginning of the thirteenth century, Leonardo da Pisa (also known as Leonardo Fibonacci), an Italian mathematician, traveler, and tradesman, discovered that the potential of algebraic computations using the Hindu (Arabic) notation for numbers far exceeded the capacities of the Roman numeral system that was standard in Europe at that time. In his writings on algebra, he dis- cussed the superiority of the symbols and concepts borne in distant lands. His writings included little original dis- coveries and were intended to illuminate pertinent ideas and problems found in his culture at that time. Unfortu- nately, his proposals were generally viewed as nothing more than interesting, and the ideas that he attempted to spread would not catch on in Europe for almost 300 years. In the late fifteenth century, an Italian named Lucas Paciolus (Lucas de Burgo) authored multiple works on arithmetic, geometry, and algebra. Though most of the mathematical elements are taken from earlier writings, his algebraic writings were integral in the development of algebraic methods because of his efficient use of symbols. Due to the invention of the printing press earlier in the century, Paciolus’ writings were among the first widely distributed algebraic texts, at long last effectively intro- ducing the benefits of algebraic reasoning and Arabic numerals. In the sixteenth century, algebra began to be used in a purely mathematical sense, with symbols and numbers completely representing general quantitative ideas. Robert Recorde—an English mathematician and originator of the symbol
for representing equality—is attributed with the first use of the term algebra in a strictly mathematical sense. François Viete made much progress in the use of symbols for representing generic numbers, which enabled mathematic ideas to be represented in a more general manner and ultimately led to the view of algebra as gener- alized arithmetic. The recognition and understanding of negative val- ues, irrational numbers, and negative roots of quadratic equations were crucial developments in the progression of algebraic theories because they opened doors to more advanced mathematical concepts. The discovery of nega- tive numbers is often attributed to Albert Girard in the early seventeenth century. Unfortunately for Girard, the work of René Descartes—another great mathematician of the time—overshadowed his findings. In the field of algebra, the most notable accomplish- ment of Descartes was the discovery of relationships between geometric measurements and algebraic meth- ods, now referred to as analytic geometry (geometry ana- lyzed using algebra). Using this analytic method of describing measurements such as lengths and angles, Descartes showed that algebraic manipulations (e.g., addition, multiplication, extraction of roots, and repre- sentation of negative values) could be represented by investigating related geometric shapes. Descartes’ fusion of algebra and geometry elucidated both mathemati- cal fields. In the more than two centuries following Descartes discoveries, mathematicians have continued to refine algebraic notation and analyze the properties of more sophisticated aspects of mathematics. Many algebraic advances enable mathematicians and scientists to investi- gate and understand real-world phenomena that were previously thought impossible or unnecessary to analyze. In the twenty-first century, it seems that there are as many types of algebra as there are problems to be solved, but all of them depend wholly on the concepts of basic algebra. Real-life Applications PERSONAL F INANCES Many people use a checkbook registry or financial software to track their income and expenses in order to make sure that they are making enough money to pay their bills and accomplish their financial goals. The reg- istry in a checkbook is basically a form that helps to perform the algebraic operations necessary to track expenses. A checkbook registry includes columns for describing transactions (including the dates on which A l g e b r a 14 R E A L - L I F E M A T H they take place), and columns for the recording amount of each transaction. There is usually a column labeled “deposits” and another column labeled “debits” so that transactions that add money and transactions that sub- tract money can be kept separate for quick and easy analysis. For example, when birthday money or a pay- check is received, it is logged in the deposits column. Things like groceries, rent, utilities, and car payments are recorded in the debits column. A “balance” column is provided for calculating the amount of money present in the bank account after each transaction. The process of recording transactions and balances in a checkbook registry basically involves performing a large, highly descriptive, ongoing algebraic equation. When a transac- tion is recorded in the column for deposits, a positive term is appended to the equation. When a transaction is recorded in the debits column, a negative term is appended to the equation. The balance column repre- sents the other side of the equation. As the terms are appended to the equation, the balance column may be updated immediately, or the various transactions can be recorded and the total can be found later; but either way, the balance is always the result of the addition and sub- traction of the terms represented by the values in the deb- its and deposits columns. Every April, millions of United States citizens must analyze their financial income from the previous year in order to determine how much income tax they are required pay to federal and state government offices. Government taxes help pay for many social benefits, such as healthcare and social security. The Internal Revenue Service (IRS) pro- vides various forms with step by step instructions for per- forming algebraic operations to calculate subtotals, and ultimately the amount of money that must be sent to the government. The various items on a tax form include the amount of taxes that are withheld from each paycheck; the amount of money taken home from each check after estimated taxes are deducted; items of personal worth such as savings, investments, and major possessions; and work- related expenses such as company lunches, office supplies, and utility bills. Many people receive money back from the government because the items representing expenses and taxes throughout the year add up to more than the total taxes due for the year. Some people end up owing taxes at the end of the year. Other people, such as self-employed workers, may or may not have taxes withheld from each paycheck. These people generally use a different type of IRS form and need to save money throughout the year to pay their taxes come tax time. Whatever form is used, the vari- ous items are added, subtracted, multiplied and divided just like the terms of an algebraic equation. In essence, an IRS tax form is an expanded algebraic equation, with the terms and operations written out as explicit, intuitive instruc- tions. The variables are described with words and a blank line or box is provided for filling in the value of each variable. COLLEGE FOOTBALL Unlike other college sports, National Collegiate Athletic Association (NCAA) football does not hold a national tournament at the end of the season to deter- mine which team is the year’s national champion. Instead, a total of 25 bowl games are held throughout the country, pitting teams with winning records against each other. The Bowl Championship Series (BCS) consists of four of these bowl games: the Orange Bowl, the Fiesta Bowl, the Sugar Bowl, and the Rose Bowl. These four bowl games feature eight of the highest rated teams of the year, and each year a different bowl game is designated as the national championship game. An invitation to any BCS game guarantees that a school will receive a hefty sum of money at the end of the year. Winning a BCS game could bring in millions of dollars. The mathematical formula used to figure out which teams make it to the BCS games (and which two teams will fight to be crowned the national champions) turns out to be a rather complicated application of statistical analysis; but algebra provides the backbone of the entire operation. Across the country every week, each team’s BCS ratings are updated according to four major factors: Computer rankings, the difficulty of the team’s schedule, opinion polls, and the team’s total number of losses. Each of these four components yields a numerical value. The computer rankings, for one, are determined by complex computer programs created by statisticians. Computer ranking programs crunch an enormous amount of statistical data, including numerical values representing a multitude of factors ranging from the score of the game, the number of turnovers, and each team’s total yardage, to the location of the game and the effects of weather. The difficulty of a team’s schedule is also determined by algebraic equations with terms accounting for the dif- ficulty of the team’s own schedule and the difficulty of the schedules of the teams that they will play throughout the season. There are two separate opinions polls: one involving national sports writers and broadcasters, and one involv- ing a select group of football coaches. Each poll results in a numerical ranking for all of the teams. For each team, an average of a these two rankings determines their national opinion poll ranking. A l g e b r a R E A L - L I F E M A T H 15 A team’s number of losses is the most straightforward factor. The number of losses is figured directly into the general mathematical model, and each loss throughout the season has a large effect on a team’s overall ranking. The four numerical values are added together to cal- culate the team’s national ranking. The top two teams at the end of the regular season are invited to the national championship BCS game. However, the selection of the six teams that are invited to the other three BCS games is not as straight forward. These other six teams are selected from the top 12 teams across the nation (excluding the top two that are automatically invited to the champi- onship game). How these 12 teams are narrowed to six depends mainly on which teams are expected to attract the most attention and, therefore, create the most profits for the hosting institution, the television and radio sta- tions that broadcast the game, and the various sponsors. These financial considerations are also modeled using algebraic formulas. UPC BARCODES Universal Product Code (UPC) barcodes are attached to almost all items purchased from mass mer- chandisers, such as department stores and grocery stores. These barcodes were originally used in grocery stores to help track inventory and speed up transactions, but shortly thereafter, UPC barcodes were appearing on all types of retail products. UPC barcodes have two components: the barcode consisting of vertical lines that can be read by special scanning devices, and a set of numbers that can be read by humans (see Figure 1). Each component represents the same 12-digit number in a different language. That is, the barcode is simply the number below it represented in the language that can be read by the barcode scanner. The language of barcode scanners is based on vertical lines of two different colors (usually black and white) and four different sizes (the skinniest lines, and lines that are two, three, and four times as thick). The UPC numbers for all items throughout the world are created and maintained by a central group called the Uniform Code Council (UCC). The first six numbers of a product’s UPC number identify the manu- facturer. Any manufacturer that wants to use UPC bar- codes must submit an application to the UCC and pay an annual fee. Every barcode found on products sold by the same manufacturer will start with the same six digits. The first digit of the manufacturer number (the first digit in the entire UPC number) organizes all manufacturers into different categories. For example, the UPC numbers for pharmaceutical items, such as medicines and soaps, begin with 3. Some numbers at the beginning of UPC numbers are reserved for special items like coupons and gift certificates. The second set of five digits represents the product itself. This five-digit product code is unique on every dif- ferent product sold by a manufacturer, even different sizes of the same product. Some larger manufacturers have secured choice manufacturer codes and product codes that contain consecutive zeros. In certain configu- rations, consecutive zeros can be left out so that the UPC barcode can be squeezed onto small products, such as packs of chewing gum. There are ways to determine the positions of missing zeros when less than 12 numbers appear; but regardless, the barcode represents all 12 digits so that a quick swipe in front of a scanner determines all of the necessary information. In any store, the price of each item is stored in a sep- arate computer, which is attached to all of the checkout registers and provides the price for each item scanned. The prices of items are not indicated on barcodes because different stores charge different prices and all stores need to be able to change prices quickly. The final digit of a UPC number is called the check digit and is used to minimize mistakes made by barcode scanners. The final digit can be calculated from the pre- ceding 11 digits using a standard set algebraic operations. Following is an explanation of the algebra involved in cal- culating the check digit of the UPC number 43938200039, which has a check digit of 9: 1. Starting with the first digit, add together all of the digits in every other position (skipping every other number): 4  9  8  0  0  9
30. In a sense, this sum is a variable in the equation for calculating the check digit because it represents values that can be changed. Figure 1: UPC bar code. KELLY QUIN. REPRODUCED BY PERMISSION. A l g e b r a 18 R E A L - L I F E M A T H To ensure safe, reliable operation, the dimensions of a parachute must be as close to perfect as possible. Luckily the specifications of all parachutes are determined and tested according to in-depth mathematical models. While these models are rather advanced applications of mathematics and physics, they rely heavily on algebraic reasoning. Most divers employ an automatic activation device (AAD), a small computation device that performs con- stant calculations in order to deploy a reserve parachute if something goes wrong. The AAD unit is turned on when the diver is on the ground, and from then on it constantly monitors the altitude of the diver. When the diver jumps out of the airplane, the AAD senses that it is falling quickly, and is programmed to recognize this as the beginning of the fall. If the diver falls past a certain alti- tude without deploying the main parachute, the AAD shoots a piece of metal into the cord that holds the reserve parachute in place, deploying it automatically. As long as the reserve parachute opens correctly, the AAD will most likely save the life of a diver that is dis- tracted or has lost consciousness. To make things more complicated, the AAD must also be programmed to dif- ferentiate a loss in altitude during free fall from a loss in altitude due to other events, such as the plane landing before the diver ever jumps out. This ensures that the AAD will only activate the reserve parachute if the diver is free falling. Every aspect of skydiving—from the altitude and timing to the lengths of all the cords and the computer assistance of the AAD—involves the addition, subtrac- tion, multiplication, and division of terms and expres- sions that represent many factors. An enormous amount of algebraic formulas helps divers, instructors, pilots, and equipment manufacturers understand the multitude of factors that must be controlled in every skydiving session. CRASH TESTS Every year, hundreds of new model cars, trucks, vans, and sports utility vehicles (SUVs) are purposely involved in controlled crashes in order to analyze the safety fea- tures of each model of automobile. The various compo- nents of these crash tests involve a seemingly endless amount of calculations. The slightest miscalculation can result in the recall of an entire model (which costs the manufacturer a substantial amount of money), and much worse, injury or death of people involved in real crashes. Therefore, the calculations involved in crash tests are checked multiple times under various conditions. The design of crash test dummies, the main focus of all crash tests, involves a great deal of algebraic calcula- tions. To ensure consistent results, all official crash tests use the same type of crash test dummy, belonging to the Hybrid III family of dummies. Various Hybrid III dum- mies are used to simulate different ages and body types for both genders. For each dummy, characteristics includ- ing height and weight are measured and factored in to the mathematical formulas used to analyze the amount of damage done to the dummy during an accident. Crash test dummies must possess rather complex structures in order to simulate all of the parts of a human body that are usually affected in car crashes. For example, an elaborate spine consisting of metal discs connected by rubber cushions is attached to sensors that collect data used to analyze the damage done to the simulated spine. Sensors for measuring how quickly different parts of the body speed up and slow down are present throughout the body of a dummy. These sensors collect data that help to analyze the potential injury sustained due to the sudden decrease in speed caused by a crash (e.g., whiplash). Other sensors are placed throughout the dummy to measure the amount of impact endured by body parts (e.g., how hard the dummy’s arm slams into the dash- board). Different colors of paint are applied to a dummy’s various body parts so that, when an impact is detected by these sensors during in a crash test, researchers can deter- mine which parts of the body collided with which parts of the car or airbag. A single sensor in a dummy’s chest measures how much the chest is compressed due to the forces applied by the seatbelt and airbag. All of the information collected by these sensors is injected into mathematical formulas in order to test and improve the timing and power of the seatbelts and airbags. For example, modern seatbelts sense abrupt decreases in an automobile’s speed, immediately lock up but allow for a small amount of movement forward, then quickly increase the tension to bring the body to a stop, and finally decrease the tension so that the seatbelt does not cause injuries. In this way, the body slows down more gradually than it would if strapped in by a constantly stiff seatbelt. If the seatbelt stopped the body from moving forward all at once, the seatbelt itself could cause sub- stantial injuries. Most cars now include airbags to supple- ment seatbelts in absorbing the forward force of the body and keeping it from slamming into anything solid. In order to effectively supplement a seatbelt, an airbag must deploy with perfect timing immediately after the seatbelt begins to lock up. Algebraic operations are integral to the mathemati- cal models used to analyze the various factors in a car crash. The wide range of problems solved using the math- ematical models found in a crash test include the realistic design of dummies and the analysis of data collected A l g e b r a R E A L - L I F E M A T H 19 from their sensors; determining the effects of modifying the weight of the automobile and the load it carries; selec- tion of the speed at which to hurl the automobile toward a concrete wall (frontal impact tests), or how fast to slam an object into the side of the automobile (side impact tests); analysis of the deliberate crunching of the materi- als that make up the automobile, which helps to absorb much of the impact; and calculation of the odds of sur- viving such a crash in real life. The rating systems used to indicate the effectiveness of an automobile’s safety fea- tures also rely on algebraic formulas. FUNDRA IS ING In any fundraiser, the planners must be sure that enough money is brought in to cover the costs of the event and meet their fundraising goals. For example, to raise money for new equipment at a local hospital, the hospital’s fundraising committee decides to sell raffle tickets for a new $30,000 car. The committee needs to raise at least $20,000 to be able to pay for the new equip- ment. To ensure that at least this amount of money is available after the paying for the car and the various com- ponents of the fundraiser, the committee decides to set up a mathematical model. To analyze the financial details of the event and decide the price of the raffle tickets and the minimum number of tickets that need to be sold, the committee prepares an algebraic formula. The formula will take into account the price of the car, the number of tickets sold, the price to be charged for each ticket, and the ultimate financial goal of raising $20,000. The for- mula they come up with is G
TP  C  E, where the variable G represents the amount of money to be raised, T is the number of tickets sold, P is the price of each ticket, C is the cost of the car, and E is the cost of the event. This equation states that the amount of money raised will be equal to the number of tickets sold multi- plied by the price of each ticket, minus the cost, the car, and the event itself. Because the purpose of this equation is to determine how many tickets to sell and at what price, the committee rewrites the equation with the term TP (representing the number of tickets multiplied by the price of each ticket) alone on the left side of the equation. By subtracting TP from both sides, the equation becomes G  TP
C  E. Next, subtracting G from both sides gives TP
G  C  E. The term TP is now alone on the left side of the equation, but notice that all of the terms are negative. By multiplying both sides of the equation by 1, all of the negative terms become positive to yield TP
G  C  E. This equation now states that the amount of money that will be taken in from the sales of the raffle tickets is equal to the financial goal of the event plus the cost of the car plus the cost of the event itself. This equation allows the committee to substitute the values for the financial goal and the costs of the car and the event in order to deter- mine the number of tickets that need to be sold, and at what price. However, the committee does not necessarily want the money brought in from the ticket sales to be exactly equal to the costs and fundraising goals; the money brought in needs to be greater than or equal to the money spent. Thus the committee makes this equation into an algebraic inequality by replacing the equal sign with the greater than or equal to symbol to get TP  G  C  E. Next, the committee begins to plug numbers into their algebraic inequality. The committee’s financial goal for the event is to raise $20,000, so G
20,000. The car costs $30,000, so C
30,000. The costs of the eventflyers, food, musical entertainment, renting a venue, and so onare estimated at $5,000, so E
5,000. By substituting these values into the equation TP  G  C  E, the com- mittee finds that TP  20,000  30,000  5,000
55,000. Since TP  55,000, the committee knows that the sale of tickets must amount to at least $55,000. At a similar fundraising event in the previous year, a little over 12,000 raffle tickets were sold. To be safe, the committee decides to predict an underestimate of 11,000 raffle tickets sold at this year’s event. Substituting 11,000 for the variable T, the inequality becomes 11,000P  55,000. Dividing through by 11,000 gives P  5; so the committee needs to charge at least $5.00 for each ticket in order to pay for the car and the event, and have enough left over to pay for the new equipment. Since 11,000 was an underestimate for the number of tickets sold, the com- mittee decides that it is safe to charge $5.00 for the tickets. BU ILD ING SKYSCRAPERS A massive amount of calculations are involved in all phases of skyscraper construction, from determining the amounts of time, manpower, money, concrete, steel, wiring, pipes, and paint needed to build the skyscraper, to determining the number of exits, bathrooms, and electri- cal outlets needed to serve the maximum capacity of people in the building. The calculations used in the actual creation of the structure are dependent on basic algebra; but long before construction can begin, more sophisti- cated mathematical formulas are developed to design a building that meets strict safety guidelines. A skyscraper towering high above a city is susceptible to many unpredictable forces and must be able to with- stand a wide range of punishing forces. These forces include large changes in weight due to people coming and going and precipitation collecting on the outside of the A l g e b r a 20 R E A L - L I F E M A T H building, fluctuations in air pressure and wind, and seis- mic activity (earthquakes). A skyscraper must even be able to endure a sizeable fire or other direct damage to the structure of the building. For example, in 1945, a United States Army B-25 bomber, whose pilot had been disori- ented by dense fog, crashed into the side of the Empire State Building, tearing gigantic holes in the walls and sup- port beams, and igniting fires on five floors. However, the nearly 1,500-foot (457 m) tall skyscraper (the tallest in the world at the time) stood and the damage was repaired. If even small miscalculations had taken place in the planning of the Empire State Building, the crash might have caused the entire building to topple. In-depth architectural specifications used to make a building visually pleasing and functionally efficient require countless algebraic systems. Formulas for model- ing the various safety issues involved in constructing such a tall building take into account all of the factors that can compromise the structure of a skyscraper. The biggest problem to overcome when attempting to design a safe skyscraper is to make the structure stable enough to with- stand wind and other forces. A skyscraper cannot be per- fectly rigid. The structure must be allowed to sway slightly in all directions or its own weight would cause the structure to snap like a dry stick when acted on by forces like wind and earthquakes. On the other hand, if the sky- scraper were allowed to sway too far from perfectly verti- cal, the building would fall over. Under normal conditions, the movement of a skyscraper is undetectable by the human eye, and unnoticed by occupants. The amount of flexibility in the structure must be controlled perfectly by the structure of each floor. Modeling the nature of a sky- scraper’s flexible components involves the use of the imaginary number, i, where i2
1. In the study of basic algebra, the value of i is not log- ical because multiplying any real number by itself results in a positive number, e.g., 22
(2)2
4. Multiples of i, such as 2i and 3i, are called imaginary numbers or complex numbers. An entire field of mathematics, known as complex analysis, is devoted to the study of the prop- erties of imaginary numbers. Although imaginary num- bers do not follow the rules of basic algebra, they are often used to simplify enormous, intricate polynomial equations—like those used to model the stability of skyscrapers—into more manageable equations. After an equation is solved using imaginary numbers, the solution can often be transformed back into real numbers. The use of imaginary numbers enables mathematicians and sci- entists to solve problems that would otherwise be unsolv- able. For example, by assuming that i exists and using it in algebraic expressions, mathematicians, physicists, chemists, statisticians, and engineers are able to model and simplify complicated phenomena. In addition to modeling the slight swaying of a skyscraper, imaginary numbers can be used to model the behavior of electrical circuits, the springs that absorb shock in automobiles, and sophisti- cated economic systems. BUY ING L IGHT BULBS Incandescent light bulbs produce light by passing electricity through a thin metal coil, called a filament (see Figure 2). When electricity is passed through the fila- ment, it glows and illuminates the light bulb. The elec- tricity also produces heat as it passes through the filament. In fact, special bulbs, called heat lamps, are intended to produce heat for purposes such as heating food and drying plants; but in most light bulbs, heat is an undesired (and unavoidable) side effect which eventually causes the filament to burn out. The amount of time that a light bulb can be turned on for before it is expected to burn out is printed on most packages so that shoppers can compare the life expectancy of the various available bulbs. It turns out that the amount of electricity that passes through the filament when the light bulb is turned on is all that is needed to predict how long a light bulb will last. By logging and analyzing the results of various experiments to test the life of light bulbs under different conditions, the life expectancy of a light bulb has been found to be inversely proportional to the voltage that is applied to the filament. That is, the life expectancy is equal to some number divided by the number of volts raised to a power; when two values are inversely propor- tional, decreasing one value causes an increase in the other value. The life expectancy of most incandescent Figure 2: Light bulb filament. ROYALTY-FREE/CORBIS. A l g e b r a R E A L - L I F E M A T H 23 stored data about the individual. When someone first uses a biometric device, physical characteristics are meas- ured, translated into mathematical formulas by a com- puting device, and stored for future comparison. The most widely developed biometric security devices include cornea and iris scanners that measure the characteristics of the parts of an individual’s eyes; face scanners that can recognize major facial features; voice scanners that meas- ure the frequencies in an individual’s voice; and finger- print scanners that read and interpret the unique curves and patterns of an individual’s fingerprints. Fingerprint scanners are widely accepted as one of the most effective forms of identification, and are becoming more common in all types of secure environments. The uses of fingerprint scanners range from the physical pro- tection of a secured room to the protection of sensitive computer files. Many computer mice and keyboard man- ufacturers integrate fingerprint scanners into their prod- ucts in hopes of replacing passwords as the most common form of identity verification for personal computers. An increasing number of automobile manufacturers have begun to incorporate fingerprint scanners into door lock- ing mechanisms and ignition systems, so that the owner of a vehicle does not need a key to lock and unlock the car, or start the engine. Banking institutions are also beginning to look to fingerprint technology in hopes of replacing bank cards and personal identification numbers (PINs). Like all biometric devices, fingerprint scanners map the unique characteristics of a fingerprint into mathe- matical formulas, which are used later to determine whether or not the fingerprint present on the scanner matches stored data. The size and relative location of the prominent features in each fingerprint are represented by the terms of mathematical formulas, so fingerprint scan- ners utilize massive amounts of algebraic operations dur- ing each security session. Potential Applications TELEPORTAT ION Throughout history, humans have invented increas- ingly advanced methods of transportation—from the invention of the wheel to the first trip into space—in order to enable and expedite the process of traveling from one physical location to another. However, even with all of the advances in transportation, no vehicle can take passengers from one point to another without traveling across the space in between. Learning how to skip the intermediate locations is the goal of scientists who are attempting to invent and perfect teleportation devices. Similar to the fantastic idea first popularized in science-fiction, teleporta- tion devices essentially collect information about an object, destroy the object, and send the information about the object to a different location, where the object is recon- structed. In a sense, this idea is similar to a fax machine, which translates a copy of a document into numerical information, and sends the information to another fax machine that uses the information to construct and print another copy. A teleportation device collects information about an object and translates it into numerical informa- tion according to mathematical formulas. Different formu- las are then used to reconstruct the object elsewhere. In 1998, a group of physicists performed the first successful teleportation experiment. In the experiment, information about a photon (a particle of energy that carries light) was collected, sent through a cable one meter in length, and used to construct a replica of the photon. When the replica was created, the original pho- ton no longer existed, so the photon is considered to have traveled instantaneously to a different location. It is uncertain whether or not this type of travel will ever be safe for living organisms. The idea of being destroyed and reconstructed elsewhere sounds rather for- eign and frightening, and it will surely be difficult to find willing subjects for experiments. However, the ability to teleport energy will likely have profound effects on com- puter networks. Instead of using cables or airwaves to transfer information between computers, information will be instantaneously teleported from machine to machine, essentially eliminating delays in the transfer of information. An bar-coded identification bracelet is scanned at Georgetown University Hospital in Washington, D.C. A/P WIDE WORLD. REPRODUCED BY PERMISSION. A l g e b r a 24 R E A L - L I F E M A T H PR IVATE SPACE TRAVEL A new form of transportation will most likely revo- lutionize the way that humans think about space travel. SpaceShipOne (see Figure 4) is the first manned space- craft project that does not depend on government funds. This privately owned and operated craft is intended to take anyone who can afford a ticket on a brief trip into space. In order to alleviate the most difficult part of any flight into space—the launch from the ground— SpaceShipOne is launched from a second aircraft, called White Night. While attached to White Night and during the launch into space, SpaceShipOne is in a contracted configuration. After the spacecraft launches and reaches its highest altitude, it spreads its wings in a con- figuration that slows its decent back into the inner atmos- phere. Finally, the craft is reconfigured again to work much like an airplane, allowing the pilot to safely steer and land. Because this project is not funded by the govern- ment, the company that designed and built Space- ShipOne and White Night had to create their innovative technology from scratch, a task that involves an unimag- inable amount of calculations, all of which rely on the fundamental concepts of algebra. Developing a rocket propulsion system alone involves a multitude of mathe- matical formulas. Designing the three different configu- rations of SpaceShipOne also involves an enormous amount of mathematical models for determining the optimal size and shape of the various parts of the space- craft. The idea of an average private citizen having the opportunity to travel into space on a regular basis is a shining example of the endless potential for using algebra to explore the real world. Where to Learn More Books Johnson, Mildred. How to Solve Word Problems in Algebra, 2nd Ed. New York, NY: McGraw-Hill, 1999. Ross, Debra Anne. Master Math: Algebra. Franklin Lakes, NJ: Career Press, 1996. Periodicals Backaitis, S.H., H.J. Mertz, “Hybrid III: The First Human-Like Crash Test Dummy.” Society of Automotive Engineers, Inter- national. Vol. PT-44 (1994): 487–494. Figure 4: SpaceShipOne: the future of travel? JIM SUGAR/CORBIS. A l g e b r a R E A L - L I F E M A T H 25 Web sites How Stuff Works. “How Skyscrapers Work.” Science Engineer- ing Department. http://science.howstuffworks.com/ skyscraper.htm (March 22, 2005). NASA. “Beginner’s Guide to Aerodynamics.” Glenn Research Center. March 4, 2004. http://www.grc.nasa.gov/WWW/ K-12/airplane/bga.html (May 27, 2005). Key Terms Algebra: A collection of rules: rules for translating words into the symbolic notation of mathematics, rules for formulating mathematical statements using symbolic notation, and rules for rewriting mathematical state- ments in a manner that leaves their truth unchanged. Arithmetic: The study of the basic mathematical opera- tions performed on numbers. Coefficient: A coefficient is any part of a term, except the whole, where term means an adding of an alge- braic expression (taking addition to include subtrac- tion as is usually done in algebra. Most commonly, however, the word coefficient refers to what is, strictly speaking, the numerical coefficient. Thus, the numerical coefficients of the expression 5xy2  3x  2y  are considered to be 5, 3, and 2. In many formulas, especially in statistics, certain numbers are considered coefficients, such as correlation coefficients. Constant: A value that does not change. Equation: A mathematical statement involving an equal sign. Exponent: Also referred to as a power, a symbol written above and to the right of a quantity to indicate how many times the quantity is multiplied by itself. Formula: A general fact, rule, or principle expressed using mathematical symbols. Term: A number, variable, or product of numbers and variables, separated in an equation by the signs of addition and equality. Variable: A symbol representing a quantity that may assume any value within a predefined range. A l g o r i t h m s 28 R E A L - L I F E M A T H make purchases totaling $150 a month, the computer’s algorithms should be able to quickly determine that a series of charges from a foreign country totaling thou- sands of dollars is “unusual.” The fact that the purchase does not fit an established pattern can be determined by algorithms and is often enough to alert bank security officials that further investigation is required before authorizing a particular purchase. Upon investigation, they may discover that the user is the authorized card- holder enjoying a vacation or semester abroad. On the other hand, investigation may reveal that the credit card number has been stolen, and that the intended purchase is unauthorized. The use of algorithms to analyze purchases can thus save the bank—or credit card holder—thousands of dollars. CRYPTOLOGY In 1977, Ronald Rivest, Adi Shamir, and Leonard Adleman published an algorithm (known as the RSA algorithm—a name derived from the first letters of the founder’s last names) that marked a major advancement in cryptology. The RSA algorithm factors very large com- posite numbers. As of 2004, the RSA algorithm was the most commonly used encryption and authentication algorithm in the world. The RSA algorithm was used in the development of Internet web browsers, spreadsheets, data analysis, email, and word processing programs. DATA M IN ING Association of data in a data mining process involves the use of algorithms that establish relationships or pat- terns in data. Such algorithms use “nested” or sub- algorithms that rely on statistics and statistical analysis to make associations between data. Usually the algorithm designer (e.g., a computer programmer) specifies desired associations or patterns to be established. Algorithms can be written, however, to perform what is termed exploratory analysis, a form of analysis where associa- tions between data are sought without a preconception or guess as to what patterns might exist. DIG I TAL AN IMAT ION AND D IG I TAL MODEL CREAT ION Moviemakers rely on mathematical algorithms to construct digital animation and models. Such algorithms relate points on known surfaces to points on a drawing (often a computer drawing) or digital model. For exam- ple, data points for the movement of an arm or leg can be obtained by actors wearing special gloves or sensors that translate movements such as waving or walking into data (sets of numbers) that can be analyzed by algorithms designed to fill in the gaps between data points. Such algorithms allow animation experts to subsequently draw and animate figures with increasingly realistic features and movement. Model makers can construct digital mod- els at a fraction of the cost needed to construct and test physical models. DNA OR GENET IC ANALYS IS Biochemists use algorithms, more commonly referred to in the laboratory as “lab procedures” to iden- tify DNA markers that allow scientists and physicians to determine genetic relatedness (identification of parents or family members) to settle a court case or find a suitable organ donor, determine a patient’s risk of disease suscep- tibility risk, or to evaluate the effectiveness (efficacy) of drug treatments. In addition to physical testing algorithms, mathe- matical and computer algorithms can be used to deter- mine or predict patterns of genetic inheritance. The pundit square used in beginning biology is a simple yet powerful use of algorithms that result in the diagram- matic representation of potential gene combinations. In some cases, the pundit square allows the calculation of the odds of having a child who might develop sickle cell anemia or be a carrier of the gene that might lead to actual sickle cell disease in their children. The task of analyzing massive amounts of data gen- erated by DNA testing is daunting even for very powerful computers. New technologies, including so-called “bio- flip” technologies use specialized computer algorithms to detect small and differences and changes in the structure of DNA (i.e., variation in genetic structure). ENCRYPT ION AND ENCRYPT ION DEV ICES Although the technology exists to allow the con- struction of cryptographic devices intended to protect private communications from unauthorized users while at the same time assuring that authorized government agencies (e.g., those agencies such as the FBI who might obtain a court order) can quickly decode (decrypt) and read messages as needed, such technologies remain highly controversial. So-called “clipper-chips” and “capstone chips” would allow use United States law and intelligence agencies to use specific algorithms to decode encrypted (coded) messages. Certain authorized agencies would then hold the algorithmic “keys” (the step-by step proce- dures and codes) to any communication using the encrypting technology. A l g o r i t h m s R E A L - L I F E M A T H 29 Use of the clipper chip was first adopted and author- ized in 1994 by the National Institute of Standards and Technology (NIST). The United States Department of the Treasury was initially designated to hold the keys (algo- rithms) to decode messages. Rules regarding access to the keys are defined in state and national security wiretap laws. The clipper chip utilizes the SKIPJACK algorithm— a symmetric cipher (code) with a fixed key length of 80 bits. A bit is shorthand for “binary digit,” a unit of infor- mation (a “1” or “0” in binary notation). A cipher uses algorithms (i.e., sets of fixed rules) to transform a legible message (clear text or plaintext) into an apparently random string of characters (ciphertext or coded text). For example, a cipher might be defined by the following rule: “For every letter of plaintext, substi- tute a two-digit number specifying the plaintext letter’s position in the alphabet plus a constant (or key) between 1 and 73 that shall be agreed upon in advance.” This would result in every letter in the alphabet being repre- sented by a number between 17 and 99 (depending on the particular constant used). For example, if 16 is the agreed-upon constant, then the plaintext word PAPA enciphers to 32173217 as follows: P  16
16  32; A  1
16  17; P  16
16  32; A  1
16  17. Real keys would, of course be longer and more complex, but the basic idea remains the same: an algorithm-based encryption key allows messages to be locked (enciphered) or unlocked (deciphered), just as a physical key fits into a lock and allows it to be locked and unlocked. Without a key, a cipher algorithm is missing its most critical part. In fact, so important is the key that many times the algo- rithm itself is widely known and distributed—it is only the keys that remain secret. For this reason, the algorithm used to code messages may remain the same for months or years, but the keys change daily. Other algorithms remain a mystery. In 1943, Alan Turing (1912–1954), Tommy Flowers (1905–1998), Harry Hinsley (1922–1998), and M. H. A. Newman at Bletchley Park, England, constructed a computational device called Colossus to crack the Nazi German encryp- tion codes created by the top secret Enigma machine used by the Germans. The decryption algorithms used by Colossus remain secret. In the late 1970s, the United States government set a specific cipher algorithm for standard use by all govern- ment departments. The digital encryption standard (DES) is a transposition-substitution algorithm that offers 256 different possible keys (a number roughly equivalent to a 1 followed by 17 zeroes). As larger a number of different keys as that number represents, modern higher speed computers might allow hackers (who also use algorithms) to too easily crack codes with this many keys, and so a new algorithm, known as the advanced encryption stan- dard, is replacing the old algorithm. Security is increasing as a function of who can develop the most sophisticated algorithms to either protect data, or hack into protected algorithmic codes. IMAG ING The digitization of images used in modern digital computers would not be possible without the use of algo- rithms to translate the images into numbers and back again into a viewable image. Digital cameras can be in a vacationer’s backpack or be mounted in satellites in orbit hundreds of miles above the Earth. Digital cameras offer higher resolution (the ability to distinguish small objects) than cameras that use light-sensitive photographic film. Digital photo manipulation has also revolutionized pho- tography, including commercial advertising, and offers new security challenges to uncover altered photos. Fractal image compression algorithms allow much greater com- pression in the storage of images. Digital cameras capture reflected light on a chip or charged coupled device (CCD). The surface of the CCD contains light-sensitive cells (photo diodes). Each cell or diode represents a pixel and so the pixel becomes a basic unit of a digital image. Light-stimulated diodes produce a signal (often using a transistor) with a voltage that corre- sponds to the light intensity recorded by the diode. An algorithm in the camera’s processing unit then translates that signal into binary code—1s and 0s—that can later be reconverted by the reverse algorithm back into a viewable image. For example, algorithms may assign a code sequence between 0 and 255 to color data (0 is black and 255 indicates an intense red color). These codes are then turned into eight digit binary code sequences (00000000 for black, 111111111 for the most intense red). Digital photo manipulation involves the alteration of the binary code (i.e., the digital 1s and 0s) that repre- sents the image. While algorithms can be used to alter photos, they can also be used to detect forgery or alter- ation. Algorithms can compare values of pixels in the background of an image and determine whether they are consistent. Other parts of an image can be protected by altering certain pixels to form a digital watermark that can only be removed by application of a particular algorithm to the image binary code. INTERNET DATA TRANSMISS ION All information sent by a computer over the Internet contains the sending computer’s hardware source address A l g o r i t h m s 30 R E A L - L I F E M A T H (MAC address). This is similar to a return address included on a piece of physical mail (snail mail). Con- versely, all the information that the computer accepts must be addressed to its unique hardware address (or often a more common “broadcast address” that is similar to a zip code used in regular mail). When an packet of data (e.g., a portion of a text) is received, the receiving computer subjects the incoming packet of data to a pro- cessing algorithm, a mathematical formula or set of pro- cedures that determines whether the address information is correct and the message intended for that computer. If the packet of data is accepted, additional algorithms are used to decode the binary message (a series of ones and zeros such as “100010110” into text, pictures, or sound). MAPPING Algorithms can analyze data measurements of height, depth, and distance to construct maps. For exam- ple, bathymetric maps (maps that depict the oceans as a function of depth) help develop a model of a body of water as depth increases. Such maps are important to fishermen and similar algorithmic programs analyze data in navigational and “fish-finding” equipment aboard many commercial and sport fishing boats. To construct a precise map of the region, whether of land or at sea, it is necessary to perform detailed meas- urements, a task increasingly performed by satellites (or in the case of bathymetric maps, by ships with echo sounding surveying equipment that bounce sound waves off the ocean floor). Such data is then set into an array (a particular grid or pattern) that are analogous to strips of a map. Algorithms perform calculations that link the data between various strips and allow the construction of a larger map of the area. THE GENET IC CODE Humans themselves are the result of an molecular algorithm that operates at the genetic and cellular level. The genetic code ultimately relates a sequence of chemi- cals called nitrogenous bases found in deoxyribonucleic acid (DNA) to the amino acid sequences of proteins (polypeptides). These proteins control the biochemistry of the body. The algorithm that describes this process allows scientists to understand the genetic and molecular basis of heredity and many genetic disorders. In humans, DNA is copied to make mRNA (messen- ger RNA), and mRNA is used as the template to make proteins. Formation of RNA is called transcription and formation of protein is called translation. This process is the fundamental control mechanism for the development (morphogenesis), growth and regulation of the body and complex physiological processes. The structure of DNA—and the sequences formed during transcription—can, for example be predicted from an algorithm (based upon the physical shape of the molecules themselves) that specifies that the nucleotide with the nitrogenous base adenine will pair only with a nucleotide that contains a thymine base (an A-T pair). Likewise, nucleotides with a cytosine base will pair only with a nucleotide that contains a guanine base (a C-G pair). The molecular algorithm allows the prediction of bases (e.g., the ATTATCGG sequences) that in triplet sequences (three base sequences) then form the backbone of genes. A sequence, such as A-T-T-C-G-C-T . . . etc., might direct a cell to make one kind of protein, while another sequence, such as G-C-T-C-T-C-G . . . etc., might code for a different kind of protein. MARKET OR SALES ANALYS IS Algorithms are routinely used to analyze buying and selling patterns. Businesses rely on algorithms to make decisions regarding which products to sell, or to which portion of the overall market advertising can be most effectively targeted (e.g., an advertising campaign designed to sell blue jeans to teenagers). Such marketing algorithms make associations between buying patterns and established demographic data (i.e., data about the age, sex, race, income groups, etc.) of the user. For example, market specialists use algorithms to study data developed from test groups. If a product receives a sufficiently favorable rating from a very small test group, a manufacturer may skip the costs and delays of further testing and move straight into produc- tion of a product. The algorithms might be simple (e.g., if 75% of the initial test group likes the product the man- ufacture knows from experience that there will be suffi- cient sales to make a profit) or complex (e.g., a complex relation or weighting of responses to the demographics of the test group). Less enthusiastic results may require fur- ther testing or a decision not to develop a particular product. Algorithms can also be used to compare observed responses of a test group to anticipated responses (or responses from other test groups) to determine which products might gain a market advantage if manufactured in a certain way. For example, algorithms can be used to analyze data to find the most favored color, size, and potential name of a skateboard and intended for sale to 10–13-year-old boys. R E A L - L I F E M A T H 33 Architectural Math Overview Architectural mathematics uses mathematical for- mulae and algorithms for designing various architectural structures. Most of these structures are buildings such as museums, galleries, sport complexes and stadiums, the- atres, churches, cathedrals, offices, houses, and so on. Architectural mathematics is also used to design open spaces in cities and towns, recreational places including gardens, parks, playgrounds, water bodies such as lakes, ponds and fountains, and a variety of physical construc- tion and development. Put simply, architectural math pertains to mathe- matical concepts that are central to architecture. These architectural math concepts are also used in several other activities that we see in our daily lives. They are exten- sively used in sports, technology, design, aviation, medi- cine, astronomy, and much more. Understanding architectural math requires knowl- edge of various two-dimensional as well as three- dimensional shapes such as square, rectangle, triangle, cube, cuboids, sphere, cone, and cylinder. There are also other concepts such as symmetry and proportion that are integral to architecture math. The pyramids of Egypt, for example, are based on these principles. Fundamental Mathematical Concepts and Terms As architectural designs are strongly inspired and implemented using various shapes and forms, basic architectural mathematics involves understanding under- lying principles that derive such shapes and forms. This requires understanding geometric equations associated with its visual representations. In other words, architec- tural mathematics is not expressed as simple numbers but rather as graphical or visual forms. What follows are some of the most widely used architectural math concepts. RAT IO A ratio is a comparison by division of two quantities expressed in the same unit of measure. In other words, you get a ratio by dividing two numbers of quantities. The ratio may be represented in words or in symbols. For example, if segment Line A is one inch long and Line B is two inches long, we say that the ratio of Line A to B is one to two. In terms of mathematical symbols, the ratio may be denoted in fractional form as 1⁄2, or it may be expressed as 1:2. A r c h i t e c t u r a l M a t h 34 R E A L - L I F E M A T H Since ancient times, Greeks and Romans were known to be the most elaborate builders. They had a flair for architecture and built structures that were pleasing to the eye. They were convinced that architectural beauty was attained by the interrelation of universally valid ratios. Frequently, complicated mathematical ratios were used by architects to accomplish their goals. Take, for example, the golden ratio (or phi)—1.618. This ratio that has applications in many areas, has been extensively used in architecture—both modern and ancient. PROPORT ION Proportion, like ratio has always been a vital compo- nent of architectural math. The ancient Greeks and Romans followed certain mathematical proportions (and ratio) to attain order, unity, and beauty in their buildings. Using simple mathematical formulae (based on propor- tion) they were able to establish a unique relationship among various parts of buildings. Such relationships have been used for generations. To better understand the concept of proportion, con- sider an example of two buildings variable in height and base, however displaying the same proportion (see Figure 1). In Figure 1, there are four terms that would define proportion of one building to another. These are 2, 3, 6, and 9. A proportion is an equation that states the ratios of comparison are equal. Thus, in the above example we would say that Building 1 is in proportion (or propor- tionate) to Building 2 if 6/9
2/3, or 9/6
3/2. This is the case, and hence the statement that the buildings are in proportion holds. If the ratios for two objects are not equal, they would not be in proportion. Proportion can also be used to calculate the ratio of the total magnitude (in this case size) of the two objects. For example, in our case 9/6 can also be expressed as 3/2  3. Thus, we can say that Building 1 is three times the size of Building 2. SYMMETRY In architecture, one way to attain balance and response while designing structures is by the use of sym- metry. Architecture is based on principles of balance. Basically, if most architectural forms are divided into two equal parts by a line in the center, the opposites sides of the dividing line would be similar (or even identical). This concept that can also be seen in all basic geometric forms (square, rectangle, circle, triangle, and so on) is known as symmetry and is extensively used architectural structures—modern and ancient. See Figure 2 to understand symmetry further. In this figure (2a), triangle ABC and triangle BCD are symmet- ric about line m. The corresponding sides and correspon- ding angles of the triangles are similar. In other words, triangle CBD is the reflection of triangle ABC, and m is the line of symmetry. Such type of symmetry is also known as symmetry by reflection. Forms can also be created using translation or sliding symmetry. This type of symmetry involves two or more similar forms of the same size and facing the same direc- tion. In figure (2b), all images are similar to each other and face the same direction. Another way to understand this is imagine that the same image has been slid on the line p (and thus the name sliding symmetry). The third method of attaining symmetry is by rota- tion. If a figure, after rotating it around a central point by less than 360, remains unchanged, then it has rotation symmetry. For example, in figure 2c, if this form is rotated from the central point B by 180, the resulting form would be the same. Thus, the figure has rotation symmetry for a rotation of 180. It is interesting to note that all geometric forms (square, rectangle, triangle, pyramid, hexagon, etc.) can have reflection, translation, and rotation symmetry. It is for this reason, they are used extensively in architecture. SCALE DRAWING For designing any structure (a building, a house, or a city), an architect is required to convert his ideas to draw- ings. These drawings provide homeowners, contractors, carpenters, and others with a small diagram of the final structure. The drawings show in detail the sizes, shapes, Height = 9 Base length = 6 Building 1 Height = 3 Base length = 2 Building 2 Figure 1. A r c h i t e c t u r a l M a t h R E A L - L I F E M A T H 35 arrangements of rooms, structural elements, windows, doors, closets, and other important details of construc- tion. For example, a drawing for a house would specify the area (length, width, and height) of every room includ- ing the living area, bedroom, and bathroom at every floor. Such miniature reproductions of the structure are called scale drawings. Scale drawings that represent parts of a structure must be in proportion to the actual structure. To do this, architects use a specific scale corre- sponding to the actual size. For example, a scale such as 1⁄4 inch
1 foot, would suggest that a length of 1⁄4 inch on the scale drawing is equal to one foot within the actual structure. Scale drawings allow an architect to visualize a struc- ture before building it. MEASUREMENT Measurement is important and is required while designing any building, right from the planning stage to the actual construction work. The instrument used to measure objects is the ruler, or a measuring tape. Architects, carpenters, and designers use measure-ments to come up with accurate scale drawings before starting construction work. Throughout the process of building any structure, measurement is extensively used. Measurement can be expressed as inches, feet, and yards (English system), or centimeters, and meters (metric system). A Brief History of Discovery and Development There is a commonality between the seventeenth- century Round Tower of Copenhagen, the thirteenth- century Leaning Tower of Pisa, Houston’s Astrodome, (the first indoor baseball stadium built in the United States), the vast dome of the Pantheon in Rome, a Chinese pagoda, and the Sydney Opera House. All these buildings were built using architectural math concepts such as scale, measurement, ratio, proportion, and symmetry. Architectural mathematics has always been a vital part of structural design. The pyramids of Egypt used basic principles of the geometric “pyramid”—a square base and an apex tapering as the elevation of the pyramid increased. The pyramid shape provides higher stability compared to other structures as it is able to counter wind forces and natural forces, such as earth- quakes, much more effectively than compared to most other shapes. The same concept of visual geometry was used while constructing the Eiffel Tower in Paris. It has a square base with a narrowing apex as one moves toward the top of the structure. One key aspect of mathematics to be consid- ered while building a pyramid structure is the use of ratio and proportion in designing the base and apex of the tower. Higher the ratio of base to the apex, the higher will be the stability of the structure to withstand the various forces. This has been kept in mind while designing the A C B 2b2a 2c D m p E C A Line p D B Figure 2. A r c h i t e c t u r a l M a t h 38 R E A L - L I F E M A T H USE OF RAT IO AND PROPORT ION Historical monuments and modern buildings, alike, have used architectural mathematics extensively. This is reflected in several landmarks. As stated earlier, the Pyra- mids of Egypt are a very good example of the use of a simple form (the geometric pyramid shape), having a square base and tapering as its height increases. These are built such that their base and height have specific ratio and proportion. This is done to impart greater strength and stability. The same is the case with the Eiffel tower in Paris. For example, the ratio of the base of the Pyramids with their height is almost 1:1.5. The ratio gives the structure higher stability. Conversely, the Eiffel tower has a ratio of 1:3 for base to height. Both these struc- tures have different ratios. However, they both impart stability to the structure due to their shape and design. (See Figure 3.) Architectural designs also use ratio and proportion to justify the dimensions of elements within the build- ings. This includes length and width of the corridor, its proportion to doors that lie within the corridor, the height of the ceiling from the floor with respect to the type of building, and the ratio of size of the steps on a staircase with respect to the total height of the staircase— all these aspects are considered vital while designing a structure. USE OF ARCH I TECTURAL SYMMETRY IN BU ILD INGS Reflection symmetry (also known as bilateral symme- try) is the most common type of symmetry in architectural designs. In bilateral symmetry, the halves of a composition mirror each other. Such symmetry exists in the Pantheon in Rome. We find the same symmetry in the mission-style architecture of the Alamo in San Antonio, Texas. Bilateral symmetry existed in several buildings built during the Roman or Greek periods. Modern architects also use such symmetry widely for various structures (see Figure 4). Additionally, translation and rotation symmetry are also employed considerably in modern architectural designs. USE OF RECTANGLE AS “GOLDEN RECTANGLE” AND “GOLDEN RAT IO” Since ancient times, architectural designs have used the golden ratio (1.618) in various ways. One of the best examples being the Parthenon, the main temple of the goddess Athena, built on the Acropolis in Athens. The front of the Parthenon is a triangular area that fits inside a rectangle whose sides are equivalent to the golden ratio (the rectangle is popularly known as the golden rectan- gle). The golden ratio and its related figures were incor- porated into every piece and detail of the Parthenon. Ratio & Proportion Eiffel tower Ratio of base Length to height is 1:3 Height 984 ft Square Base, Side is 330 ft long Ratio & Proportion Pyramid of Cheops Ratio of Base length to Height is 1:1.5 Height of Apex is 715 ft length of one side of the base is 475 ft A E B FG D H C 51"50' 51"50' Figure 3. A r c h i t e c t u r a l M a t h R E A L - L I F E M A T H 39 Symmetry of building elements in elevation Multiple symmetry in building plan Figure 4. The same math principles that allowed the construction of the Arch of Constantine in Rome also allow designers to shape modern home interiors. The arch distributes load. TRAVELSITE/DAGLI ORTI. REPRODUCED BY PERMISSION. A r c h i t e c t u r a l M a t h 40 R E A L - L I F E M A T H The Triumphal Arch of Constantine, and the Colosseum—an amphitheater in Rome built in around A.D. 75 (both in Rome)—are other great examples of ancient use of golden relationships in architecture. The main idea behind employing this ratio was to make the structure visually appealing and also more stable. USE OF BAS IC FORMS AND SHAPES OF GEOMETRY Apart from mathematical concepts such as ratio, proportion, and symmetry, most architectural designs are based on basic geometric shapes and forms includ- ing triangle, rectangles, pyramids, cones, cylinders, and more. Although, when viewed as a whole these struc- tures would have basic shapes, their interiors can always be represented by the above mentioned mathematical concepts. The Taj Mahal in India is an example of the use of a basic shape or form—the cube. The Taj Mahal was built as a cube, where the four minarets and the center burial tomb of the queen all are contained in a perfect cube. The length, breadth, and height of all sides are equal in dimension. Additionally, the sense of ratio, pro- portion, and symmetry of this structure is precise and spell-binding. A modern example of the use of basic shapes is the Pentagon, in Washington, D.C. The Pentagon’s five sides are equal in length, denoting a perfect pentagon. Within the main structure, there are five concentric pentagons of corridors and offices. Again, these internal pentagons are symmetric and in proportion to each other. SPORTS Geometric shapes and forms, symmetry, ratio, and proportion have found a place in sports as well. Practi- cally, every field sport uses architectural math principles. A tennis court, basketball court, football, hockey, soccer Decorating Numerous symmetrical shapes and forms are used while decorating furniture and home accessories ranging from a flower vase to the kitchen sink. The use of architectural shapes and concepts is clearly visible in every decora- tive aspect of the complete design. Besides the arrangement of these shapes and forms (see figures below), the concept of symmetry also plays a vital role in the layout of these. For example, while decorating a room, most interior designers would ensure that the entire layout of the room (and how all elements within the room are placed) is based on symmetry principles. The main purpose is to give a dec- orative touch to the room to make it visually more appealing. Home accessories, especially decorative artwork (vase, glassware, china pottery, and so on) are often made of wood or ceramics. These decorative pieces more often than not are also based on principles of symmetry. Their shape, exterior designs, and colors are amazingly symmetrical, and in many cases are based on basic geometric forms and shapes—much like designs in architecture. A r c h i t e c t u r a l M a t h R E A L - L I F E M A T H 43 Cathedrals are a common example, where the architecture is inspired by arranging materials and objects in symme- try, similar to that in ornaments and jewelry. ASTRONOMY Fundamentals of architectural math including dis- tance, size, and proportion are also visible in various astronomical advancements. The telescope is one such example. Telescopes are used to view stars and planets located in far away galaxies. The distance is measured in light years. The distance traveled by light in one year is known as a light year (light travels at a speed of 186,000 miles per second). This gives an indication of the distances between the Earth and some of the stars and planets. Telescopes are used to magnify the image of these objects. This is done by using different lenses. Larger tel- escopes, such as the Hubble telescope, are able to magnify objects situated at a larger distance. Smaller telescopes in comparison have lower magnification implying lower visibility and clarity. One of the basic mathematical principles of tele- scopes is scaling—a concept extremely common in archi- tecture. Just like architects draw scale diagrams using ratio and proportion, telescopes use the same principles to magnify objects situated at large distances. In other words, telescopes present a scale model of an object that is not otherwise visible (or too tiny) with the naked eye. Although, larger telescopes magnify objects that are further away, as compared to the smaller telescopes, the degree of magnification (of both types of telescopes) is always in proportion. TEXT I LE AND FABR ICS Cloth or fabrics are used for a variety of purposes. This includes bed sheets, covers, clothes, apparels, wipes, and more. Fabrics are textile products that require knit- ting. These are made from fibers of cotton, nylon, or other types. However, most of these fabrics do not have any value until a design is printed or woven on them. In other words, fabric prints carry considerable value to a plain piece of fabric or cloth. People would usually buy fabrics with visually appealing prints, rather than those that are plain. Symmetry, which is used commonly used in archi- tecture, is often reflected in fabric or cloth designs. Most fabric designs are composed of motifs. Motifs are repeti- tive use of a single design concept, style, or shape—Motifs signify symmetry (translation symmetry). The type of motifs could range from a leaf or a flower of same color or style repeated over the entire fabric print. The design varies depending on the final use of the fabric. Bed sheets, clothes, fashion apparels, and so on have different sym- metry motifs depending the type of fabric, their manu- facturing price, and quality of print. Motifs are not limited to floral or color patterns but are often extend to lines, simple geometric shapes (squares, circles, rectangle, etc.), blocks, and much more. In some cases, once the fabric is cut or is stitched to make the final product, the symmetry may be lost. Nev- ertheless, the design is still based on the very principle of symmetry. This is one of the most common applications in daily life that uses mathematical concepts of architecture in a very different way. ARCH I TECTURAL CONCEPTS IN WHEELS Commuting has become an integral part of our daily life. We drive (on our own or in public transportation) to work, to school, to attend meetings, to go shopping or buy groceries. We require transportation to reach differ- ent places. Today, transportation is seen as a necessity. Transportation is facilitated by public buses, rail- ways, airplanes, and cars. All of these use wheels. A wheel, be it of rubber, magnet, or iron, is a vital component of any automobile. The wheel consists of a bar in its center known as the axle. The width of the axle is governed by the width of the carriage (weight of the automobile) required. Subsequently, the width varies in trains, buses, and cars. While designing wheels, engineers must ensure that the size of the wheel and the axle is in proportion to the total weight of the vehicle (including the people it car- ries) as well as the speed at which the vehicle can travel. Ratio and proportion play a very important role in defining the diameter, width and the number of wheels that have to be attached with a vehicle. Higher the load to be carried, the more number of wheels (and even stronger wheels) will be required. Similarly, the longer the length of the vehicle, more the number of wheels required. Airplanes do not travel on wheels but require them to land and take off. However, the proportion of their wheels is much greater when compared with other vehicles as the amount of load is much higher. Besides, the size of the plane is also much larger when compared with other vehicles. In short, wheels have to compliment the size of the vehicle and its intended purpose. Automobile design uses mathematical concepts of ratio and proportion, similar to those used in architecture. These are also based on ergonomical standards (see section on Ergonomics). A r c h i t e c t u r a l M a t h 44 R E A L - L I F E M A T H Where to Learn More Books Rossi, Corinna. Architecture and Mathematics in Ancient Egypt. Cambridge University Press, 2004. Williams, Kim. Nexus III: Architecture and Mathematics. Pacini Editore, 2000. Web sites University College London, Department of Geography. “Fractals New Ways of Looking at Cities” http://www.geog.ucl.ac .uk/casa/nature.html(April 9, 2005). Yale New Haven Teachers Institute. “Some Mathematical Princi- ples of Architecture” http://www.cis.yale.edu/ynhti/ curriculum/units/1983/1/83.01.12.x.html (April 9, 2005). Key Terms Proportion: An equality between two ratios. Ratio: The ratio of a to b is a way to convey the idea of relative magnitude of two amounts. Thus if the number a is always twice the number b, we can say that the ratio of a to b is “2 to 1.” This ratio is sometimes written 2:1. Today, however, it is more common to write a ratio as a fraction, in this case 2/1. Scale: The ratio of the size of an object to the size of its representation. Symmetry: An object that is left unchanged by an oper- ation has a symmetry. R E A L - L I F E M A T H 45 Area Overview An area is a measurement of a defined surface such as a face, plane, or side. Conceptually, an object’s area can be compared quantitatively to the amount of paint needed to cover the object completely. However, in con- trast to measures of volume in pints, liters, or gallons, area measurements are expressed in units such as square feet, square meters, or square miles. Calculations of area are basic to science, engineering, business, buying and selling land, medicine, and building. Fundamental Mathematical Concepts and Terms AREA OF A RECTANGLE Every real-world object and every geometrical figure that is not a point or a line has a surface. The amount or size of that surface is the object’s or figure’s area. There are many standard formulas for calculating areas, the simplest and most commonly used being the formula for the area of a rectangle. To find the area of a rectangle, first measure the lengths of its sides. If the rectangle is W cen- timeters (cm) wide and H cm high, then its area, A, is given by A
W cm  H cm. Centimeters are used here only as an example. The units used to measure length—centimeters, inches, kilo- meters, miles, or anything else—do not change the basic formula: area equals width times height. So, for example, a typical sheet of typing paper, which is 8.5 inches wide and 11 inches high, has area A
8.5 inches  11 inches
93.5 square inches. UNITS OF AREA Area has now been explained in terms of “square inches” (or centimeters). This means that on the right- hand side of the formula A
W cm  H cm, four terms are multiplied: W, H, and cm (twice). These four terms can be reordered to give W  H  cm  cm. It is cus- tomary in mathematics to use the square notation when a term is multiplied by itself, so cm  cm is always writ- ten cm2, which is centimeters squared, or square centime- ters. Another way of writing the rectangle area formula is, therefore, A
WH cm2. Area is therefore measured in units of square centimeters—or square inches, square feet, square kilometers, square miles, or any other length measure squared. For example, a square with edges 1 foot long has an area of 1 square foot. When talking about physical materials such as cloth, land, sheet steel, plywood, or the like, it is important to A r e a 48 R E A L - L I F E M A T H causes AIDS), hepatitis B, cancer, and some other dis- eases, doctors do not use the patient’s weight but instead use the patient’s body surface area (BSA). They do so because BSA is a better guide to how quickly the kidneys will clear the drug out of the body. Doctors can measure skin area of patients directly using molds, but this is practical only for special research studies. Rather than measuring a patient’s skin area, doc- tors use formulas that give an approximate value for BSA based on the patient’s weight and height. These are simi- lar in principle to the standard geometric formulas that give the area of a sphere or cone based on its dimensions, but less exact (because people are all shaped differently). Several formulas are in use. In the West, an equation called the DuBois formula is most often used; in Japan, the Fujimoto formula is standard. The DuBois formula estimates BSA in units of square meters based on the patient’s weight in kilograms, Wt, and height in centime- ters, Ht : BSA = .007184Wt .425Ht .725 In recent years, doctors have debated whether setting drug doses according to BSA really is the best method. Some research shows that BSA is useful for calculating doses of drugs such as lamivudine, given to treat the hep- atitis B virus, which is transmitted by blood, dirty nee- dles, and unprotected sex. (Teenagers are a high-risk group for this virus.) Other research shows that drug dos- ing based on BSA does not work as well in some kinds of cancer therapy. BUY ING BY AREA Besides addition and subtraction to keep track of money, perhaps no other mathematical operation is per- formed so often by so many ordinary people as the calcu- lation of areas. This is because the price of so many common materials depends on area: carpeting, floor tile, construction materials such as sheetrock, plywood, exte- rior siding, wallpaper, and paint, whole cloth, land, and much more. In deciding how much paint it takes to paint a room, for example, a painter measures the dimensions of the walls, windows, floor, and doors. The walls (and ceil- ing or floor, if either of those is to be painted) are basically rectangles, so the area of each is calculated by multiplying its height by its width. Window and door areas are calcu- lated the same way. The amount of area that is to be painted is, then, the sum of the wall areas (plus ceiling or floor) minus the areas of the windows and doors. For each kind of paint or stain, manufacturers specify how much area each gallon will cover, the spread rate. This usually ranges from 200 to 600 square feet per gallon, depending on the product and on the smoothness of the surface being painted. (Rough surfaces have greater actual surface area, just as the lid of an egg carton has more surface area than a flat piece of cardboard of the same width and length.) Dividing the area to be painted by the spread rate gives the number of gallons of paint needed. F I LTER ING Surface area is important in chemistry and filtering because chemical reactions take place only when sub- stances can make contact with each other, and this only happens on the surfaces of objects: the outside of a mar- ble can be touched, but not the center of it (unless the marble is cut in half, in which case the center is now exposed on a new surface). Therefore a basic way to take a lump of material, like a crystal of sugar, and make it react more quickly with other chemicals is to break it into smaller pieces. The amount of material stays the same, but the surface area increases. But don’t larger cubes or spheres have more surface area than small ones? Of course they do, but a group of small objects has much more surface area than a single large object of the same total volume. Imagine a cube having sides of length L. Its area is L
6L2. If the cube is cut in half by a knife, there are now two rectangular bricks. All the outside surfaces of the original cube are still there, but now there are two additional surfaces—the ones that have appeared where the knife blade cut. Each of these surfaces is the same size as any of the cube’s orig- inal faces, so by cutting the cube in half there has added 2L2 to the total area of the material. Further cuts will increase the total surface area even more. Increasing reaction area by breaking solid material down into smaller pieces, or by filling it full of holes like a sponge, is used throughout industrial chemistry to make reactions happen faster. It is also used in filtering, especially with activated charcoal. Charcoal is solid carbon; activated charcoal is solid carbon that has been treated to fill it with billions of tiny holes, making it spongelike. When water is passed through activated charcoal, chemi- cals in the water stick to the carbon. A single teaspoonful of activated charcoal can contain about 10,000 square feet of surface area (930 square meters, the size of an Ameri- can football field). About a fourth of the expensive bot- tled water sold in stores is actually city tap water that has been passed through activated charcoal filters. CLOUD AND ICE AREA AND GLOBAL WARMING Climate change is a good example of the importance of area measurements in earth science. For almost 200 years, human beings, especially those in Europe, the A r e a R E A L - L I F E M A T H 49 United States, and other industrialized countries, have been burning massive quantities of fossil fuels such as coal, natural gas, and oil (from which gasoline is made). The carbon in these fuels combines with oxygen in the air to form carbon dioxide, which is a greenhouse gas. A greenhouse gas allows energy from the Sun get to the sur- face of the Earth, but keeps heat from escaping (like the glass panels of a greenhouse). This can melt glaciers and ice caps, thus raising sea levels and flooding low-lying lands, and can change weather patterns, possibly making fertile areas dry and causing violent weather disasters to happen more often. Scientists are constantly trying to make better predictions of how the world’s climate will change as a result of the greenhouse effect. Among other data that scientists collect to study global warming, they measure areas. In particular, they measure the areas of clouds and ice-covered areas. Clouds are important because they can either speed or slow global climate change: high, wispy clouds act as greenhouse fil- ters, warming Earth, while low, puffy clouds act to reflect sunlight back into space, cooling Earth. If global warming produces more low clouds, it may slow climate change; if it produces more high wispy clouds, it may speed climate change. Cloud areas are measured by having computers count bright areas in satellite photographs. Cloud areas help predict how fast the world will get warmer; tracking ice area helps to verify how fast the world has already been getting warmer. Most glaciers around the world have been melting much faster over the last century—but scientists need to know exactly how much faster. To find out, they first take a satellite photo of a glacier. Then they measure its outline, from which they can calculate its area. If the area is shrinking, then the gla- cier is melting; this is itself an important piece of knowl- edge. Scientists also measure the area of the glacier’s accumulation zone, which is the high-altitude part of the glacier where snow is adding to its mass. Knowing the total area of the glacier and the area of the accumulation zone, scientists can calculate the accumulation area ratio, which is the area of the glacier’s accumulation zone divided by its total area. The mass balance of a glacier— whether it is growing or shrinking—can be estimated using the accumulation area ratio and other information. CAR RAD IATORS Chemical reactions are not the only things that hap- pen at surfaces; heat is also gained or lost at an object’s surface. To cool an object faster, therefore, surface area needs to be increased. This is why elephants have big ears: they have a large volume for their body surface area, and their large, flat ears help them radiate extra heat. It is also why we hug ourselves with our arms and curl up when we are cold: we are trying to decrease our surface area. And it is how cars engines are kept cool. A car engine is sup- posed to turn the energy in fuel into mechanical motion, but about half of it is actually turned into heat. Some of this heat can be useful, as in cold weather, but most of it must simply be expelled. This is done by passing a liquid (consisting mostly of water) through channels in the engine and then pumping the hot liquid from the engine through a radiator. A radiator is full of holes, which increase its surface area. The more surface area a radiator has, the more cool air it can touch and the more quickly the metal (heated by the flowing liquid inside) can get rid of heat. When the liquid has given up heat to the outside world through the large surface area of the radiator, the liquid is cooler and is pumped back through the engine to pick up more waste heat. Car designers must size radiator surface area to engine heat output in order to produce cars that do not overheat. SURVEY ING If a parcel of land is rectangular, calculating its area is simple: length  width. But, how do surveyors find the area of an irregularly shaped piece of land—one that has crooked boundaries, or maybe even a winding river along one side? If the piece of land is very large or its boundaries very curvy, the surveyor can plot it out on a map marked with grid squares and count how many squares fit in the par- cel. If an exact area measurement is needed and the par- cel’s boundary is made up of straight line segments, which is usually the case, the surveyor can divide a draw- ing of the piece of land into rectangles, trapezoids, trian- gles. The area of each of these can be calculated separately using a standard formula, and the total area found as the Figure 2. A r e a 50 R E A L - L I F E M A T H sum of the parts. Figure 2 depicts an irregular piece of property that has been divided into four triangles and one trapezoid. Today, it is also possible to take global positioning system readings of locations around the boundary of a piece of property and have a computer estimate the inside area automatically. This is still not as accurate as an area estimate based on a true survey, because global position- ing systems are as yet only accurate to within a meter or so at best. Error in measuring the boundary leads to error in calculating the area. SOLAR PANELS Solar panels are flat electronic devices that turn part of the energy of sunlight that falls on them—anywhere from 1% or 2% to almost 40%—into electricity. Solar panels, which are getting cheaper every year, can be installed on the roofs of houses to produce electricity to run refrigerators, computers, TVs, lights, and other machines. The amount of electricity produced by a col- lection of solar panels depends on their area: the more area, the more electricity. Therefore, whether a system of solar panels can meet all the electricity demands of a household depends on three things: (1) how much elec- tricity the household uses, (2) how efficient the solar pan- els are (that is, how much of the sun energy that falls on them is turned into electricity), and (3) how much area is available on the roof of the house. The average U.S. household uses about 9,000 kWh of electricity per year. A kWh, or kilowatt-hour, is the amount of electricity used by a 100-watt light bulb burn- ing for 10 hours. That’s equal to 1,040 watts of around- the-clock use, which is the amount of electricity used by ten 100-watt bulbs burning constantly. A typical square meter of land in the United States receives from the Sun about 150 watts of power per square meter (W/m2), aver- aged around the clock, so using solar panels with an effi- ciency of 20% we could harvest about 30 watts per square meter of panel (on average, around the clock). To get 1,040 watts, therefore, we need 1,040 W / 30 W/m2
34 m2 of solar panels. At a more realistic 10% panel effi- ciency, we would need twice as much panel area, about 68 m2. This would be a square 8.2 meters on a side (27 feet). Many household rooftops in the United States could accommodate a solar system of this size, but it would be a tight fit. In Europe and Japan, where the aver- age household uses about half as much electricity as the average U.S. household, it would be easier to meet all of a household’s electricity demands using a solar panel sys- tem. Of course, it might still a good idea to meet some of a household’s electricity needs using solar panels, even where it is not practical to meet them completely that way. Where to Learn More Web sites Math.com. “Area Formulas.” 2005. http://www.math.com/ tables/geometry/areas.htm (March 9, 2005). Math.com. “Area of Polygons and Circles.” 2005. http://www.math.com/school/subject3/lessons/S3U2L4 GL.html (March 9, 2005). O’Connor, J.J., E.F. Robertson. “An Overview of Egyptian Mathematics.” December 2000. http://www-groups.dcs .st-and.ac.uk/~history/HistTopics/Egyptian_mathematics .html (March 9, 2005). O’Neill, Dennis. “Adapting to Climate Extremes.” http:// anthro.palomar.edu/adapt/adapt_2.htm (March 9, 2005). A v e r a g e R E A L - L I F E M A T H 53 this would be somewhat ridiculous. It is more reasonable to say simply that the room contains a 50-gram mouse and a 1,000,000-gram elephant and forget about averag- ing altogether in this case. If the room contains a thou- sand mice and a thousand elephants, it might be useful to talk about the mean weight of the mice and the mean weight of the elephants, but it would still probably not make sense to average the mice and the elephants together. The weights of the mice and elephants belong on different lists because mice and elephants are such dif- ferent creatures. These two lists will have different means. In general, the average or arithmetic mean of a list of numbers is meaningful only if all the numbers belong on that list. A Brief History of Discovery and Development The concept of the average or mean first appeared in ancient times in problems of estimation. When making an estimate, we seek an approximate figure for some number of objects that cannot be counted directly: the number of leaves on a tree, soldiers in an attacking army, galaxies in the universe, jellybeans in a jar. A realistic way to get such a figure—sometimes the only realistic way— is to pick a typical part of the larger whole, then count how many leaves, soldiers, galaxies, or jellybeans appear in that fragment, then multiply this figure by the number of times that the part fits into the whole. This gives an estimate for the total number. If there are 100 leaves on a typical branch, for instance, then we can estimate that on a tree with 1,000 branches there will be 100,000 leaves. By a “typical” branch, we really mean a branch with a num- ber of leaves on it equal to the average or mean number of leaves per branch. The idea of the average is therefore embedded in the idea of estimation from typical parts. The ancient king Rituparna, as described in Hindu texts at least 3,000 years old, estimated the number of leaves on a tree in just this way. This shows that an intuitive grasp of averages existed at least that long ago. By 2,500 years ago, the Greeks, too, understood esti- mation using averages. They had also discovered the idea of the arithmetic mean, possibly to help in spreading out losses when a ship full of goods sank. By 300 B.C., the Greeks had discovered not only the arithmetic mean but the geometric mean, the median, and at least nine other forms of average value. Yet they understood these aver- ages only for cases involving two numbers. For example, the philosopher Aristotle (384–322 B.C.) understood that the arithmetic mean of 2 and 10 was 6 (because 2 plus 10 divided by 2 equals 6), but could not have calculated the average height of the five students in the example used earlier. It was not until the 1500s that mathematicians realized that the arithmetic mean could be calculated for lists of three or more numbers. This important fact was discovered by astronomers who realized that they could make several measurements of a star’s position, with each individual measurement suffering from some unknown, ever-changing error, and then average the measurements to make the errors cancel out. From the late 1500s on, averaging to reduce measurement error spread to other fields of study from astronomy. By the nineteenth century averaging was being used widely in business, insurance, and finance. Today it is still used for all these purposes and more, including the calculation of grade-point aver- ages in schools. Real-life Applications BATT ING AVERAGES A batting average is a three-digit number that tells how often a baseball player has managed to hit the ball during a game, season, or career. A player’s batting aver- age is calculated by dividing the number of hits the player gets by the number of times they have been at bat (although this is not the number of times they have stepped up to the plate to hit because there are also spe- cial rules as to what constitutes a legal “at bat” to be used in calculating a player’s batting average). Say a player goes to bat 3 times and gets 0 hits the first time, 1 the second, and 0 the third (this is actually pretty good). Their batting average is then (0  1  0) / 3  .333. (A batting average is always rounded off to three decimal places.) A batting average cannot be higher than 1, because a player’s turn at bat is over once they get a hit: if a player went up three times and got three hits, their batting average would (1  1  1) / 3  1.000. But this would be superhumanly high. Not even the greatest hitters in the Baseball Hall of Fame got a hit every time they went to bat—or even half the time they went to bat. Ty Cobb, for instance, got 4,191 hits in 11,429 turns at bat for a batting average of .367, the highest career bat- ting average ever. The highest batting average for a single season, .485, was achieved by Tip O’Neill in 1887. In cricket, popular in much of the world outside the United States, a batsman’s batting average is determined by the number of runs they have scored divided by the number of times they have been out. A “bowling average” is calculated for bowlers (the cricket equivalent of pitch- ers) as the number of runs scored against the bowler divided by the number of wickets they have taken. The A v e r a g e 54 R E A L - L I F E M A T H higher a cricket player’s batting average, the better; the lower a player’s bowling average, the better. GRADES In school, averages are an everyday fact of life: an English or algebra grade for the marking period is calcu- lated as an average of all the students’ test scores. For example, if you do four assignments in the course of the marking period for a certain class and get the scores 95, 87, 82, and 91, then your grade for the marking period is In many schools that assign letter grades, all grades between 80 and 90 are considered Bs. In such a school, your grade for the marking period in this case would be a B. 95 + 87 + 82 + 91 4 = 88.75 WEIGHTED AVERAGES IN GRAD ING What if some of the assignments in a course are more important than the others? It would not be fair to count them all the same when averaging scores to calculate your grade from the marking period, would it? To make score- averaging meaningful when not all scores stand for equally important work, teachers use the weighted- average method. Calculation of a weighted average assigns a weight or multiplying factor to each grade. For example, quizzes might be assigned a weight of 1 and tests a weight of 2 to signify that they are twice as important (in this particular class). The weighted average is then cal- culated as the sum of the grades—each grade multiplied by its weight—divided by the sum of the weights. So if during a marking period you take two quizzes (grades 82 and 87) and two tests (grades 95 and 91), your grade for the marking period will be Because you did better on the tests than on the quizzes, and the tests are weighted more heavily than the quizzes, your grade is higher than if all the scores had been worth the same. In most colleges and some high schools, weighted averaging is also used to assign a single number to aca- demic performance, the famous (or perhaps infamous) grade point average, or GPA. Like individual tests, some classes require more work and must be given a heavier weight when calculating the GPA. WEIGHTED AVERAGES IN BUS INESS Weighted averages are also used in business. If in the course of a month a store sells different amounts of five kinds of cheese, some more expensive than others, the owner can use weighted averaging to calculate the average income per pound of cheese sold. Here the “weight” assigned to the sales figure for each kind of cheese is the price per pound of that cheese: more expensive cheeses are weighted more heavily. Weighted averaging is also used to calculate how expensive it is to borrow capital (money for doing business) from various lenders that all charge different interest rates: a higher interest rate means that the borrower has to pay more for each dollar bor- rowed, so money from a higher-interest-rate source costs more. When a business wants to know what an average dollar of capital costs, it calculates a weighted average of borrowing costs. This commonly calculated figure is known in business as the weighted average cost of capital. Spread- sheet software packages sold to businesses for calculating 82 + 87 + (2 × 95) + (2 × 91) 1 + 1 + 2 + 2 541 6 = = 90.2 A motorcyclist soars high during motocross freestyle practice at the 2000 X Games in San Francisco. Riders and coaches make calculations of average “hang time” and length of jumps at various speeds so that they know what tricks are safe to land. AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. A v e r a g e R E A L - L I F E M A T H 55 profit and loss routinely include a weighted-averaging option. AVERAG ING FOR ACCURACY How long does it take a rat to get sick after eating a gram of Chemical X? Exactly how bright is Star Y? Each rat and each photograph of a star is a little different from every other, so there is no final answer to either of these questions, or to any other question of measurement in science. But by performing experiments on more than one rat (or taking more than one picture of a star, or tak- ing any other measurement more than once) and averag- ing the results, scientists can get a better answer than if they look at just one measurement. This is done con- stantly in all kinds of science. In medical research, for instance, nobody performs an experiment or gathers data on just one patient. An observation is performed as many times as is practical, and the measurements are averaged to get a more accurate result. It is also standard practice to look at how much the measurements tend to spread out around the average value—the “standard deviation.” How does averaging increase accuracy? Imagine weighing a restless cat. You weigh the cat four times, but because it won’t hold still you get a scale reading each time that is a little too high or a little too low: 5.103 lb, 5.093 lb, 5.101 lb, 5.099 lb. In this case, the cat’s real weight is 5.1 lb. The error in the first reading, therefore, is .003 lb, because 5.1  .003  5.103. Likewise, the other three errors are .003, .001, and .001 lb. The average of these errors is 0: The average of the four weights is therefore the true weight of the cat: Although in real life the errors rarely cancel out to exactly zero, the average error is usually much smaller than any of the individual errors. Whenever measurement errors are equally likely to be positive and negative, averaging improves accuracy. In astronomy, this principle has been used for the star pictures taken by the International Ultraviolet Explorer satellite, which took pictures of stars from 1978 to 1996. To make final images for a standard star atlas (a collection of images of the whole sky), two or three images for each star were combined by averaging. In fact, a weighted average was calculated, with each image being 5.103 + 5.093 + 5.101 + 5.099 4 20.4 4 = = 5.1 .003 + (−.003) +.001 + (−.001) 4 0 4 = = 0 weighted by its exposure: short-exposure images were dimmer, and were given a heavier weight to compensate. The resulting star atlas is more accurate than it would have been without averaging. HOW MANY GALAX IES? As scientists discovered in the early twentieth century, the Universe does not go on forever. It is finite in size, like a very large room (only without walls, and other strange properties). There cannot, therefore, be an infinite num- ber of galaxies because there is not an infinite space. Scientists use averages to estimate such large num- bers. Galaxies, like leaves on a large tree, are hard to count. Many galaxies are so faint and far away that even the powerful Hubble Space Telescope must gaze for days a small patch of sky to see them. It would take many years to examine the whole sky this way, so instead the Hubble takes a picture of just one part of the sky—an area about as big as a dime 75 ft (22.86 m) away. Scientists assume that the number of galaxies in this small area of the sky is about the same as in any other area of the same size. That is, they assume that the number of galaxies in the observed area is equal to the average for all areas of the same size. By counting the number of galaxies in that small area and multiplying to account for the size of the whole sky, they can estimate the number of galaxies in the Universe. In 2004, the Hubble took a picture called the Ultra Deep Field, gazing for 300 straight hours at one six- millionth of the sky. The Ultra Deep Field found over 10,000 galaxies in that tiny area. If this is a fair average for any equal-sized part of the sky, then there are at least twenty billion galaxies in the universe. Most galaxies con- tain several hundred billion stars. THE “AVERAGE” FAMILY Any list of numbers has an average, but an average that has been calculated for a list of numbers that does not cluster around a central value can be meaningless or misleading. In such a case, the “distribution” of the numbers—how they are clumped or spread out on the number line—can be important. This knowledge is lost when the numbers are squashed down into a single num- ber, the average. In politics, numbers about income, taxes, spending, and debt are often named. It is sometimes necessary to talk about averages when talking about these numbers, but some averages are misleading. Sometimes politicians, financial experts, and columnists quote averages in a way that creates a false impression. A v e r a g e 58 R E A L - L I F E M A T H animals and their offspring. The most famous example of observed evolutionary changes is the research done by the biologists Peter and Rosemary Grant on the Galapagos Islands off the west coast of South America. Fourteen or 15 closely related species of finches live in the Galapagos. The Grants have been watching these finches carefully for decades, taking exact measurements of their beaks. They average these measurements together because they are interested in how each finch population as a whole is evolv- ing, rather than in how the individual birds differ from each other. The individual differences, like random measure- ment errors, tend to cancel each other out when the beak measurements are averaged. When a list of data is averaged like this, the resulting mean is called a “sample mean.” The Grants’ measurements show that the average beak for each finch species changes shape depending on what kind of food the finches can get. When mostly large, tough seeds are available, birds with large, seed-cracking beaks get more food and leave more offspring. The next generation of birds has, on average, larger, tougher beaks. This is exactly what the Darwinian theory of evolution predicts: slight, inherited differences between individual animals enable them to take advantage of changing conditions, like food supply. Those birds whose beaks just happen to be better suited to the food supply leave more offspring, and future generations become more like those successful birds. Where to Learn More Books Tanur, Judith M., et al. Statistics: A Guide to the Unknown. Belmont, CA: Wadsworth Publishing Co., 1989. Wheater, C. Philip, and Penny A. Cook. Using Statistics to Under- stand the Environment. New York: Routledge, 2000. Web sites Insurance Institute for Highway Safety. “Q7&A: Teenagers: Gen- eral.” March 9, 2004. http://www.iihs.org/safety_facts/ qanda/teens.htm#2 (February 15, 2005). Mathworld. “Arithmetic mean.” Wolfram Research. 1999. http://mathworld.wolfram.com/ArithmeticMean.html (February 15, 2005). Wikelsky, Martin. “Natural Selection and Darwin’s Finches.” Pearson Education. 2003. http://wps.prenhall.com/esm _ freeman_evol_3/0,8018,849374-,00.html (February 15, 2005). Key Terms Mean: Any measure of the central tendency of a group of numbers. Median: When arranging numbers in order of ascending size, the median is the value in the middle of the list. R E A L - L I F E M A T H 59 Base Overview In everyday life, a base is something that provides support. A house would crumble if not for the support of its base. So it is too with math. Various bases are the foun- dation of the various ways we humans have devised to count things. Counting things (enumeration) is an essen- tial part of our everyday life. Enumeration would be impossible if not for based valued numbers. Fundamental Mathematical Concepts and Terms In numbering systems, the base is the positive integer that is equal to the value of 1 in the second highest count- ing place or column. For example, in base 10, the value of a 1 in the “tens” column or place is 10. A Brief History of Discovery and Development The various base numbering systems that have arisen since before recorded history have been vital to our exis- tence and have been one of the keys that drove the for- mation of societies. Without the ability to quantify information, much of our everyday world would simply be unmanageable. Base numbering systems are indeed an important facet of real life math. The concept of the base has been part of mathemat- ics since primitive humans began counting. For example, animal bones that are about 37,000 years old have been found in Africa. That is not the remarkable thing. The remarkable thing is that the bones have human-made notches on them. Scientists argue that each notch repre- sented a night when the moon was visible. This base 1 (1, 2, 3, 4, 5, . . .) system allowed the cave dwellers to chart the moon’s appearance. So, the bones were a sort of calendar or record of the how frequent the nights were moonlit. This knowledge may have been important in determining when the best was to hunt (sneaking up on game under a full moon is less successful than when there is no moon). Another base system that is rooted in the deep past is base 5. Most of us are familiar with base 5 when we chart numbers on paper, a whiteboard or even in the dirt, by making four vertical marks and then a diagonal line across these. The base 5-tally system likely arose because of the construction of our hands. Typically, a hand has four fingers and a thumb. It is our own carry-around base 5 counting system. B a s e 60 R E A L - L I F E M A T H In base 5 tallying, the number 7 would be repre- sented as depicted in Figure 1. Of course, since typically we have two hands and a total of ten digits, we can also count in multiples of 10. So, most of us also naturally carry around with us a conven- ient base 10 (or decimal) counting system. Counting in multiples of 5 and 10 has been common for thousands of years. Examples can be found in the hieroglyphics that adorn the walls of structures built by Egyptians before the time of Christ. In their system, the powers of 10 (ones, tens, hundreds, thousands, and so on) were represented by different symbols. One thousand might be a frog, one hundred a line, ten a flower and one a circle. So, the number 5,473 would be a hieroglyphic that, from left to right, would be a pattern of five frogs, four lines, seven flowers and three circles. There are many other base systems. Base 2 or binary (which we will talk about in more detail in the next section) is at the heart of modern computer languages and applications. Numbering in terms of groups of 8 is a base-8 (octal) system. Base 8 is also very important in computer languages and programming. Others include base 12 (duodecimal), base 16 (hexidecimal), base 20 (vigesimal) and base 60 (sexagesimal). The latter system is also very old, evidence shows its presence in ancient Babylon. Whether the Babylonians created this numbering system outright, or modified it from earlier civilizations is not clear. As well, it is unclear why a base 60 system ever came about. It seems like a cumbersome system, as compared with the base 5 and 10 systems that could literally rely on the fingers and some scratches in the dirt to keep track of really big numbers. Even a base 20 system could be done manually, using both fingers and toes. Scholars have tried to unravel the mystery of base 60’s origin. Theories include a relationship between num- bers and geometry, astronomical events and the system of weights and measures that was used at the time. The real explanation is likely lost in the mists of time. Real-life Applications BASE 2 AND COMPUTERS Base 2 is a two digit numbering system. The two dig- its are 0 and 1. Each of these is used alternately as num- bers grow from ones to tens to hundreds to thousands and upwards. Put another way, the base 2 pattern looks like this: 0, 1, 10, 11, 100, 101, 110, 111, 1000, . . . (0, 1, 2, 3, 4, 5, 6, 7, 8, . . .). The roots of base 2 are thought to go back to ancient China but base 2 is as also fresh and relevant because it is perfect for the expression of information in computer lan- guages. This is because, for all their sophistication, com- puter language is pretty rudimentary. Being driven by electricity, language is either happening as electricity flows (on) or it is not (off). In the binary world of a com- puter, on is represented by 1 and off is represented by 0. As an example, consider the sequence depicted in Figure 2. Figure 1: Counting to seven in a base 5 tally system. off-off-on-off-on-on-on-off-on-on Figure 2: Information series. 0010111011 Figure 3: Information series translated to Base 2. In the base 2 world, this sequence would be written as depicted in Figure 3. View the fundamental code for a computer program and you will see line upon line of 0s and 1s. Base 2 in action! Each 0 or 1 is known as a bit of information. An arrangement of four bits is called a nibble and an arrange- ment of 8 bits is called a byte (more on this arrangement below, in the section on base 8). A base 2 numbering system can also involve digits other than 0 and 1, with the arrangement of the numbers being the important facet. In this arrangement, each number is double the preceding number. This base 2 pat- tern looks like this: 1, 2, 4, 8, 16, 32, 64, 128, 256, . . . It is also evident that in this series, from one number to the next, the numbers of the power also double. For example, compare the numbers 64 and 128. In the larger number, 12 is the double of 6 and 8 is the double of 4. B u s i n e s s M a t h R E A L - L I F E M A T H 63 Fundamental Mathematical Concepts and Terms Business math is a very broad subject, but the most fundamental areas include budgets, accounting, payroll, profits and earnings, and interest. BUDGETS All successful businesses of any size, from single indi- viduals to world-class corporations, manage everything according to a budget. A budget is a plan that considers the amount of money to be spent over a specific time schedule, what it is to be spent on, how that money is to be obtained, and what it is expected to deliver in return. Though this sounds simple, it is a very compli- cated concept. Businesses and governments rise and fall on their ability to perform reliably according to their budgets. Budgets include detailed estimates of money and all related activities in a format that enables the state of progress toward established goals and objectives to be monitored on a regular basis through various business reports. The reports provide the information necessary for management to identify opportunity and areas of concern or changing conditions so that proper adjust- ments may be made and put into action in timely fashion to improve the likelihood of success or warn of impend- ing failure to meet expectations. In a budget, all actions, events, activities, and project outcomes are quantified in terms of money. The basic components of any budget are capital investments, operating expense and revenue generation. Capital investments include building offices, plants and factories, and purchasing land or equipment and the related goods and services for new projects, including the cost of acquiring the money to invest in these projects. Expense outlays include personnel wages, personnel ben- efits, operating goods and services, advertising, rents, roy- alties, and taxes. Budgets are prepared by identifying and quantifying the cost and contributions from all ongoing projects, as well as new projects being put in place and potential new projects and opportunities expected to be begun during the planning cycle. Typically, budgets cover both the immediate year and a longer view of the next three to five years. Historical trends are derived by taking an after-look at the actual results of prior period budgets compared to their respective plan projections. Quite often the numer- ical data is converted to graphs and charts to aid in spot- ting trends and changes over time. A simple budget is represented by Figure 1. The math involved in this simplistic example budget is addition, subtraction, and multiplication, where Revenue from shoe and sandal sales
Number of pairs of sold mul- tiplied by the price received; Personnel Expense
Number of people employed each month multiplied by individual monthly wages; Federal Taxes
The applicable published tax rate multiplied times Income Before Tax. As the year progresses, a second report would be pre- pared to compare the projections above with the actual performance. If seasonal shoe sales fall below plan, then the company knows that they need to improve the prod- uct or find out why it is not selling as expected. If shoe sales are better than expected, they may need to consider building another factory to meet increasing demand or acquire additional shoes elsewhere. This somewhat boring exercise is essential to the A.Z. Neuman Shoe Factory to know if it is making or losing money and if it is a healthy company or not. This infor- mation also helps potential investors decide if the com- pany is worth investing money in to help grow, to possibly buy the company itself, or to sell if they own any part of it. As a single year look at the company, A.Z. Neuman seems to be doing fine. To really know how well the company is doing, one would have to look at similar combined reports over the past history of the company, its outstanding debts, and similar information on its competitors. ACCOUNT ING Accounting is a method of recordkeeping, commonly referred to as bookkeeping, that maintains a financial record of the business transactions and prepares various statements and reports concerning the assets, liabilities, and operating performance of a business. In the case of the A.Z. Neuman Shoe Factory, transactions include the sale of shoes and sandals, the purchase of supplies, machines, and the building of a new store as shown in the budget. Other transactions not shown in detail in the budget might include the sale of stocks and bonds or loans taken to raise the necessary money to buy the machines or build the new store if the company did not have the money on hand from prior years’ profits to do so. People who perform the work of accounting are called accountants. Their job is to collect the numbers related to every aspect of the business and put them in proper order so that management can review how the company is performing and make necessary adjustments. Accountants usually write narratives or stories that serve to explain the numbers. Computing systems help gather and sort the numbers and information, and it is very important that the accountant understand where the B u s i n e s s M a t h 64 R E A L - L I F E M A T H A. Z. Neuman Shoe Factory - Projected Annual Budget – Figures rounded to $MM (millions) Months: J F M A M J J A S O N D Total Revenue Shoe sales 3 4 4 16 14 2 2 3 18 4 3 1 74 Sandal sales 0 1 1 3 4 3 3 2 1 1 0 0 19 Total 3 5 5 19 18 5 5 5 19 5 3 1 93 Operating Expense Personnel 1 2 2 2 1 1 1 1 1 1 1 1 15 Supplies 2 2 2 2 2 2 2 2 1 1 1 1 20 Electricity 1 1 1 1 1 0 1 0 1 0 1 0 8 Local Taxes 0 0 0 1 0 0 1 0 0 0 0 1 3 Total 4 5 5 6 4 3 5 3 3 2 3 3 46 Net Contribution (Revenue – OpExp.) –1 0 0 13 14 2 0 2 16 3 0 –2 47 Capital Investments Machines 1 3 3 4 0 0 0 0 0 0 0 0 11 New Store 0 0 0 0 5 0 0 0 0 0 0 0 5 Total 1 3 3 4 5 0 0 0 0 0 0 0 16 Income Before Tax (IBT = Net Contribution – Capital) –2 –3 –3 9 9 2 0 2 16 3 0 –2 31 State & Federal Tax (Minus = credit) –1 –1 –2 3 3 1 0 0 5 1 0 –1 8 Income After Tax (IAT = IBT – S&FT) –1 –2 –1 6 6 1 0 2 11 2 0 –1 23 Figure 1: A simple budget. computing system got its information and what mathe- matical functions were performed to produce the tables, charts, and figures in order to verify that the information is true and correct. Management must understand the accounting and everything involved in it before it can fully understand how well the company is doing. When this level of understanding is not achieved for any reason, the performance of the company is not likely B u s i n e s s M a t h R E A L - L I F E M A T H 65 to be as expected. It would be like trying to ride a bicycle with blinders on: one hopes to make to the corner with- out crashing, but odds are they will not. Recent and his- torical news articles are full of stories of successful companies that achieved positive outcomes because they were aware of what they were doing and managed it well. However, there are almost as many stories of companies that did not do well because they did not understand what they were truly doing and mismanaged themselves or misrepresented their performance to investors and legal authorities. If they only mismanage themselves, companies go out of business and jobs are lost and past investments possibly wasted. If a company misrepresents itself either because it did not keep its records properly, did not do its accounting accurately, or altered the facts and calculations in any untruthful way, people can go to jail. The truth begins with honest mathematics and numbers. PAYROLL Payroll is the accounting process of paying employees for the work performed and gathering the information for budget preparation and monitoring. An employee sees how much money is received at the end of a pay period, while the employer sees how much it is spending each pay period and the two perspectives do not see the same num- ber. Why? A.Z. Neuman wants to attract quality employ- ees so it pays competitive wages and provides certain benefits. Tom Smith operates a high-tech machine that is critical to the shoe factory on a regular 40-hour-per-week schedule, has been with the company a few years, and has three dependents to care for. How much money does Tom take home and what does it cost A.Z. Neuman each month? Figure 2 lays out the details. This is just an example. Not all companies offer such benefits, and the relative split in shared cost may vary considerably if the cost is shared at all. If Tom is a mem- ber of a labor union, dues would also be withheld. As is shown in Figure 2, the company has to spend approxi- mately $2 for every $1 Tom takes home as disposable income to live on. Correspondingly, Tom will take home only about half of any raise or bonus he receives from the company. At the end of each tax year, Tom then has to file both State and Federal income tax and may discover that Troy McConnell, founder of Batanga.com at his office. The center broadcasts alternative Hispanic music on dedicated Internet channels to consumers between 12 and 33 years of age. In addition, studies at the center include all aspects of business math. AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. B u s i n e s s M a t h 68 R E A L - L I F E M A T H The advantage of a bond to the company is that owner- ship is not being shared among the buyers, the upside potential of the company remains owned by the com- pany, and the interest rate paid out is usually less than the interest rate that would have to be paid by the company on a loan. The benefit to the buyer is that bonds are not as risky as stock and, while the return is limited by the established interest rate, the initial investment is not at as great a risk of loss. Bonds are safer investments than stocks in that they tend to have guaranteed earnings, even if considerably lower than the growth potential of stock without the downside risk of loss. Companies pay the interest on loans, the interest on bonds, and any dividends to stockholders out of their earnings; thus, the rate of return as mentioned earlier is an important indicator to potential investors of all types. The assessment of business risks and opportunity can only be performed through extensive mathematical evaluation, and the individuals performing these evaluations and using them to consider investments must possess a high degree of math skills. In the end, the primary difference between evaluating a business and balancing one’s own per- sonal checkbook is the magnitude of the numbers. Where to Learn More Books Boyer, Carl B. A History of Mathematics. New York: Wiley and Sons, 1991. Bybee, L. Math Formulas for Everyday Living. Uptime Publica- tions, 2002. Devlin, Keith. Life by the Numbers. New York: Wiley and Sons, 1998. Westbrook, P. Math Smart for Business: Essentials of Managerial Finance. Princeton Review, 1997. Key Terms Balance: An amount left over, such as the portion of a credit card bill that remains unpaid and is carried over until the following billing period. Bankruptcy: A legal declaration that one’s debts are larger than one’s assets; in common language, when one is unable to pay his bills and seeks relief from the legal system. Interest: Money paid for a loan, or for the privilege of using another’s money. R E A L - L I F E M A T H 69 Calculator Math Overview A calculator is a tool that performs mathematical operations on numbers. Some of the simplest calculators can only perform addition, subtraction, multiplication, and division. More sophisticated calculators can find roots, perform exponential and logarithmic operations, and evaluate trigonometric functions in a fraction of a second. Some calculators perform all of these operations using repeated processes of addition. Basic calculators come in sizes from as small as a credit card to as large as a coffee table. Some specialized calcula- tors involve groups of computing machines that can take up an entire room. A wide variety of calculators around the world perform tasks ranging from adding up bills at retail stores to figuring out the best route when launching satel- lites into orbit. Calculators, in some form or another, have been important tools for mankind throughout history. Throughout the ages, calculators have progressed from pebbles in sand used for solving basic counting problems to modern digital calculators that come in handy when solving a homework problem or balancing a checkbook. People regularly use calculators to aid in everyday calculations. Some common types of modern digital cal- culators include basic calculators (capable of addition, subtraction, multiplication, and division), scientific cal- culators (for dealing with more advanced mathematics), and graphing calculators. Scientific calculators have more buttons than more basic calculators because they can perform many more types of tasks. Graphing calculators generally have more buttons and larger screens allowing them to display graphs of information provided by the user. In addition to providing a convenient means for working out mathematical problems, calculators also offer one of the best ways to verify work performed by hand. Fundamental Mathematical Concepts and Terms Modern calculators generally include buttons, an internal computing mechanism, and a screen. The inter- nal computing mechanism (usually a single chip made of silicon and wires, called a microprocessor, central pro- cessing unit, or CPU) provides the brains of the calcula- tor. The microprocessor takes the numbers entered using the buttons, translates them into its own language, com- putes the answer to the problem, translates the answer back into our numbering system, and displays the answer on the screen. What is even more impressive is that it usu- ally does all of this in a fraction of a second. C a l c u l a t o r M a t h 70 R E A L - L I F E M A T H The easiest way to understand the language of a cal- culator is to compare it to our numbering system, which is a base ten system. This is due to the fact that we have ten fingers and ten toes. For example, consider how humans count to 34 using fingers. You basically keep track of how many times you count to ten until you get to three, and then count four more fingers. This idea is represented in our numbering system. There is a three in the tens col- umn and a four in the ones column. The tens column represents how many times we have to go through a set of ten fingers, and the ones column represents the rest of the fingers required. A calculator counts in a similar way, but its numbering system is based on the number two instead of ten. This is known as a binary numbering system, meaning that it is based on the number two. Our ten-based numbering system is known as the decimal numbering system. Much in the same way that each column of a decimal number represents one of the ten numbers between zero and nine, a number in binary form is represented by a series of zeros and ones. Though binary numbers may seem unintuitive and confusing, they are simpler than decimal numbers in many ways, allowing complex calculations to be carried out on tiny microprocessor chips. The columns (places) in the decimal numbering sys- tem each represent multiples of ten: ones, tens, hundreds, thousands, and so forth. After the value of a column reaches nine, the next column is increased. Similarly, the columns in binary numbers represent multiples of two: ones, twos, fours, eights, and so on. Counting from zero, binary numbers go 0, 1, 10, 11, 100, 101, 110, 111, 1000, etc. 110 represents six because it has a one in the fours column, a one in the twos column, and a zero in the ones column. Because binary notation only involves two values in different columns, it is common to think of each col- umn either being on or off. If a column has a 1 in it, then the value represented by the column (1, 2, 4, 8, 16, 32, and so on) is included in the number. So a 1 can be seen to mean that the column is on, and a 0 can be seen to mean that the column is off. This is the essence of the binary numbering system that a calculator uses to perform mathematical operations. As an example, add the numbers 6 and 7 together. Using fingers to count in decimal numbers, count 6 fin- gers and then count 7 more fingers. When all ten fingers are used, make a mental tally in the tens column, and then count the last three fingers to get a single tally in the tens column and three in the ones column. This represents one ten and three ones, or 13. When you input 6 plus 7 into a calculator, the calculator firsts translates the two numbers into binary notation. In binary notation, 6 is represented by 110 (a one in the fours column, a one in the twos column, and a zero in the ones column) and 7 is represented by 111 (a one in the fours column, a one in the twos column, and a one in the ones column). Next, the two numbers are added together by adding the columns together. First, adding up the values in the ones column (0 and 1) results in a one in the ones column. Next, adding the values in the twos column results in a 2 so the twos column of the sum get a 0 and the next col- umn over, the fours column, is increased by one (just like the next column in the decimal numbering system is increased when a column goes beyond nine). Adding this to the other values in the fours column results in a 3 in the fours column (because the two numbers being added together each have a 1 in the fours column), so the eights column now has a 1 in it, and a 1 is still left in the fours column. Now listing the columns together reveals the answer in the binary form: 1101. Finally, the calculator translates this answer back into decimal form and dis- plays it on the screen: 8
4
0
1  13. As illustrated A student works on his Texas Instruments graphing calculator. American students have been using graphing calculators for over a decade, and Texas Instruments accounts for more than 80% of those sales, according to an industry research firm. Texas Instruments faces what may turn out to be a more serious challenge: software that turns handheld computers into graphing calculators. AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. C a l c u l a t o r M a t h R E A L - L I F E M A T H 73 feel of a handheld calculator, with buttons that can be clicked with the mouse. The main difference between computers and calculators is that computers are capable of handling complex logical expressions involving unknown values. This basically means that computers are capable of processing more types of information and performing a wider variety of tasks. Making the jump from calculators to computers is an important technological milestone. Just as people a thou- sand years ago could not have imagined a small battery- operated mathematical tool, it is difficult to imagine a technology that will replace electronic calculators and computers. Real-life Applications F INANC IAL TRANSACT IONS When it comes to personal finances, electronic calcu- lating devices have gone far beyond helping people bal- ance checkbooks. Cash registers and automatic teller machines (ATMs) have shaped how people trade money for products and services. Cash Registers A cash register can be thought of as a large calculator with a secured drawer that holds money. The cash register was originally invented in 1879 to pre- vent employee theft. The drawer on most cash registers can only be opened after a sales transaction has taken place so that employees can not purposely fail to record a transaction and pocket the money. Manually opening the drawer requires either a secret code or a key that is kept safe by the store manager or owner. The buttons on a cash register are different from the buttons on calculators intended for personal use. The basic buttons of a calcula- tor that are applicable to money (e.g., the numbers and the decimal point) are present on a cash register; but the remaining buttons can usually be customized to fit the needs of the organization that uses it. For example, a restaurant can program a group of buttons to store the prices of their various menu items; or cash registers in certain geographic locations might have buttons for com- puting the regional sales tax. The screen can usually be turned so that the merchant and the customer can both see the prices, taxes (if any), and total. Like many calcula- tors, a cash register has a roll of paper and a printing device used for creating printed records of calculations (called receipts in the case of monetary transactions). The inside of a cash register works (and always has worked) almost exactly like a calculator. Modern cash registers include electronic microprocessors similar to those found in handheld calculators; but when calculators were powered by the turning of mechanical gears, cash registers were also powered by similar gear mechanisms. ATM Machines Automatic teller machines (ATMs) were first used in 1960 when a few machines were placed in bank lobbies to allow customers to quickly pay bills with- out talking to a bank teller. Later in the decade, the first cash dispensing ATMs were introduced, followed by ATMs that could accept and read bank cards. The fact that ATMs are unmanned requires that they possess greater security. To ensure the safety of the bank’s money, the materials that make up the ATM and connect it to a building are precisely constructed and physically strong. To thwart attempts to pose as another person in order to take that person’s money out of an ATM, transactions require two forms of identification: physical possession of a bank card and knowledge of a personal identification number (PIN). While the inner workings of an ATM are more complicated than that of a cash register, the technology and concepts of the electronic calculator provide the basis for computing the values of every transaction. The introduction of check cards has combined the technological benefits of cash registers and ATMs to fur- ther facilitate the storage and expenditure of money. A check card can be used to make purchases using money that is stored in a checking account at a bank in another location. Other advancements in technology (e.g., scan- ners that quickly scan barcodes on items, self-checkout stations that allow customers to scan their own items, and secure Internet transactions that use calculators operat- ing on a computer thousands of miles away from the computer being used by the customer) continue to revo- lutionize how humans buy and sell products and services. However, none of these accomplishments would have been possible without tools that automatically perform the mathematical operations that take place in every monetary transaction. NAUT ICAL NAV IGAT ION For hundreds of years, sailors used celestial naviga- tion: navigating sea vessels by keeping track of the relative positions of stars in the sky. Through the ages, a wide variety of tools have been created to help a navigators navigate boats and ships from one point to another in a safe and timely manner. Different colored buoys warn of shallow waters or fishing nets, and ensure that ships do not collide when nearing docks and harbors. A compass is an essential tool for determining and maintaining directional bearings. Tables of tides and detailed nautical maps help to determine the quickest and safest route and foresee potential obstacles and dangers. For centuries, C a l c u l a t o r M a t h 74 R E A L - L I F E M A T H navigation of the seas required an in-depth understand- ing of trigonometry (relationships between lengths and angles) and intensive calculations performed by hand; and, as many navigators have discovered the hard way, small directional errors can result in devastating miscal- culations over a trip of thousands of miles. Handheld electronic calculators have proven to be an essential navigational aid since they became reasonably affordable. They are often used aboard sea vessels as either the pri- mary tool for calculating directions and distances on the water or the secondary tool for double-checking calcula- tions carried out by hand. For every type of navigational problem that can be solved with the help of a handheld electronic calculator, there is also a specialized calculator for solving the spe- cific problem. Often found either on a sea vessel or on the Internet, several calculators have been programmed to take a few pertinent values and find a specific answer. One example of a specialized nautical calculator is a speed-distance-time calculator for finding the time that it will take to get from one point to another if traveling at a certain speed. Most of these calculators require two of the three values (speed, distance, and time) in order to calculate the third value. The time that it takes to get from one point to another is the product of the distance between the two points and the speed at which the ship is traveling (time is equal to distance multiplied by speed). Similarly, to figure out how fast the ship needs to travel in order to get from one point to another in a specified amount of time requires dividing the distance by the desired time of travel (speed is equal to distance divided by time). Finally, to figure out far a ship will go if traveling at a given speed for a specified amount of time, the speed and time must be multiplied together (distance is equal to speed multiplied by time). Due to the fact that all of these operations involve only multipli- cation and division, this type of calculator only needs to be capable of multiplication and division. More sophis- ticated navigation calculators exist to quickly determine values that help a ship’s navigator make crucial decisions. These decisions range from determining the fuel neces- sary for completing a trip and planning appropriate stops for refueling, and finding the true direction in which to steer the ship in order to maintain a desired heading (direction) while taking into account forces such as wind and the current of the water. Specialized calculators are also often used to ensure that a ship is built properly. One such calculator measures a ship’s resistance to capsizing (turning upside-down in the water) based on the width of the widest part of the ship and the weight of the ship. Although modern global positioning system (GPS) technology allows precise and accurate position measure- ments, calculators (whether external or internal) are used determine vectors (directions and distance) to execute course changes or to determine the best path. COMPOUND INTEREST Banking can be a highly profitable business. For example, a bank can use the money in a savings account for other investments as long as the money is stored at the bank; so the more money present in the bank’s various accounts at any given time, the more money the bank can earn on its own investments. As an incentive for banking customers to store their money with a bank, savings accounts earn compound interest. That is, the bank pays a savings account holder a relatively small amount of money based on the amount of money in the savings account. The basic idea that drives this investment chain is that the bank makes more money in its own invest- ments than it pays out to its account holders. The amount of money that is earned on a savings account containing a given amount of money is deter- mined by a compound interest formula. Compound interest is an example of exponential growth: the larger the number becomes, the faster the number grows. The term compound refers to the idea that the growth depends both on how much money is deposited into the account as well as the amount of interest already earned in past growth periods. These growth periods are referred Beat the Abacus Contests throughout the world have pitted individu- als equipped with an abacus against individuals equipped with a handheld digital calculator. In most cases, the person with the abacus wins, no matter how complicated the mathematical operations involved. This, of course, does not mean that even the most skilled person with an abacus can make calculations faster than a calculator; the time that it takes to press the buttons accounts for most of the time that it takes to use a calculator to solve a prob- lem. Nonetheless, even in operations as compli- cated as multiplying and dividing 100 pairs of numbers with up to 12 digits (trillions), a proficient abacus user beats a skilled calculator user almost every time. C a l c u l a t o r M a t h R E A L - L I F E M A T H 75 to as compounding periods. Interest is typically com- pounded annually or monthly, but may also be com- pounded weekly, or even daily. More frequent compounding benefits the account holder and may be offered to attract more account holders in order to increase the bank’s profits. Determining the amount of interest earned and pre- dicting future account values requires calculations of inverses (1 divided by a number) and exponents (one number raised to the power of another number), both of which are usually rather messy operations, especially when performed by hand. A handheld scientific calcula- tor allows account holders to calculate these values quickly and accurately in order to compare banks and track earnings with ease. MEASUREMENT CALCULAT IONS How calculators function to solve an array of meas- urement and conversion problems is perhaps best illus- trated by example. Imagine a local high school is hosting a regional basketball tournament. On the day of the tour- nament, the athletic director discovers that the supply closet has been vandalized and all of the basketballs have been damaged. As the athletic director begins to make the announcement that the tournament will have to be delayed due to the lack of basketballs in the building, a student in the crowd reveals that she has a basketball in her backpack and throws it down to the court. Before the tournament can resume, the officials must determine whether or not the ball is regulation size. All of the writ- ing, including the size of the ball, has been worn off by years of use. Fortunately, one of the referees knows that the diameter of a full-sized basketball (the distance from one side of the ball to the other measured through the center of the ball) is about 9.4 inches. The high school home economics teacher, who happens to be in the crowd, quickly produces a tape measure from her purse, hoping to be of assistance. However, an accurate meas- urement of the diameter of the basketball cannot be determined with a tape measure. The referee measures the circumference of the ball (the longest distance around the surface of the ball) and finds that it is 29.5 inches. Not knowing the circumference of a regulation-size basket- ball, the referee asks if anyone in the crowd might know how to solve this problem. A student speaks up, stating that he has been study- ing circles and spheres in his math class. He was able to recall an important fact that would help to determine the diameter of the basketball: the circumference of a sphere (such as a basketball) is equal to the diameter of the sphere multiplied by pi. So the diameter of the basketball is 29.5 divided by pi. The student cannot remember a good approximation of the value of pi, but his scientific calculator has a button for recalling the value of pi (approximated to the ten digits that his calculator can dis- play). He enters 29.5, presses the / button (for division), recalls the value of pi (which displays 3.141592654), and presses the  button (the equal sign). The calculator dis- plays the answer as 9.390141642. This value rounds to 9.4, which is the value that the referee indicated as the diameter of a regulation basketball. The ball is accepted by the officials and the tournament continues. RANDOM NUMBER GENERATOR When conducting scientific experiments, it is often necessary to generate a random number (or a set of mul- tiple random numbers) in order to simulate real-life situ- ations. For example, a group of scientists attempting to model the way that fire spreads in a forest need to account for the fact that a burning tree may or may not ignite a nearby tree. Unpredictable factors like shifting winds and seasonal levels of moisture make incorporating the probability of fire spreading in a certain direction into models next to impossible because the nature of wildfires is seemingly random. However, this randomness can be loosely accounted for in scientific wildfire models by strategically inserting random numbers into the mathe- matical formulas that are used to describe the nature of the fire. These models are often run repeatedly in order to evaluate how well they fit real-world observations. Each time the formula is used, different random numbers are generated and inserted into the formula. Cryptography Another important area of study that ben- efits from the generation of random numbers is cryptog- raphy, in which messages are encrypted (scrambled) so that they cannot be understood if they are intercepted by an unauthorized party. A message is encrypted according to mathematical formulas. Most of these encryption for- mulas incorporate random numbers in order to create keys that must be used to decrypt (unscramble) the mes- sage. The decryption key is available only to the message sender and the intended message reader. Random number generators are important tools in many other scientific endeavors, from population model- ing to sports predictions. Fortunately, most scientific cal- culators and graphing calculators include buttons for generating random numbers. Some calculators have a single button (often labeled RAND or RND) for generat- ing a random three-digit number, between 000 and 999. Each time the button is pressed, a new random number is generated. Other calculators also allow the user to adjust the number of digits and the placement of the decimal