Theory and application of infinite series

(Parte 1 de 2)

TIGHT BINDING BOOKPage No 104 missing TIGHT BINDING BOOKPage No 104 missing

CO > OU 164019 CO > OU 164019

BLACKIE & SON LIMITED16/18 William IV Street, Charing Crosi,, LONDON V C 217 Stanhope Street, GLASGOWBLACKJE & SON (INDIA) LIM1TKD103/5 Fort Street, BOMBAYBLACKIH & SON (CANADA) LIMITEDTORONTO BLACKIE & SON LIMITED16/18 William IV Street, Charing Crosi,, LONDON V C 217 Stanhope Street, GLASGOWBLACKJE & SON (INDIA) LIM1TKD103/5 Fort Street, BOMBAYBLACKIH & SON (CANADA) LIMITEDTORONTO

BYDR. KONRAD KNOPPPROFESSOR OF MATHTCMATICb AT THEUNIVLRSITY OF TUBINGEN Translatedfrom the Second German Editionand revisedin accordance with the Fourth byMiss R. C. H. Young, Ph.D., L.esSc.

First issued ioa8Reprinted 1044, 1946Second Publish Fdition, translatedft o,the Fourth German Edition, 1951Rfpntitrd TO 54 Printed in Great Britain by Blackie & Son, Ltd., Glasgow

From the preface to the first (German) edition.There is no general agreement as to where an account of the theoryof infinite series should begin, what its main outlines should be, or whatit should include. On the one hand, the whole of higher analysis maybe regarded as a field for the application of this theory, for all limitingprocesses including differentiation and integration are based onthe investigation of infinite sequences or of infinite series. On the otherhand, in the strictest (and therefore narrowest) sense, the only mattersthat arc in place in a textbook on infinite series are their definition, themanipulation of the symbolism connected with them, and the theoryof convergence.In his "Vorlesungen uber Zahlcn- und Funktioncnlehre", Vol. 1,Part 2, A. Pringsheim has treated the subject with these limitations.There was no question of offering anything similar in the present book.My aim was quite different: namely, to give a comprehensiveaccount of all the investigations of higher analysis in which infinite seriesare the chief object of interest, the treatment to be as free from assump-tions as possible and to start at the very beginning and lead on to theextensive frontiers of present-day research. To set all this forth in asinteresting and intelligible a way as possible, but of course without inthe least abandoning exactness, with the object of providing the studentwith a convenient introduction to the subject and of giving him an ideaof its rich and fascinating variety such was my vision.The material grew in my hands, however, and resisted my effortsto put it into shape. In order to make a convenient and useful book,the field had to be restricted. But I was guided throughout by the ex-perience I have gained in teaching I have covered the whole of theground several times in the general course of my work and in lecturesat the universities of Berlin and Konigsbcrg and also by the aimof the book. It was to give a thorough and reliable treatment which wouldbe of assistance to the student attending lectures and which would at thesame time be adaptedfor private study.The latter aim was particularly dear to me, and this accounts forthe form in which I have presented the subject-matter. Since it is gener-ally easier especially for beginners to prove a deduction in puremathematics than to recognize the restrictions to which the train ofreasoning is subject, I have always dwelt on theoretical difficulties, and From the preface to the first (German) edition.There is no general agreement as to where an account of the theoryof infinite series should begin, what its main outlines should be, or whatit should include. On the one hand, the whole of higher analysis maybe regarded as a field for the application of this theory, for all limitingprocesses including differentiation and integration are based onthe investigation of infinite sequences or of infinite series. On the otherhand, in the strictest (and therefore narrowest) sense, the only mattersthat arc in place in a textbook on infinite series are their definition, themanipulation of the symbolism connected with them, and the theoryof convergence.In his "Vorlesungen uber Zahlcn- und Funktioncnlehre", Vol. 1,Part 2, A. Pringsheim has treated the subject with these limitations.There was no question of offering anything similar in the present book.My aim was quite different: namely, to give a comprehensiveaccount of all the investigations of higher analysis in which infinite seriesare the chief object of interest, the treatment to be as free from assump-tions as possible and to start at the very beginning and lead on to theextensive frontiers of present-day research. To set all this forth in asinteresting and intelligible a way as possible, but of course without inthe least abandoning exactness, with the object of providing the studentwith a convenient introduction to the subject and of giving him an ideaof its rich and fascinating variety such was my vision.The material grew in my hands, however, and resisted my effortsto put it into shape. In order to make a convenient and useful book,the field had to be restricted. But I was guided throughout by the ex-perience I have gained in teaching I have covered the whole of theground several times in the general course of my work and in lecturesat the universities of Berlin and Konigsbcrg and also by the aimof the book. It was to give a thorough and reliable treatment which wouldbe of assistance to the student attending lectures and which would at thesame time be adaptedfor private study.The latter aim was particularly dear to me, and this accounts forthe form in which I have presented the subject-matter. Since it is gener-ally easier especially for beginners to prove a deduction in puremathematics than to recognize the restrictions to which the train ofreasoning is subject, I have always dwelt on theoretical difficulties, and

Contents. PageIntroduction ., 1Part I.Real numbers and sequences.Chapter 1.Principles of the theory of real numbers.1. The system of rational numbers and its gaps 32 Sequences of rational numbers 143 Irrational numbers 234. Completeness and uniqueness of the system of real numbers . . . Jj35. Radix fractions and the Ded'-kind section 37Exercises on Chapter 1 (1 8) 42Chapter I.Sequences of real numbers.6. Arbitrary sequences and arbitrary null sequences 437. Powers, roots, and logarithms Special null sequences 498. Convergent sequences 64\$ The two main criteria ... ... . 7810 Limiting1 points and upper and lower limits 8911. Infinite series, infinite products, and infinite continued fractions . 98Exercises on Chapter I (933) . . . . 106Part I.Foundations of the theory of infinite series./ Chapter 1.* Series of positive terms.12. The first principal criterion and the two comparison tests .... 11013. The root test and the ratio test 11614 Series of positive, monotone decreasing terms 120Exercises on Chapter I (3444) 125** (CJ51) Contents. PageIntroduction ., 1Part I.Real numbers and sequences.Chapter 1.Principles of the theory of real numbers.1. The system of rational numbers and its gaps 32 Sequences of rational numbers 143 Irrational numbers 234. Completeness and uniqueness of the system of real numbers . . . Jj35. Radix fractions and the Ded'-kind section 37Exercises on Chapter 1 (1 8) 42Chapter I.Sequences of real numbers.6. Arbitrary sequences and arbitrary null sequences 437. Powers, roots, and logarithms Special null sequences 498. Convergent sequences 64\$ The two main criteria ... ... . 7810 Limiting1 points and upper and lower limits 8911. Infinite series, infinite products, and infinite continued fractions . 98Exercises on Chapter I (933) . . . . 106Part I.Foundations of the theory of infinite series./ Chapter 1.* Series of positive terms.12. The first principal criterion and the two comparison tests .... 11013. The root test and the ratio test 11614 Series of positive, monotone decreasing terms 120Exercises on Chapter I (3444) 125** (CJ51)

X Contents.Chapter IV. PageSeries of arbitrary terms.15. The second principal criterion and the algebra of convergent series 12616. Absolute convergence. Derangement of series 13617. Multiplication of infinite series 146Exercises on Chapter IV (4503) 149/ Chapter V.Power series.18. The radius of convergence 15119. Functions of a real variable . . 15820. Principal piopertics of functions leprcsented by power series . . . 17121. The algebra of power series 179Exercises on Chapter V (64 -73) 188Chapter VIThe expansions of the so-called elementary functions.2. The rational functions 18923. The exponential function 19124. The trigonometrical functions 19825. The binomial series 20826. The logarithmic series 21127. The cyclometrical functions 213Exercises on Chapter VI (74 -84) 215Chapter VII.Infinite products.28. Products with positive terms 21829. Products with arbitrary terms. Absolute convergence ... . 2130. Connection between series and products. Conditional and unconditionalconvergence . 26Exercises on Chapter Vll (8599) 28Chapter VIII.s Closed and numerical expressions for the sums of series.31. Statement of the problem -J3032. Evaluation of the sum of a series by means of a closed expression 23233. Transformation of series 24034. Numerical evaluations 24735. Applications of the transformation of series to numerical evaluations 260Exercises on Chapter VIIF (100132) 267 X Contents.Chapter IV. PageSeries of arbitrary terms.15. The second principal criterion and the algebra of convergent series 12616. Absolute convergence. Derangement of series 13617. Multiplication of infinite series 146Exercises on Chapter IV (4503) 149/ Chapter V.Power series.18. The radius of convergence 15119. Functions of a real variable . . 15820. Principal piopertics of functions leprcsented by power series . . . 17121. The algebra of power series 179Exercises on Chapter V (64 -73) 188Chapter VIThe expansions of the so-called elementary functions.2. The rational functions 18923. The exponential function 19124. The trigonometrical functions 19825. The binomial series 20826. The logarithmic series 21127. The cyclometrical functions 213Exercises on Chapter VI (74 -84) 215Chapter VII.Infinite products.28. Products with positive terms 21829. Products with arbitrary terms. Absolute convergence ... . 2130. Connection between series and products. Conditional and unconditionalconvergence . 26Exercises on Chapter Vll (8599) 28Chapter VIII.s Closed and numerical expressions for the sums of series.31. Statement of the problem -J3032. Evaluation of the sum of a series by means of a closed expression 23233. Transformation of series 24034. Numerical evaluations 24735. Applications of the transformation of series to numerical evaluations 260Exercises on Chapter VIIF (100132) 267

Contents.Part IIIDevelopment of the theory.y Chapter IX.Series of positive terms.36. Detailed study of the two comparison tests . 27437. The logarithmic scales . . 27838. Special comparison tests of the second kind 28439. Theorems of Abel, Dint, and Prinheimt and their application to afresh deduction of the logarithmic scale of comparison tests . . . 29040. Series of monotonely diminishing positive terms 2941. General remarks on the theory of the convergence and divergenceof series of positive terms 29842. Systernati/ation of the general theory of convergence 305Exercises on Chapter IX (138141) 311Chapter X.\/ Series of arbitrary terms.\ 43. Tests of convergence for series ot arbitrary terms 312I 4. Rearrangement of conditionally convergent series 318| 45. Multiplication of conditionally convergent series 320Exercises on Chapter X (142153; 324, Chapter XI.Series of variable terms (Sequences of functions).46. Uniform convergence 32647. Passage to the limit term by term 33848 Tests of uniform convergence 3449. Fourier scries 350A. Euler's formulae 350B. Dinchlet's integral 356C. Conditions of convergence 36450. Applications of the theory of Fourier series 37251. Products with variable terms 380Exercises on Chapter XI (154 173J 385Chapter XII.Series of complex terms.52. Complex numbers and sequences 38853. Series of complex terras 39654. Power series. Analytic functions 401 Contents.Part IIIDevelopment of the theory.y Chapter IX.Series of positive terms.36. Detailed study of the two comparison tests . 27437. The logarithmic scales . . 27838. Special comparison tests of the second kind 28439. Theorems of Abel, Dint, and Prinheimt and their application to afresh deduction of the logarithmic scale of comparison tests . . . 29040. Series of monotonely diminishing positive terms 2941. General remarks on the theory of the convergence and divergenceof series of positive terms 29842. Systernati/ation of the general theory of convergence 305Exercises on Chapter IX (138141) 311Chapter X.\/ Series of arbitrary terms.\ 43. Tests of convergence for series ot arbitrary terms 312I 4. Rearrangement of conditionally convergent series 318| 45. Multiplication of conditionally convergent series 320Exercises on Chapter X (142153; 324, Chapter XI.Series of variable terms (Sequences of functions).46. Uniform convergence 32647. Passage to the limit term by term 33848 Tests of uniform convergence 3449. Fourier scries 350A. Euler's formulae 350B. Dinchlet's integral 356C. Conditions of convergence 36450. Applications of the theory of Fourier series 37251. Products with variable terms 380Exercises on Chapter XI (154 173J 385Chapter XII.Series of complex terms.52. Complex numbers and sequences 38853. Series of complex terras 39654. Power series. Analytic functions 401

XII Contents. Page.5. The elementary analytic functions 410I. Rational functions 410II. The exponential function 411III. The functions cosz and sin z 414IV. The functions cot z and tan* 417V. The logarithmic scries 419VI. The inverse sine series 4*21VII. The inverse tang-cnt series 422VIII. The binomial series 42356. Series of variable terms. Uniform convergence. Weierstrass* theo-rem on double series *- 42857. Products with complex terms 43458. Special classes of series of analytic functions 441A. Dinchlet's series 441B. Faculty series 446C. Lambert's series . 448Exercises on Chapter XII (174199) 452Chapter XI11.Divergent series.59. General remarks on divergent series and the processes of limitation 45760. The C- and H- processes 47861. Application of Ct- summation to the theory of Fourier series . . . 49262. The A- process 49863. The E- process 507Exercises on Chapter XIII (200216) 516Chapter XIV.Euler's summation formula and asymptotic expansions.64. Euler's summation formula . 518A. The summation formula 518B. Applications 525C. The evaluation of renviinders 53165. Asymptotic scries 53566. Special cases of asymptotic expansions 543A. Examples of the expansion problem 543B. Examples of the summation problem 548Exercises on Chapter XIV (217-225) 53Bibliography 56Name and subject index ........ 57 XII Contents. Page.5. The elementary analytic functions 410I. Rational functions 410II. The exponential function 411III. The functions cosz and sin z 414IV. The functions cot z and tan* 417V. The logarithmic scries 419VI. The inverse sine series 4*21VII. The inverse tang-cnt series 422VIII. The binomial series 42356. Series of variable terms. Uniform convergence. Weierstrass* theo-rem on double series *- 42857. Products with complex terms 43458. Special classes of series of analytic functions 441A. Dinchlet's series 441B. Faculty series 446C. Lambert's series . 448Exercises on Chapter XII (174199) 452Chapter XI11.Divergent series.59. General remarks on divergent series and the processes of limitation 45760. The C- and H- processes 47861. Application of Ct- summation to the theory of Fourier series . . . 49262. The A- process 49863. The E- process 507Exercises on Chapter XIII (200216) 516Chapter XIV.Euler's summation formula and asymptotic expansions.64. Euler's summation formula . 518A. The summation formula 518B. Applications 525C. The evaluation of renviinders 53165. Asymptotic scries 53566. Special cases of asymptotic expansions 543A. Examples of the expansion problem 543B. Examples of the summation problem 548Exercises on Chapter XIV (217-225) 53Bibliography 56Name and subject index ........ 57

Introduction.The foundation on which the structure of higher analysis rests isthe theory of real numbers. Any strict treatment of the foundations ofthe differential and integral calculus and of related subjects must in-evitably start from there; and the same is true even for e. g. the cal-culation of roots and logarithms. The theory of real numbers first createsthe material on which Arithmetic and Analysis can subsequently build,and with which they deal almost exclusively.The necessity for this has not always been realized. The greatcreators of the infinitesimal calculus Leibniz and Newton l andthe no less famous men who developed it, of whom Eider 2 is the chief,were too intoxicated by the mighty stream of learning springing fromthe newly-discovered sources to feel obliged to criticize fundamentals.To them the results of the new methods were sufficient evidence forthe security of their foundations. It was only when the stream beganto ebb that critical analysis ventured to examine the fundamental con-ceptions. About the end of the 18th century such efforts became strongerand stronger, chiefly owing to the powerful influence of Gauss 3. Nearlya century had to pass, however, before the most essential matters couldbe considered thoroughly cleared up.Nowadays rigour in connection with the underlying number conceptis the most important requirement in the treatment of any mathematicalsubject. Ever since the later decades of the past century the last wordon the matter has been uttered, so to speak, by Weierstrass 4 in thesixties, and by Cantor 5 and Dedekind 6 in 1872. No lecture or treatise1 Gottfried Wilhelm Leibniz, born in Leipzig in 1646, died in Hanover in1716. Isaac Neivton, born at Woolsthorpe in 1642, died in London in 1727. Eachdiscovered the foundations of the infinitesimal calculus independently of the other.2 Leonhard Eider, born in Basle in 1707, died in St. Petersburg in 1783.3 Karl Friedrich Gauss, born at Brunswick in 1777, died at Gottingen in 1853.4 Karl Weierstrass, born at Ostenfelde in 1815, died in Berlin in 1897. Thefirst rigorous account of the theory of real numbers which Weierstrass had expoundedin his lectures since 1860 was given by G. Mittag-Leffler, one of his pupils, in hisessay: Die Zahl, Einleitung zur Theone der analytischen Funktionen, The TohokuMathematical Journal, Vol. 17, p. 157209. 1920.5 Georg Cantor, born in St. Petersburg in 1845, died at Halle in 1918: cf.Mathem. Annalen, Vol. 5, p. 123. 1872.6 Richard Dedekind, born at Brunswick in 1831, died there in 1916: cf. hisbook: Stetigkeit und irrationaJe Zahlen, Brunsuick 1872.1 Introduction.The foundation on which the structure of higher analysis rests isthe theory of real numbers. Any strict treatment of the foundations ofthe differential and integral calculus and of related subjects must in-evitably start from there; and the same is true even for e. g. the cal-culation of roots and logarithms. The theory of real numbers first createsthe material on which Arithmetic and Analysis can subsequently build,and with which they deal almost exclusively.The necessity for this has not always been realized. The greatcreators of the infinitesimal calculus Leibniz and Newton l andthe no less famous men who developed it, of whom Eider 2 is the chief,were too intoxicated by the mighty stream of learning springing fromthe newly-discovered sources to feel obliged to criticize fundamentals.To them the results of the new methods were sufficient evidence forthe security of their foundations. It was only when the stream beganto ebb that critical analysis ventured to examine the fundamental con-ceptions. About the end of the 18th century such efforts became strongerand stronger, chiefly owing to the powerful influence of Gauss 3. Nearlya century had to pass, however, before the most essential matters couldbe considered thoroughly cleared up.Nowadays rigour in connection with the underlying number conceptis the most important requirement in the treatment of any mathematicalsubject. Ever since the later decades of the past century the last wordon the matter has been uttered, so to speak, by Weierstrass 4 in thesixties, and by Cantor 5 and Dedekind 6 in 1872. No lecture or treatise1 Gottfried Wilhelm Leibniz, born in Leipzig in 1646, died in Hanover in1716. Isaac Neivton, born at Woolsthorpe in 1642, died in London in 1727. Eachdiscovered the foundations of the infinitesimal calculus independently of the other.2 Leonhard Eider, born in Basle in 1707, died in St. Petersburg in 1783.3 Karl Friedrich Gauss, born at Brunswick in 1777, died at Gottingen in 1853.4 Karl Weierstrass, born at Ostenfelde in 1815, died in Berlin in 1897. Thefirst rigorous account of the theory of real numbers which Weierstrass had expoundedin his lectures since 1860 was given by G. Mittag-Leffler, one of his pupils, in hisessay: Die Zahl, Einleitung zur Theone der analytischen Funktionen, The TohokuMathematical Journal, Vol. 17, p. 157209. 1920.5 Georg Cantor, born in St. Petersburg in 1845, died at Halle in 1918: cf.Mathem. Annalen, Vol. 5, p. 123. 1872.6 Richard Dedekind, born at Brunswick in 1831, died there in 1916: cf. hisbook: Stetigkeit und irrationaJe Zahlen, Brunsuick 1872.1

2 Introduction.dealing with the fundamental parts of higher analysis can claim validityunless it takes the refined concept of the real number as its starting-point.Hence the theory of real numbers has been stated so often andin so many different ways since that time that it might seem superfluousto give another very detailed exposition 7 : for in this book (at least inthe later chapters) we wish to address ourselves only to those alreadyacquainted with the elements of the differential and integral calculus.Yet it would scarcely suffice merely to point to accounts given elsewhere.For a theory of infinite series, as will be sufficiently clear from laterdevelopments, would be up in the clouds throughout, if it were notfirmly based on the system of real numbers, the only possible foundation.On account of this, and in order to leave not the slightest uncertaintyas to the hypotheses on which we shill build, we shall discuss in thefollowing pages those idsas and data from the theory of real numberswhich we shall need further on. We have no intention, however, of con-structing a statement of the theory compressed into smaller space butotherwise complete. We merely wish to make the main ideas, the mostimportant questions, and the answers to them, as clear and prominentas possible. So far as the latter are concerned, our treatment throughoutwill certainly be detailed and without omissions; it is only in the casesof details of subsidiary importance, and of questions as to the complete-ness and uniqueness of the system of real numbers which lie outside theplan of this book, that we shall content ourselves with shorter indications.7 An account which is easy to follow and which includes all the essentialsis given by H. v. Mangoldt, Einfuhrung in die hohere Mathematik, Vol. I, 8th edition(by K. Knop), Leipzig 1944. The treatment of G. Kozvalezvski, Grundziigeder Differential- und Integralrechnung, 6th edition, Leipzig 1929, is accurate andconcise. A rigorous construction of the system of real numbers, which goes intothe minutest details, is to be found in A. Loezvy, Lehrbuch der Algebra, Part I,Leipzig 1915, in A. Pnngsheim, Vorlesungen uber Zahlen- und Funktionenlehre,Vol. I, Part I, 2n(1 edition, Leipzig 1923 (cf. also the review of the latter work byH. Hahn, Gott. gel. Anzeigen 1919, p. 321 47), and in a book by E. Landauexclusively devoted to this purpose, Grundlagen der Analysis (Das Rechnen mitganzen, rationalen, irrationalen, komplexen Zahlen), Leipzig 1930. A critical accountof the whole problem is to be found in the article by F. Bachmann, Aufbau desZahlensystems, in the Enzyklopadie d. math. Wissensch., Vol. I, 2nii edition, Part I,article 3, Leipzig and Berlin 1938. 2 Introduction.dealing with the fundamental parts of higher analysis can claim validityunless it takes the refined concept of the real number as its starting-point.Hence the theory of real numbers has been stated so often andin so many different ways since that time that it might seem superfluousto give another very detailed exposition 7 : for in this book (at least inthe later chapters) we wish to address ourselves only to those alreadyacquainted with the elements of the differential and integral calculus.Yet it would scarcely suffice merely to point to accounts given elsewhere.For a theory of infinite series, as will be sufficiently clear from laterdevelopments, would be up in the clouds throughout, if it were notfirmly based on the system of real numbers, the only possible foundation.On account of this, and in order to leave not the slightest uncertaintyas to the hypotheses on which we shill build, we shall discuss in thefollowing pages those idsas and data from the theory of real numberswhich we shall need further on. We have no intention, however, of con-structing a statement of the theory compressed into smaller space butotherwise complete. We merely wish to make the main ideas, the mostimportant questions, and the answers to them, as clear and prominentas possible. So far as the latter are concerned, our treatment throughoutwill certainly be detailed and without omissions; it is only in the casesof details of subsidiary importance, and of questions as to the complete-ness and uniqueness of the system of real numbers which lie outside theplan of this book, that we shall content ourselves with shorter indications.7 An account which is easy to follow and which includes all the essentialsis given by H. v. Mangoldt, Einfuhrung in die hohere Mathematik, Vol. I, 8th edition(by K. Knop), Leipzig 1944. The treatment of G. Kozvalezvski, Grundziigeder Differential- und Integralrechnung, 6th edition, Leipzig 1929, is accurate andconcise. A rigorous construction of the system of real numbers, which goes intothe minutest details, is to be found in A. Loezvy, Lehrbuch der Algebra, Part I,Leipzig 1915, in A. Pnngsheim, Vorlesungen uber Zahlen- und Funktionenlehre,Vol. I, Part I, 2n(1 edition, Leipzig 1923 (cf. also the review of the latter work byH. Hahn, Gott. gel. Anzeigen 1919, p. 321 47), and in a book by E. Landauexclusively devoted to this purpose, Grundlagen der Analysis (Das Rechnen mitganzen, rationalen, irrationalen, komplexen Zahlen), Leipzig 1930. A critical accountof the whole problem is to be found in the article by F. Bachmann, Aufbau desZahlensystems, in the Enzyklopadie d. math. Wissensch., Vol. I, 2nii edition, Part I,article 3, Leipzig and Berlin 1938.

Part I.Real numbers and sequences.Chapter I.Principles of the theory of real numbers.1. The system of rational numbers and its gaps.What do we mean by saying that a particular number is "known"or "given" or may be "calculated"? What does one mean by sayingthat he knows the value of 1/2 or n>> or lnat ne can calculate 1/5?A question like this is easier to ask than to answer. Were I to saythat \/2 = l-414, I should obviously be wrong, since, on multi-plying out, 1-414 X 1-414 does not give 2. If I assert, with greatercaution, that 1/2 = 1-4 142 135 and so on, even that is no tenableanswer, and indeed in the first instance it is entirely meaningless. Thequestion is, after all, how we are to go on, and this, without furtherindication, we cannot tell. Nor is the position improved by carryingthe decimal further, even to hundreds of places. In this sense itmay well be said that no one has ever beheld the whole of "V/2,not held it completely in his own hands, so to speak whilst astatement that 1/9 = 3 or that 35-7-7 = 5 has a finished and thorough-ly satisfactory appearance. The position is no better as regardsthe number n, or a logarithm or sine or cosine from the tables.Yet we feel certain that 1/2 and n and log 5 really do have quite definitevalues, and even that we actually know these values. But a clearnotion of what these impressions exactly amount to or imply we donot as yet possess. Let us endeavour to form such an idea.Having raised doubts as to the justification for such statementsas "I know 1/2", we must, to be consistent, proceed to examinehow far one is justified even in asserting that he knows the number or is given (for some specific calculation) the number ~. Naymore, the significance of such statements as "I know the number 97"or "for such and such a calculation I am given a = 2 and 6 = 5" would Part I.Real numbers and sequences.Chapter I.Principles of the theory of real numbers.1. The system of rational numbers and its gaps.What do we mean by saying that a particular number is "known"or "given" or may be "calculated"? What does one mean by sayingthat he knows the value of 1/2 or n>> or lnat ne can calculate 1/5?A question like this is easier to ask than to answer. Were I to saythat \/2 = l-414, I should obviously be wrong, since, on multi-plying out, 1-414 X 1-414 does not give 2. If I assert, with greatercaution, that 1/2 = 1-4 142 135 and so on, even that is no tenableanswer, and indeed in the first instance it is entirely meaningless. Thequestion is, after all, how we are to go on, and this, without furtherindication, we cannot tell. Nor is the position improved by carryingthe decimal further, even to hundreds of places. In this sense itmay well be said that no one has ever beheld the whole of "V/2,not held it completely in his own hands, so to speak whilst astatement that 1/9 = 3 or that 35-7-7 = 5 has a finished and thorough-ly satisfactory appearance. The position is no better as regardsthe number n, or a logarithm or sine or cosine from the tables.Yet we feel certain that 1/2 and n and log 5 really do have quite definitevalues, and even that we actually know these values. But a clearnotion of what these impressions exactly amount to or imply we donot as yet possess. Let us endeavour to form such an idea.Having raised doubts as to the justification for such statementsas "I know 1/2", we must, to be consistent, proceed to examinehow far one is justified even in asserting that he knows the number or is given (for some specific calculation) the number ~. Naymore, the significance of such statements as "I know the number 97"or "for such and such a calculation I am given a = 2 and 6 = 5" would

1. The system of rational numbers and its gaps. 51. Rational numbers form an ordered aggregate; meaning thatbetween any two, say a and 6, one and only one of the three relationsa < b. a = b, a > bnecessarily holds 2; and these relations of "order" between rationalnumbers are subject to a set of quite simple laws, which we assume known,the only essential ones for our purposes being theFundamental Laws of Order.1. Invariably 3 a a.2. a b always implies b - a.3. a = by b c implies a = c.4. a by b < c, or a < b, b < cy implies 4 a < c.2. Any two rational numbers may be combined in four distinctways, referred to respectively as the four processes (or basic operations)of Addition, Subtraction, Multiplication, and Division. These operationscan always be carried out to one definite result, with the single exceptionof division by 0, which is undefined and should be regarded as an entirelyimpossible or meaningless process; the four processes also obey a numberof simple laws, the so-called Fundamental Laws of Arithmetic, and furtherrules cleducible therefrom.These too we shall regard as known, and state, concisely, thoseFundamental Laws or Axioms of Arithmetic from which all the others maybe inferred, by purely formal rules (i. e. by the laws of pure logic).I. Addition. 1. Every pair of numbers a and b has invariably associ-ated with it a third, c, called their sum and denoted by a + b.2. a = a', b b' always implv a \ b -- a' + b'.3. Invariably, a + b b + (Commutative Law).4. Invariably, (a + b) + c = a + (b + c) (Associative Law).5. a < b always implies a + c < b + c (Law of Monotony).I. Subtraction.To every pair of numbers a and b there corresponds a third numberc, such that a + c b.8 a > b and b < a are merely two different expressions of the same relation.Strictly speaking, the one symbol "<" would therefore suffice.3 With regard to this seemingly trivial "law" cf. footnote 1, p. 9, remark 1 , p. 28,and footnote 24, p. 29.4 To express that one of the relations of order: a < b, a 6, or a > b, doesnot hold, we write, respectively, ab ("greater than or equal to", "at least equalto", "not less than"), a -t= b ("unequal to", "different from") or a *- 6. Kach ofthese statements (negations) definitely excludes one of the three relations and leavesundecided which of the other two holds good. 1. The system of rational numbers and its gaps. 51. Rational numbers form an ordered aggregate; meaning thatbetween any two, say a and 6, one and only one of the three relationsa < b. a = b, a > bnecessarily holds 2; and these relations of "order" between rationalnumbers are subject to a set of quite simple laws, which we assume known,the only essential ones for our purposes being theFundamental Laws of Order.1. Invariably 3 a a.2. a b always implies b - a.3. a = by b c implies a = c.4. a by b < c, or a < b, b < cy implies 4 a < c.2. Any two rational numbers may be combined in four distinctways, referred to respectively as the four processes (or basic operations)of Addition, Subtraction, Multiplication, and Division. These operationscan always be carried out to one definite result, with the single exceptionof division by 0, which is undefined and should be regarded as an entirelyimpossible or meaningless process; the four processes also obey a numberof simple laws, the so-called Fundamental Laws of Arithmetic, and furtherrules cleducible therefrom.These too we shall regard as known, and state, concisely, thoseFundamental Laws or Axioms of Arithmetic from which all the others maybe inferred, by purely formal rules (i. e. by the laws of pure logic).I. Addition. 1. Every pair of numbers a and b has invariably associ-ated with it a third, c, called their sum and denoted by a + b.2. a = a', b b' always implv a \ b -- a' + b'.3. Invariably, a + b b + (Commutative Law).4. Invariably, (a + b) + c = a + (b + c) (Associative Law).5. a < b always implies a + c < b + c (Law of Monotony).I. Subtraction.To every pair of numbers a and b there corresponds a third numberc, such that a + c b.8 a > b and b < a are merely two different expressions of the same relation.Strictly speaking, the one symbol "<" would therefore suffice.3 With regard to this seemingly trivial "law" cf. footnote 1, p. 9, remark 1 , p. 28,and footnote 24, p. 29.4 To express that one of the relations of order: a < b, a 6, or a > b, doesnot hold, we write, respectively, ab ("greater than or equal to", "at least equalto", "not less than"), a -t= b ("unequal to", "different from") or a *- 6. Kach ofthese statements (negations) definitely excludes one of the three relations and leavesundecided which of the other two holds good.

6 Chapter I. Principles of the theory of real numbers.I. Multiplication.1. To every pair of numbers a and b there corresponds a thirdnumber c, called their product and denoted by a b.2. a a', b b' always implies a b = a' b'.3. In all cases ab = ba (Commutative Law).4. In all cases (ab) c =-- a (b c) (Associative Law).5. In all cases (a + b) c a c + b c (Distributive Law).6. a < b implies, provided c is positive, a c <.b c (Law of Mono-tony).IV. Division.To every pair of numbers a and b of which the first is not therecorresponds a third number c, such that a c = b.As already remarked, all the known rules of arithmetic, andhence ultimately all mathematical results, are deduced from thesefew laws, with the help of the laws of pure logic alone. Among theselaws, one is distinguished by its primarily mathematical character, namelythe V. Law of Induction, which may be reckoned among the fundamentallaws of arithmetic and is normally stated as follows:If a set S3)t of natural numbers includes the number 1, and if, everytime a certain natural number n and all those less than n can be taken tobelong to the aggregate, the number (n h 1) rniy be inferred also to belongto it, then \$)J includes all the natural numbers.This law of induction itself follows quite easily from the followingtheorem, which appears even more obvious and is therefore normallycalled the fundamental law of the natural numbers :Law of the Natural Numbers. In every set of natural numbers thatis not "empty" there is always a number less than all the rest.For if, according to the hypotheses of the Induction Law, we con-sider the set 9i of natural numbers not belonging to \$)?, this set W mustbe "empty", that is, \$ft must contain all the natural numbers. For other-wise, by the law of the natural numbers, 1U would include a number lessthan all the rest. This least number would exceed 1, for it was assumedthat 1 belongs to sl)i; hence it could be denoted by n + 1. Then n wouldbelong to 3)i, but (n + 1) would not, which contradicts the hypothesesin the law of induction.5In applications it is usually an advantage to be able to make state-ments not merely about the natural numbers but about any whole numbers.6 The following rather more general form of the law of induction can bededuced in exactly the same way from the fundamental law of the natural numbers.If set >j.)j of natural numbers includes the number 1, and if the number (n -|- 1)can be proved to belong to the aggregate provided the number n does, then Wl con-tains all the natural numbers. 6 Chapter I. Principles of the theory of real numbers.I. Multiplication.1. To every pair of numbers a and b there corresponds a thirdnumber c, called their product and denoted by a b.2. a a', b b' always implies a b = a' b'.3. In all cases ab = ba (Commutative Law).4. In all cases (ab) c =-- a (b c) (Associative Law).5. In all cases (a + b) c a c + b c (Distributive Law).6. a < b implies, provided c is positive, a c <.b c (Law of Mono-tony).IV. Division.To every pair of numbers a and b of which the first is not therecorresponds a third number c, such that a c = b.As already remarked, all the known rules of arithmetic, andhence ultimately all mathematical results, are deduced from thesefew laws, with the help of the laws of pure logic alone. Among theselaws, one is distinguished by its primarily mathematical character, namelythe V. Law of Induction, which may be reckoned among the fundamentallaws of arithmetic and is normally stated as follows:If a set S3)t of natural numbers includes the number 1, and if, everytime a certain natural number n and all those less than n can be taken tobelong to the aggregate, the number (n h 1) rniy be inferred also to belongto it, then \$)J includes all the natural numbers.This law of induction itself follows quite easily from the followingtheorem, which appears even more obvious and is therefore normallycalled the fundamental law of the natural numbers :Law of the Natural Numbers. In every set of natural numbers thatis not "empty" there is always a number less than all the rest.For if, according to the hypotheses of the Induction Law, we con-sider the set 9i of natural numbers not belonging to \$)?, this set W mustbe "empty", that is, \$ft must contain all the natural numbers. For other-wise, by the law of the natural numbers, 1U would include a number lessthan all the rest. This least number would exceed 1, for it was assumedthat 1 belongs to sl)i; hence it could be denoted by n + 1. Then n wouldbelong to 3)i, but (n + 1) would not, which contradicts the hypothesesin the law of induction.5In applications it is usually an advantage to be able to make state-ments not merely about the natural numbers but about any whole numbers.6 The following rather more general form of the law of induction can bededuced in exactly the same way from the fundamental law of the natural numbers.If set >j.)j of natural numbers includes the number 1, and if the number (n -|- 1)can be proved to belong to the aggregate provided the number n does, then Wl con-tains all the natural numbers.

1. The system of rational numbers and its gaps. 7The laws then take the following forms, obviously equivalent to thoseabove :Law of Induction. If a statement involves a natural number n (e. g."if n 10, then 2W > n*", or the like) and ifa) this statement is correct for n = ptand b) its correctness for n = p, p -{- I, . . . , k (where k is any naturalnumber >; p) always implies its correctness for n = k -f- 1, then thestatement is correct for every natural number p.Law of Integers. In every set of integers all r p that is not "empty",there is always a number less than all the rest.6We will lastly mention a theorem susceptible, in the domain ofrational numbers, of immediate proof, although it becomes axiomaticin character very soon after this domain is left; namely theVI. Theorem of Eudoxus.If a and b are any two positive rational numbers, then a naturalnumber n always exists 7 such that n b > a.The four ways of combining two rational numbers give in everycase as the result another rational number. In this sense the systemof rational numbers forms a closed aggregate (naturlicher Rationalitats-bereich or number corpus). This property of forming a closed system \\ithrespect to the four rules is obviously not possessed by the aggregate ofall natural numbers, or of all positive and negative integers. These are,so to speak, too sparsely sown to meet all the demands which the fourrules make upon them.This closed aggregate of all rational numbers and the laws which holdin it, are then all that we regard as given, known, secured.As that type of argument which makes use of inequalities and absolute values 3.may be a little unfamiliar to some, its most important rules may be set down here,briefly and without proof:I. Inequalities. Here all follows from the laws of order and monotony.In particular1. The statements in the laws of monotony are reversible; e. g. a -f- c< b -|- c always implies a < 6; and so does a c < b c, provided c > 0.2. a < b, c < d always implies a -f c < b -f d.3. a < b, c < d implies, provided b and c are positive, a c < b d.4. a < b a!ways implies b < a, . . . 11and also, provided a is positive, , < -.b aTo reduce these forms of the laws to the previous ones, we need only con-sider the natural numbers m such that, in the one case, the statement in questionis correct for n (p 1) -f m, or, in the other, that (p 1) -f m belongs to thenon-"empty" set under consideration.7 This theorem is usually, but incorrectly, ascribed to Archimedes ; it is alreadyto be found in Euclid, Elements, Book V, Def. 4.

8 Chapter I. Principles of the theory of real numbers.Also these theorems, as well as the laws of order and monotony, hold (withappropriate modifications) when the signs "S", "-*", "_-" and <c= l> are sub-stituted for "<", provided we maintain the assumptions that c> b and a are posi-tive, in 1, 3, and 4 respectively.I. Absolute values. Definition: By \ a |, the absolute value (or modulus)of a, is meant that one of the two numbers -\-a and a which is positive, sup-posing a 3= 0; and the number 0, if a 0. (Hence | | - and if a = 0, | a \ > 0.)The following theorems hold, amongst others:3. a\ --- \ - a\. 2. | ab \ =-1a a , provided a =f= 0.J 4. \ a + b [ :_j, \ a\ + \ b \; |a + 6||a|- |6|, and indeed | a + b \ \a\ -|6|(.5. The two relations | a \ < r and r < a < r are exactly equivalent;similarly for | x a \ < r and a r <. x < a -\- r.0. | a b | is the distance between the points a and b, with the represen-tation of numbers on a straight line described immediately below.Proof of the first relation in 4: a \ a |, b < | b |, so that by 3, I, 2,(a -\ b) | a | -f- | 6 |, and hence | a -\ b \ | a \ -}- | b |.We also assume it to be known how the relations of magnitudebetween rational numbers may be illustrated graphically by relationsof positions between points on a straight line. On a straight line ornumber-axis, any two distinct points arc marked, one O, the origin (0)and one U9 the unit point (1). The point P which is to represent a numbera = *- (q > 0, p 0, both integers) is obtained by marking off on theaxis, | p | times in succession, beginning at O, the <?th part of the dis-tance O U (immediately constructed by elementary geometry) either inthe direction O U, if p > 0, or if p is negative, in the opposite direction.This point 8 we call for brevity the point a, and the totality of pointscorresponding in this way to all rational numbers we shall referto as the rational points of the axis. The straight line is usuallythought of as drawn from left to right and U chosen to the right of O.In this case, the words positive and negative obviously become equiva-lents of the phrases: to the right of O and to the left of O, respectively;and, more generally, a < b signifies that a lies to the left of b, b to theright of a. This mode of expression may often assist us in illustratingabstract relations between numbers.8 The position of this point is independent of the particular representationof the number at i. e. if a p'/q' is another representation with </' *> and p' both integers, and if the construction is performed with q', p' in place of qt p, thesame point P is obtained. 8 Chapter I. Principles of the theory of real numbers.Also these theorems, as well as the laws of order and monotony, hold (withappropriate modifications) when the signs "S", "-*", "_-" and <c= l> are sub-stituted for "<", provided we maintain the assumptions that c> b and a are posi-tive, in 1, 3, and 4 respectively.I. Absolute values. Definition: By \ a |, the absolute value (or modulus)of a, is meant that one of the two numbers -\-a and a which is positive, sup-posing a 3= 0; and the number 0, if a 0. (Hence | | - and if a = 0, | a \ > 0.)The following theorems hold, amongst others:3. a\ --- \ - a\. 2. | ab \ =-1a a , provided a =f= 0.J 4. \ a + b [ :_j, \ a\ + \ b \; |a + 6||a|- |6|, and indeed | a + b \ \a\ -|6|(.5. The two relations | a \ < r and r < a < r are exactly equivalent;similarly for | x a \ < r and a r <. x < a -\- r.0. | a b | is the distance between the points a and b, with the represen-tation of numbers on a straight line described immediately below.Proof of the first relation in 4: a \ a |, b < | b |, so that by 3, I, 2,(a -\ b) | a | -f- | 6 |, and hence | a -\ b \ | a \ -}- | b |.We also assume it to be known how the relations of magnitudebetween rational numbers may be illustrated graphically by relationsof positions between points on a straight line. On a straight line ornumber-axis, any two distinct points arc marked, one O, the origin (0)and one U9 the unit point (1). The point P which is to represent a numbera = *- (q > 0, p 0, both integers) is obtained by marking off on theaxis, | p | times in succession, beginning at O, the <?th part of the dis-tance O U (immediately constructed by elementary geometry) either inthe direction O U, if p > 0, or if p is negative, in the opposite direction.This point 8 we call for brevity the point a, and the totality of pointscorresponding in this way to all rational numbers we shall referto as the rational points of the axis. The straight line is usuallythought of as drawn from left to right and U chosen to the right of O.In this case, the words positive and negative obviously become equiva-lents of the phrases: to the right of O and to the left of O, respectively;and, more generally, a < b signifies that a lies to the left of b, b to theright of a. This mode of expression may often assist us in illustratingabstract relations between numbers.8 The position of this point is independent of the particular representationof the number at i. e. if a p'/q' is another representation with </' *> and p' both integers, and if the construction is performed with q', p' in place of qt p, thesame point P is obtained.

1. The system of rational numbers and its gaps. 9This completes the sketch of what we propose to take as thepreviously secured foundation of our subject. We shall now regardthe description of these foundations as characterizing the concept ofnumber; in other words, we shall call any system of conceptually well-distinguished objects (elements, symbols) a number system, and itselements numbers, if to put it quite briefly for the moment wecan operate with them in essentially the same ways as we do with rationalnumbers.We proceed to give this somewhat inaccurate statement a preciseformulation.We consider a system S of well-distinguished objects, which wedenote by a, /?,.... S will be called a number system and its elementsa, j3, . . . will be called numbers if, besides being capable of definitionexclusively by means of rational numbers (i. c. ultimately by means ofnatural numbers alone) 9, these symbols a, jS, . . . satisfy the following fourconditions :1. Between any two elements a and /3 of S one and only one of thethree relations 10 a < 0, a = a >necessarily holds (this is expressed briefly by saying that S is an orderedsystem) and these relations of order between the elements of S are subjectto the same fundamental laws 1 as their analogues in the system of rationalnumbers u.2. Four distinct methods of combining any two elements of S aredefined, called Addition, Subtraction, Multiplication and Division. Witha single exception, to be mentioned immediately (3.), these processescan always be carried out to one definite result, and obey the same Fun-damental Laws 2, I IV, as their analogues in the system of the rational9 We shall come across actual examples m 3 and 5; for the moment, wen.ay think of decimal fractions, or similar symbols constructed from rational numbers.See also footnote 10, p. 12.10 Cf. also footnotes 2 and 4.1 As to what we may call the practical meaning of these relations, nothingIs implied; "<" may as usual stand for "less than'*, but it may equally well mean"before", "to the left of", "higher than", "lower than", "subsequent to", in factmay express any relation of order (including "greater than"). This meaning merelyhas to be defined without ambiguity and kept consistent. Similarly, "equality"need not imply identity. Thus, for example, within the system of symbols of theform p/q, where/), q are integers and q =4= 0, the symbols 3/4, 0/8, I)/ 12 aregenerally said to be "equal"; that is, for certain purposes (calculating, measuring,and so on) we define equality within our system of symbols in such a way that 3/4 -=6/8-= -9/-12, although 3/4, 0/8, -9/-12 are in the first instance differentelements of that system (see also 14, note 1). 1. The system of rational numbers and its gaps. 9This completes the sketch of what we propose to take as thepreviously secured foundation of our subject. We shall now regardthe description of these foundations as characterizing the concept ofnumber; in other words, we shall call any system of conceptually well-distinguished objects (elements, symbols) a number system, and itselements numbers, if to put it quite briefly for the moment wecan operate with them in essentially the same ways as we do with rationalnumbers.We proceed to give this somewhat inaccurate statement a preciseformulation.We consider a system S of well-distinguished objects, which wedenote by a, /?,.... S will be called a number system and its elementsa, j3, . . . will be called numbers if, besides being capable of definitionexclusively by means of rational numbers (i. c. ultimately by means ofnatural numbers alone) 9, these symbols a, jS, . . . satisfy the following fourconditions :1. Between any two elements a and /3 of S one and only one of thethree relations 10 a < 0, a = a >necessarily holds (this is expressed briefly by saying that S is an orderedsystem) and these relations of order between the elements of S are subjectto the same fundamental laws 1 as their analogues in the system of rationalnumbers u.2. Four distinct methods of combining any two elements of S aredefined, called Addition, Subtraction, Multiplication and Division. Witha single exception, to be mentioned immediately (3.), these processescan always be carried out to one definite result, and obey the same Fun-damental Laws 2, I IV, as their analogues in the system of the rational9 We shall come across actual examples m 3 and 5; for the moment, wen.ay think of decimal fractions, or similar symbols constructed from rational numbers.See also footnote 10, p. 12.10 Cf. also footnotes 2 and 4.1 As to what we may call the practical meaning of these relations, nothingIs implied; "<" may as usual stand for "less than'*, but it may equally well mean"before", "to the left of", "higher than", "lower than", "subsequent to", in factmay express any relation of order (including "greater than"). This meaning merelyhas to be defined without ambiguity and kept consistent. Similarly, "equality"need not imply identity. Thus, for example, within the system of symbols of theform p/q, where/), q are integers and q =4= 0, the symbols 3/4, 0/8, I)/ 12 aregenerally said to be "equal"; that is, for certain purposes (calculating, measuring,and so on) we define equality within our system of symbols in such a way that 3/4 -=6/8-= -9/-12, although 3/4, 0/8, -9/-12 are in the first instance differentelements of that system (see also 14, note 1).

10 Chapter I. Principles of the theory of real numbers.numbers 12. (The "zero" of the system, which must be known in orderthat the elements can be divided into positive and negative, is to be definedas explained in footnote 14 below.)3. With every rational number we can associate an element of S(and all others "equal'' to it) in such a manner that, if a and b denoterational numbers, a, ft their associates from S:a) the relation 1. holding between a and ft is of the same form asthat holding between a and b.b) the element resulting from a combination of a and ft (i. e. a + ft,a ft, a ft, or a -f- ft) has for its associated rational number the resultof the similar combination of a and b (i. e. a + b, a b, a b, or a - brespectively).[This is also expressed, more shortly, by saying that the system Scontains a sub-system S' sivnilar and isomorphous to the systemof rational numbers. Such a sub-system is in fact constituted by thoseelements of S which we have associated with rational numbers 13.]In such a correspondence, an element of S associated with the rationalnumber zero, and all elements equal to it, may be shortly referred to asthe "zero" of the system of elements. The exception mentioned in 2.then relates to division by zero 14.12 With reference to these four processes it should be noted, as in the caseof the symbols < and -, that no practical interpretation is implied. We alsodraw attention to the fact that subtraction is already completely denned in termsof addition, and division in terms of multiplication, so that, properly speaking,only two modes of combining elements need be assumed known.13 Two ordered systems are similar if it is possible to associate each elementof the one \\ith an element of the other in such a way that the same one of therelations 4, 1 as holds between two elements of the one system also holds betweenthe two associated elements of the other, they are tsomorfihous relatively to thepossible modes of combining their elements, if the element resulting from a com-bination of two elements of the one system is associated with that resulting fromthe similar combination of the two associated elements of the other system.14 The third of the stipulations by means of which we here characterise theconcept of number is fulfilled, moreover, as a consequence of the first arid second.For our purposes, this fact is not essential; but as it is significant from a systematicpoint of view, we briefly indicate its proof as follows' By 4, 2, there is an elementfor which a -f- a. From the fundamental laws 2, 1, it then quite eastl> followtha one and the same element of S satisfies a -I- - a, for every a. This element, with all elements equal to it, is called the neutral element relatively to the processof addition, or for brevity the "zero" in S. If a is different from this "zero", thereis, further, an element for which a e a; and it again appears thit this elementis the same as that satisfying n - a for any other a in S. This e, with all elementsequal to it, is called the neutral element relatively to the process of multiplication,or, briefly, the "unit" in S. The elements of S produced bv repeated addition orsubtraction of this "unit", and any others equal to them, are then called "integers"of S. All further elements of S (and all equal to them) which result fiom theseby the process of division then form the sub-system S' of S in question; that itis similar and iamorphous to the system of all rational numbers is in fact easilydeduced from 4, i and 4, 2. Thus, as asserted, our concept of number is alreadydetermined by the requirements of 4, 1, 2 and 4. 10 Chapter I. Principles of the theory of real numbers.numbers 12. (The "zero" of the system, which must be known in orderthat the elements can be divided into positive and negative, is to be definedas explained in footnote 14 below.)3. With every rational number we can associate an element of S(and all others "equal'' to it) in such a manner that, if a and b denoterational numbers, a, ft their associates from S:a) the relation 1. holding between a and ft is of the same form asthat holding between a and b.b) the element resulting from a combination of a and ft (i. e. a + ft,a ft, a ft, or a -f- ft) has for its associated rational number the resultof the similar combination of a and b (i. e. a + b, a b, a b, or a - brespectively).[This is also expressed, more shortly, by saying that the system Scontains a sub-system S' sivnilar and isomorphous to the systemof rational numbers. Such a sub-system is in fact constituted by thoseelements of S which we have associated with rational numbers 13.]In such a correspondence, an element of S associated with the rationalnumber zero, and all elements equal to it, may be shortly referred to asthe "zero" of the system of elements. The exception mentioned in 2.then relates to division by zero 14.12 With reference to these four processes it should be noted, as in the caseof the symbols < and -, that no practical interpretation is implied. We alsodraw attention to the fact that subtraction is already completely denned in termsof addition, and division in terms of multiplication, so that, properly speaking,only two modes of combining elements need be assumed known.13 Two ordered systems are similar if it is possible to associate each elementof the one \\ith an element of the other in such a way that the same one of therelations 4, 1 as holds between two elements of the one system also holds betweenthe two associated elements of the other, they are tsomorfihous relatively to thepossible modes of combining their elements, if the element resulting from a com-bination of two elements of the one system is associated with that resulting fromthe similar combination of the two associated elements of the other system.14 The third of the stipulations by means of which we here characterise theconcept of number is fulfilled, moreover, as a consequence of the first arid second.For our purposes, this fact is not essential; but as it is significant from a systematicpoint of view, we briefly indicate its proof as follows' By 4, 2, there is an elementfor which a -f- a. From the fundamental laws 2, 1, it then quite eastl> followtha one and the same element of S satisfies a -I- - a, for every a. This element, with all elements equal to it, is called the neutral element relatively to the processof addition, or for brevity the "zero" in S. If a is different from this "zero", thereis, further, an element for which a e a; and it again appears thit this elementis the same as that satisfying n - a for any other a in S. This e, with all elementsequal to it, is called the neutral element relatively to the process of multiplication,or, briefly, the "unit" in S. The elements of S produced bv repeated addition orsubtraction of this "unit", and any others equal to them, are then called "integers"of S. All further elements of S (and all equal to them) which result fiom theseby the process of division then form the sub-system S' of S in question; that itis similar and iamorphous to the system of all rational numbers is in fact easilydeduced from 4, i and 4, 2. Thus, as asserted, our concept of number is alreadydetermined by the requirements of 4, 1, 2 and 4.

1. The system ot rational numbers and its gaps. 114. For any two elements a and /3 of S both standing in the relation">" to the "zero" of the system, there exists a natural number n forwhich n j8 > a. Here n )3 denotes the sum ]8 -f- jf? + . . . -|- ]8 containingthe element ]8 w times. (Postulate of Eudoxus; cf. 2, VI.)To this abstract characterisation of the concept of number wewill append the following remark l5 : If the system S contains no otherelements than those corresponding to rational numbers as specifiedin 3, then our system does not differ in any essential feature from thesystem of rational numbers, but only in the (purely external) designationof the elements by symbols, or in the (purely practical) interpretationwhich we give to these symbols; differences almost as irrelevant,at bottom, as those which occur when we write figures at one time inArabic characters, at another, in Roman or Chinese, or take them todenote now temperature, now velocity or electric charge. Disregardingexternal characteristics of notation and practical interpretation, weshould thus be perfectly justified in considering the system S as identicalwith the system of rational numbers and in this sense we may put a = a,b --.&....If, however, the system S contains other elements besides the abovementioned, then we shall say that S includes the system of rationalnumbers, and is an extension of it. Whether a system of this more com-prehensive kind exists at all, remains for the moment an open question;15 We have defined the concept of number by a set of properties characterisingit. A critical construction of the foundations of arithmetic, which is quite outof the question within the limits of this volume, would have to comprise a strictinvestigation as to the extent to which these properties are independent of oneanother, i. e. whether any one of them can or cannot be deduced from the rest asa provable fact. Further, t would have to be shuwn that none of these fundamentalstipulations is in contradiction with any other and other matters too wouldrequire consideration. These investigations are tedious and have not yet reached afinal conclusion.In the treatment by E. Landau mentioned on p. 2, footnote 7, it is proved withabsolute rigour that the fundamental laws of arithmetic which we have set upcan all be deduced from the following 5 axioms relating to the natural numbers:Axiom 1 : 1 is a natural number.Axiom 2: For every natural number n there is just one other numberthat is called the successor of n. (Let it be denoted by n'.)Axiom 3: We have always n' 1.Axiom 4: From m' ~~ n't it follows that m n.Axiom 5: The induction law V is valid (in its first form).These 5 axioms, first formulated as here by G. Peano, but in substance set upby R. Dedektnd, assume that the natural numbers as a whole are regarded as given,that a relation of equality (and hence also inequality) is defined between them,and that this equality satisfies the relations 1, 1, 2, 3 (which belong to purelogic). 1. The system ot rational numbers and its gaps. 114. For any two elements a and /3 of S both standing in the relation">" to the "zero" of the system, there exists a natural number n forwhich n j8 > a. Here n )3 denotes the sum ]8 -f- jf? + . . . -|- ]8 containingthe element ]8 w times. (Postulate of Eudoxus; cf. 2, VI.)To this abstract characterisation of the concept of number wewill append the following remark l5 : If the system S contains no otherelements than those corresponding to rational numbers as specifiedin 3, then our system does not differ in any essential feature from thesystem of rational numbers, but only in the (purely external) designationof the elements by symbols, or in the (purely practical) interpretationwhich we give to these symbols; differences almost as irrelevant,at bottom, as those which occur when we write figures at one time inArabic characters, at another, in Roman or Chinese, or take them todenote now temperature, now velocity or electric charge. Disregardingexternal characteristics of notation and practical interpretation, weshould thus be perfectly justified in considering the system S as identicalwith the system of rational numbers and in this sense we may put a = a,b --.&....If, however, the system S contains other elements besides the abovementioned, then we shall say that S includes the system of rationalnumbers, and is an extension of it. Whether a system of this more com-prehensive kind exists at all, remains for the moment an open question;15 We have defined the concept of number by a set of properties characterisingit. A critical construction of the foundations of arithmetic, which is quite outof the question within the limits of this volume, would have to comprise a strictinvestigation as to the extent to which these properties are independent of oneanother, i. e. whether any one of them can or cannot be deduced from the rest asa provable fact. Further, t would have to be shuwn that none of these fundamentalstipulations is in contradiction with any other and other matters too wouldrequire consideration. These investigations are tedious and have not yet reached afinal conclusion.In the treatment by E. Landau mentioned on p. 2, footnote 7, it is proved withabsolute rigour that the fundamental laws of arithmetic which we have set upcan all be deduced from the following 5 axioms relating to the natural numbers:Axiom 1 : 1 is a natural number.Axiom 2: For every natural number n there is just one other numberthat is called the successor of n. (Let it be denoted by n'.)Axiom 3: We have always n' 1.Axiom 4: From m' ~~ n't it follows that m n.Axiom 5: The induction law V is valid (in its first form).These 5 axioms, first formulated as here by G. Peano, but in substance set upby R. Dedektnd, assume that the natural numbers as a whole are regarded as given,that a relation of equality (and hence also inequality) is defined between them,and that this equality satisfies the relations 1, 1, 2, 3 (which belong to purelogic).

12 Chapter I. Principles of the theory of real numbers.but an example will come before our notice presently in the system ofreal numbers 16.Having thus agreed as to the amount of preliminary assumptionwe require, we may now drop all argument on the subject, and againraise the question: What do we mean by saying that we know the numberV2 or T?It must in the first instance be termed altogether paradoxical thata number having its square equal to 2 does not exist in the system sofar constructed 17, or, in geometrical language, that the point A ofthe number-axis, whose distance from O equals the diagonal of thesquare of side O U, coincides with none of the "rational points". Forthe rational numbers are dense, i. e. between any two of them (whichare distinct) we can point out as many more as we please (since, if a byfo athe n rational numbers given by a + v , for v = 1, 2, . . . , n, evi-n -|- 1dently all lie between a and b and are distinct from these and from oneanother); but they are not, as we might say, dense enough to symboliseall conceivable points. Rather, as the aggregate of all integers provedtoo scanty to meet the requirements of the four processes of arithmetic,16 The mode of defining the number-concept given in 4 is of course notthe only possible one. Frequently the designation of number is still ascribed toobjects which fail to satisfy some one or other of the requirements there laid down.Thus for instance we may relinquish the condition that the objects under con-sideration should be constructively developed from rational numbers, regardingany entities (for instance points, or distances, or such like) as numbers, providedonly they satisfy the conditions 4, 1 4, or, in short, are similar and isomorphousto the system we have just set up. This conception of the notion of number,in accordance with which all isomoiphous systems must be regarded as in the ab-stract sense identical, is perfectly justified from a mathematical point of view, butobjections necessarily arise in connection with the theory of knowledge. Weshall encounter another modification of the number -concept when we come todeal with complex numbers.17 Proof'. There is certainly no natural number of square equal to 2, asI2 - 1 and all other integers have their squares 4. Thus V2 could only be a(positive) fraction , where q may be taken 2 and prime to p (i. e. the fractionis in its lowest terms). But if - is in its lowest terms, so is ( - J , which there-Q W/ Q ' qfore cannot reduce to the whole number 2. In a slightly different form: For anytwo natural numbers p and q without common factor, we have necessarily />2 4- 2 q~.For since two integers without common factors cannot both be even, either p isodd, or else p is even and q odd. In the first case />2 is again odd, hence cannotequal an even integer 2 q2. In the second case p2 = (2 p'Y is divisible by 4, but 2 qzis not, since it is double an odd number. So p'2 =1= 2 r/2 again. This Pythagoras issaid to have already known (cf. M. Cantor, Gesch. d. Mathem., Vol. 1, 2 lj ed., p.142 and 169. 1894). 12 Chapter I. Principles of the theory of real numbers.but an example will come before our notice presently in the system ofreal numbers 16.Having thus agreed as to the amount of preliminary assumptionwe require, we may now drop all argument on the subject, and againraise the question: What do we mean by saying that we know the numberV2 or T?It must in the first instance be termed altogether paradoxical thata number having its square equal to 2 does not exist in the system sofar constructed 17, or, in geometrical language, that the point A ofthe number-axis, whose distance from O equals the diagonal of thesquare of side O U, coincides with none of the "rational points". Forthe rational numbers are dense, i. e. between any two of them (whichare distinct) we can point out as many more as we please (since, if a byfo athe n rational numbers given by a + v , for v = 1, 2, . . . , n, evi-n -|- 1dently all lie between a and b and are distinct from these and from oneanother); but they are not, as we might say, dense enough to symboliseall conceivable points. Rather, as the aggregate of all integers provedtoo scanty to meet the requirements of the four processes of arithmetic,16 The mode of defining the number-concept given in 4 is of course notthe only possible one. Frequently the designation of number is still ascribed toobjects which fail to satisfy some one or other of the requirements there laid down.Thus for instance we may relinquish the condition that the objects under con-sideration should be constructively developed from rational numbers, regardingany entities (for instance points, or distances, or such like) as numbers, providedonly they satisfy the conditions 4, 1 4, or, in short, are similar and isomorphousto the system we have just set up. This conception of the notion of number,in accordance with which all isomoiphous systems must be regarded as in the ab-stract sense identical, is perfectly justified from a mathematical point of view, butobjections necessarily arise in connection with the theory of knowledge. Weshall encounter another modification of the number -concept when we come todeal with complex numbers.17 Proof'. There is certainly no natural number of square equal to 2, asI2 - 1 and all other integers have their squares 4. Thus V2 could only be a(positive) fraction , where q may be taken 2 and prime to p (i. e. the fractionis in its lowest terms). But if - is in its lowest terms, so is ( - J , which there-Q W/ Q ' qfore cannot reduce to the whole number 2. In a slightly different form: For anytwo natural numbers p and q without common factor, we have necessarily />2 4- 2 q~.For since two integers without common factors cannot both be even, either p isodd, or else p is even and q odd. In the first case />2 is again odd, hence cannotequal an even integer 2 q2. In the second case p2 = (2 p'Y is divisible by 4, but 2 qzis not, since it is double an odd number. So p'2 =1= 2 r/2 again. This Pythagoras issaid to have already known (cf. M. Cantor, Gesch. d. Mathem., Vol. 1, 2 lj ed., p.142 and 169. 1894).

14 Chapter I. Principles of the theory of real numbers.statement of these matters, we insert a discussion of sequences of rationalnumbers, provisional in character, but nevertheless of fundamental im-portance for all that comes after.2. Sequences of rational numbers1.In the process indicated above for calculating V2, successive well-defined rational numbers were constructed; their expression in decimalform was material in the description; from this form we now proposeto free it, and start with the following5. Definition. If, by means of any suitable process of construction, wecan form successively a first, a second, a third, . . . (rational) number andif to every positive integer n one and only one well-defined (rational) numberxn thus corresponds, then the numbersXl> X2> X'3> > Xm(in this order, corresponding to the natural order of the integers 1 , 2, 3, ...n, . . .) are said to form a sequence. We denote it for brevity by (xn)or (*!, *2, . . .).O Examples.i u i*n ~~ ] '* C* secluence > or ]> 2' 3'2. xn - 2"; i. e. the sequence 2, 4, 8, 16, ...3. xn an; i. e. the sequence a, a2, a3, . . . , where a is a given number.- 4. xn ~ H1 - (- 1)71}; 1- e. the sequence 1, 0, 1, 0, 1, 0, ...6. xn = the decimal fraction for V2, terminated at the wth digit./ iyi i 1116. xn - L_._ ; i. e. the sequence 1, - i, + * - ' . . .n & j *7. Let x1 = 1, x2 = 1, #3 = xl + #2 ~ and, generally, for n > 3, letxn ~ xn-i + xn-z- We thus obtain the sequence 1, 1, 2, 3, 5, 8, 13, 21, . . . , ubuallycalled Fibonacci's sequence.8. l,2,},-8,-J,S,J,-3,-J,...o 3 4 5 + IA 2,3,3,..., -n ....10 1 2 3 4 - 10- U'2'3'4' n"""1. xn the wth prime number 2 ; i. e. the sequence 2, 3, 5, 7, 1, 13, ... \12. The sequence 1, |, , g, m wh.ch * = (l + J + . . . + i)1 In this section all literal symbols will continue to stand for rational numbersonly. 2 Euclid proved that there is an infinity of primes. If plt p2, . . . , pk are anyprime numbers, then the integer m -= (/>,/>2 . . . pk) + 1 is either a prime differentfrom pi, pi, . . . , pk, or else a product of such primes. Hence no finite set of primenumbers can include all primes.

2. Sequences of rational numbers. 15Remarks.1. The law of formation may be quite arbitrary; it need not, in particular,be embodied in any explicit formula enabling us to obtain xn, for a given nt bydirect calculation. In examples 6, 5, 7 and 1, clearly no such formula can be im-mediately written down. If the terms of the sequence are individualy given, neitherthe law of formation (cf. 6, 5 and 12) nor any other kind of regularity (cf. 6, l)among the successive numbers is necessarily apparent.2. It is sometimes advantageous to start the sequence with a "0th" term x ,or even with a ( l)th or ( 2)th term, x__lt #_2. Occasionaly, it pays better to startindexing with 2 or 3. The only essential is that there should be an integer m such that xn is defined for every n m. The term xm is then caled the initial termof the sequence. We wil however, even then, continue to designate as the nih termlhat which bears the index n. In 6, 2, 3 and 4, for instance, we can without furtherdifficulties take a th term or even ( l)t}l or ( 2)<h to head the sequence. The "firstterm" of a sequence is then not necessarily the term with which the sequence begins.The notation wil be preferably (x0> *i> ) or (#-i #o> ) etc., as the case may be,unless it is either quite clear or irrelevant where our enumeration begins, and theabbreviated notation (xn) can be adopted.3. A sequence is frequently characterised as infinite. The epithet is thenmerely intended to emphasize the fact that every term is succeeded by other terms.It is also said that there is an infinite number of terms. More generaly, there issaid to be a finite number or an infinite number of things under consideration accord-ing as the number of these things can be indicated by a definite integral numberor not. And we may remark here that the word infinite, when otherwise used inthe sequel, wil have a symbolic significance only, intended as a concise expressionof some perfectly definite (and usualy quite simple) circumstance.4. If al the terms of a sequence have one and the same value c, the sequenceis said to be identicaly equal to c, and in symbols (xn) ~ c. More generaly, we shalwrite (xn) == (xn') if the two sequences (xn) and (xn') agree term for term, i. e. forevery index in question xn ~ xn'.5. It is often helpful and convenient to represent a sequence graphicalyby marking off its terms on the number-axis, or to think of them as so marked.We thus obtain a sequence of point*. But in doing this it should be borne in mindthat, in a sequence, one and the same number may occur repeatedly, even "in-finitely often" (cf. 6, 4); the corresponding point has then to be counted (i. e. con-sidered as a term of the sequence of points) repeatedly, or infinitely often, as thecase may be.0. A graphical representation of a different kind is obtained by marking,with respect to a pair of rectangular coordinate axes, the points whose coordinatesare (w, xn) for w = 1, 2, 3, ... and joining consecutive points by straight segments.The broken line so constructed gives a picture (diagram, or graph) of the sequence.To consider from the most diverse points of view the sequences herebyintroduced, and the real sequences that wil shortly be defined, wil be themain object of the folowing chapters. We shal be interested more par-ticularly in properties which hold, or are stipulated to hold, for al theterms of the sequence, or at least for al terms beyond (or folowing) somedefinite term 3. With reference to this last restriction, it may sometimes8 E. g. al the terms of the sequence 6, 9 are > 1. Or, al the terms of thesequence 6, 2 after the 6th are > 100 (or more shortly: for n > 6, xn > 100). 2. Sequences of rational numbers. 15Remarks.1. The law of formation may be quite arbitrary; it need not, in particular,be embodied in any explicit formula enabling us to obtain xn, for a given nt bydirect calculation. In examples 6, 5, 7 and 1, clearly no such formula can be im-mediately written down. If the terms of the sequence are individualy given, neitherthe law of formation (cf. 6, 5 and 12) nor any other kind of regularity (cf. 6, l)among the successive numbers is necessarily apparent.2. It is sometimes advantageous to start the sequence with a "0th" term x ,or even with a ( l)th or ( 2)th term, x__lt #_2. Occasionaly, it pays better to startindexing with 2 or 3. The only essential is that there should be an integer m such that xn is defined for every n m. The term xm is then caled the initial termof the sequence. We wil however, even then, continue to designate as the nih termlhat which bears the index n. In 6, 2, 3 and 4, for instance, we can without furtherdifficulties take a th term or even ( l)t}l or ( 2)<h to head the sequence. The "firstterm" of a sequence is then not necessarily the term with which the sequence begins.The notation wil be preferably (x0> *i> ) or (#-i #o> ) etc., as the case may be,unless it is either quite clear or irrelevant where our enumeration begins, and theabbreviated notation (xn) can be adopted.3. A sequence is frequently characterised as infinite. The epithet is thenmerely intended to emphasize the fact that every term is succeeded by other terms.It is also said that there is an infinite number of terms. More generaly, there issaid to be a finite number or an infinite number of things under consideration accord-ing as the number of these things can be indicated by a definite integral numberor not. And we may remark here that the word infinite, when otherwise used inthe sequel, wil have a symbolic significance only, intended as a concise expressionof some perfectly definite (and usualy quite simple) circumstance.4. If al the terms of a sequence have one and the same value c, the sequenceis said to be identicaly equal to c, and in symbols (xn) ~ c. More generaly, we shalwrite (xn) == (xn') if the two sequences (xn) and (xn') agree term for term, i. e. forevery index in question xn ~ xn'.5. It is often helpful and convenient to represent a sequence graphicalyby marking off its terms on the number-axis, or to think of them as so marked.We thus obtain a sequence of point*. But in doing this it should be borne in mindthat, in a sequence, one and the same number may occur repeatedly, even "in-finitely often" (cf. 6, 4); the corresponding point has then to be counted (i. e. con-sidered as a term of the sequence of points) repeatedly, or infinitely often, as thecase may be.0. A graphical representation of a different kind is obtained by marking,with respect to a pair of rectangular coordinate axes, the points whose coordinatesare (w, xn) for w = 1, 2, 3, ... and joining consecutive points by straight segments.The broken line so constructed gives a picture (diagram, or graph) of the sequence.To consider from the most diverse points of view the sequences herebyintroduced, and the real sequences that wil shortly be defined, wil be themain object of the folowing chapters. We shal be interested more par-ticularly in properties which hold, or are stipulated to hold, for al theterms of the sequence, or at least for al terms beyond (or folowing) somedefinite term 3. With reference to this last restriction, it may sometimes8 E. g. al the terms of the sequence 6, 9 are > 1. Or, al the terms of thesequence 6, 2 after the 6th are > 100 (or more shortly: for n > 6, xn > 100).

16 Chapter I. Principles of the theory of real numbers.be said that particular considerations in hand are valid "a finite numberof terms being disregarded", or only concern the ultimate behaviour ofthe sequence. Our first examples of considerations of the kind referredto are afforded by the following definitions:Definitions. I. A sequence is said to be bounded*, if there is apositive number K such that each term xn of the sequence satisfies theinequality xn K orThe number K is then called a bound of the sequence.Remarks and Examples.1. In definition 8, it is a matter of practical indifference whether we write"" or "<K". For if | xn \ K holds always (i. e. for every n in question),then we can also find a constant K' such that \ xn \ < K' holds always; indeed,clearly any K.' > K will serve the purpose. Conversely, if | xn \ < K. always, thena fortiori \ xn \ K. When the exact magnitude of the bound comes in of coursethe distinction may be essential.2. If K is a bound of (xn)t then so is any larger number K'.3. The sequences 6, 1, 4, 5, 6, 9, 10 are evidently bounded; so is 6, 3, pro-vided | a | Si 1. The sequences 6, 2, 7, 8, 1 are certainly not so. Whether 6, 3for every \a\ >1, or 6, 12, is bounded or not, i> not immediately obvious.4. If all we know is the existence of a constant Klt such that xn < Klt forevery nt then the sequence is said to be bounded on the right (or above) and Kl iscalled a bound above (or a right hand bound) of the sequence.If there is a constant K2 such that xn > K2 always, then (xn) is said to bebounded on the left (or below) and K2 is called a bound below (or a left hand bound)of the sequence.Here K and K2 need not be positive.5. Supposing a given sequence is bounded on the right, it may still happenthat among its numbers none is the greatest. For instance, 6, 10 is bounded onthe right, yet every term of this sequence is exceeded by all that follow it, and nonecan be the greatest 6. Similarly, a sequence bounded on the left need contain noleast term; cf. 6, 1 and 0. (With this fact, which will appear at first sight para-doxical, the beginner should make himself thoroughly familiar.)Among a finite number of values there is of course always both a greatest anda least, i. e. a value not exceeded by any of the others, and one which none of theothers falls below. (There may, however, be several equal to this greatest or leastvalue.)(5. The property of boundedness of a sequence xn (though not the actual valueof one of the bounds) is a property of the tail-end of the sequence ; it is unaffectedby any alteration to an isolated term of the sequence. (Proof?)4 This nomenclature appears to have been introduced by C. Jordan, Coursd'analyse, Vol. 1, p. 2. Paris 1893.6 The beginner should guard against modes of expression such as these,which may often be heard: "for n infinitely large, xn 1"; "1 is the greatestnumber of the sequence". Anything of this sort is sheer nonsense (cf. on this point7, 3). For the terms of the sequence are 0, ,],},... and none of these is -- 1, onthe contrary all of them are < 1. And there is no such thing as an "infinitely large n". 16 Chapter I. Principles of the theory of real numbers.be said that particular considerations in hand are valid "a finite numberof terms being disregarded", or only concern the ultimate behaviour ofthe sequence. Our first examples of considerations of the kind referredto are afforded by the following definitions:Definitions. I. A sequence is said to be bounded*, if there is apositive number K such that each term xn of the sequence satisfies theinequality xn K orThe number K is then called a bound of the sequence.Remarks and Examples.1. In definition 8, it is a matter of practical indifference whether we write"" or "<K". For if | xn \ K holds always (i. e. for every n in question),then we can also find a constant K' such that \ xn \ < K' holds always; indeed,clearly any K.' > K will serve the purpose. Conversely, if | xn \ < K. always, thena fortiori \ xn \ K. When the exact magnitude of the bound comes in of coursethe distinction may be essential.2. If K is a bound of (xn)t then so is any larger number K'.3. The sequences 6, 1, 4, 5, 6, 9, 10 are evidently bounded; so is 6, 3, pro-vided | a | Si 1. The sequences 6, 2, 7, 8, 1 are certainly not so. Whether 6, 3for every \a\ >1, or 6, 12, is bounded or not, i> not immediately obvious.4. If all we know is the existence of a constant Klt such that xn < Klt forevery nt then the sequence is said to be bounded on the right (or above) and Kl iscalled a bound above (or a right hand bound) of the sequence.If there is a constant K2 such that xn > K2 always, then (xn) is said to bebounded on the left (or below) and K2 is called a bound below (or a left hand bound)of the sequence.Here K and K2 need not be positive.5. Supposing a given sequence is bounded on the right, it may still happenthat among its numbers none is the greatest. For instance, 6, 10 is bounded onthe right, yet every term of this sequence is exceeded by all that follow it, and nonecan be the greatest 6. Similarly, a sequence bounded on the left need contain noleast term; cf. 6, 1 and 0. (With this fact, which will appear at first sight para-doxical, the beginner should make himself thoroughly familiar.)Among a finite number of values there is of course always both a greatest anda least, i. e. a value not exceeded by any of the others, and one which none of theothers falls below. (There may, however, be several equal to this greatest or leastvalue.)(5. The property of boundedness of a sequence xn (though not the actual valueof one of the bounds) is a property of the tail-end of the sequence ; it is unaffectedby any alteration to an isolated term of the sequence. (Proof?)4 This nomenclature appears to have been introduced by C. Jordan, Coursd'analyse, Vol. 1, p. 2. Paris 1893.6 The beginner should guard against modes of expression such as these,which may often be heard: "for n infinitely large, xn 1"; "1 is the greatestnumber of the sequence". Anything of this sort is sheer nonsense (cf. on this point7, 3). For the terms of the sequence are 0, ,],},... and none of these is -- 1, onthe contrary all of them are < 1. And there is no such thing as an "infinitely large n".

2. Sequences of rational numbers. 17II. A sequence is said to be monotone ascending or increasing 9.if, for every value of n, Xn Xn+ilit is said to be monotone descending or decreasing if, for every n,xn S Xn+l*Both kinds will also be referred to as monotone sequences.Remarks and Examples.1. A sequence need not of course be either monotone increasing, or mono-tone decreasing; cf. 6, 4, 6, 8. Monotone sequences are, however, extremely com-mon, and usually easier to deal with than those which are not monotone. Thatis why it is convenient to give them a distinguishing name.2. Instead of "ascending" we should more strictly say "non-descending",and instead of "descending", "non-ascending". This, however, is not customary.If in any special instance the sign of equality is excluded, so that xn < xni l orvn > xn} ,, as the case may be, for every nt then the sequence is said to be strictlymonotone (increasing or decreasing).3. The sequences 6, 2, 5, 7, 10, 1, 12 and 6, 1, 9 are monotone; the first-named ascending, the others descending. 6, 3 is monotone descending, if a 1,but monotone ascending if a " . 1 ; for a < 0, it is not monotone.4. The designation of "monotone" is due to C. Neumann (Ober die nachKteis-, Kugel- und Zylmderfunktionen fortschreitenden Entwickelungen, p. 2(5,27. Leipzig 1881).We now come to a definition to which the reader should paythe greatest attention, sparing no effort to make himself master of itsmeaning and all that it implies.I. A sequence will be called a null sequence if it possesses the fol- 10lowing property: given any arbitrary positive (rational) number e, the in-equality | xn | < cis satisfied by all the terms, with at most a finite number 6 of exceptions. Inother words : an arbitrary positive number e being chosen, it is always possibleto designate a term xm of the sequence, beyond which the terms are less thane in absolute value. Or a number nQ can always be found, such that|*|< forRemarks and Examples.1. If, in a given sequence, these conditions are fulfilled for a particular e,they will certainly be fulfilled for every greater e (cf. 8, 1), but not necessarily forany smaller e. (In 6, 10, for instance, the conditions are fulfilled for e = 1 and there-fore for every larger e, if we put n =0; for e - } it is not possible to satisfy them.)In the case of a null sequence, the conditions have to be fulfilled for every positive8 Cf. 7, 3. 2. Sequences of rational numbers. 17II. A sequence is said to be monotone ascending or increasing 9.if, for every value of n, Xn Xn+ilit is said to be monotone descending or decreasing if, for every n,xn S Xn+l*Both kinds will also be referred to as monotone sequences.Remarks and Examples.1. A sequence need not of course be either monotone increasing, or mono-tone decreasing; cf. 6, 4, 6, 8. Monotone sequences are, however, extremely com-mon, and usually easier to deal with than those which are not monotone. Thatis why it is convenient to give them a distinguishing name.2. Instead of "ascending" we should more strictly say "non-descending",and instead of "descending", "non-ascending". This, however, is not customary.If in any special instance the sign of equality is excluded, so that xn < xni l orvn > xn} ,, as the case may be, for every nt then the sequence is said to be strictlymonotone (increasing or decreasing).3. The sequences 6, 2, 5, 7, 10, 1, 12 and 6, 1, 9 are monotone; the first-named ascending, the others descending. 6, 3 is monotone descending, if a 1,but monotone ascending if a " . 1 ; for a < 0, it is not monotone.4. The designation of "monotone" is due to C. Neumann (Ober die nachKteis-, Kugel- und Zylmderfunktionen fortschreitenden Entwickelungen, p. 2(5,27. Leipzig 1881).We now come to a definition to which the reader should paythe greatest attention, sparing no effort to make himself master of itsmeaning and all that it implies.I. A sequence will be called a null sequence if it possesses the fol- 10lowing property: given any arbitrary positive (rational) number e, the in-equality | xn | < cis satisfied by all the terms, with at most a finite number 6 of exceptions. Inother words : an arbitrary positive number e being chosen, it is always possibleto designate a term xm of the sequence, beyond which the terms are less thane in absolute value. Or a number nQ can always be found, such that|*|< forRemarks and Examples.1. If, in a given sequence, these conditions are fulfilled for a particular e,they will certainly be fulfilled for every greater e (cf. 8, 1), but not necessarily forany smaller e. (In 6, 10, for instance, the conditions are fulfilled for e = 1 and there-fore for every larger e, if we put n =0; for e - } it is not possible to satisfy them.)In the case of a null sequence, the conditions have to be fulfilled for every positive8 Cf. 7, 3.

18 Chapter I. Principles of the theory of real numbers., and in particular, therefore, for every very small e > 0. On this account, it isusual to formulate the definition somewhat more emphatically as follows: (xn)is a null sequence if, to every > 0, however small, there corresponds a numbern such that | xn | < c for every n > n .

I xn | < c, provided n >whatever be the value of e. It is thus sufficient to put na

Here w() need not be an integer.2. The sequence 6, 1 is clearly a null sequence; for -,

.3. The place in a given sequence beyond which the terms remain numeri-cally < e, will naturally depend in general on the magnitude of e; speaking broadly,it will lie further and further to the right (i. e. n will be larger and larger), thesmaller the given c is (cf. 2). This dependence of the number n on e is oftenemphasised by saying explicitly: "To each given corresponds a number nQ w (t)such that ..."4. The positive number below which | xn \ is to he from some stage onwardsneed not always be denoted by c. Any positive number, however designated, mayserve. In the sequel, where e, a, Kt . . . , denoting any given positive numbers, wemay often use instead , , , e2, a e, ta, etc.5. The sign of xn plays no part here, since | xn \ = | xn \. Accordingly6, is also a null sequence.6. In a null sequence, no term need be equal to zero. But all terms, whoseindex is very large, must be very small. For if I choose e = 10~~, say, then for cver\n > a certain n0t ( xn \ must be < 10~'5. Similarly for e - - 10~10 and for any other e.7. The sequence (an) specified in 6, 3 is also a null sequence provided \ a \ < 1.Proof. If a 0, the assertion is trivial, since then, for every > 0, | xn \ <for every n. If < | a \ < 1, then (by 3, 1,4). ---. > 1. If therefore we putI * I* = 1 4- pt then p > 0.I a \But in that case, for every n 2, we have(a) <l + #)n >! + #For when n = 2, we have (1 4- />)2 1 + 2/> -f pz > 1 -f 2p; the stated relationtherefore holds in that case. If, for n k 2,(!+/>)*> 1-1- kp,then by 2, I, 6therefore our relation, assumed true for n = kt is true for w = & + 1. By 2, Vit therefore holds 7 for every n 2.7 The proof shows moreover that (a) is valid for n 2 provided only 1 4- P> 0, i. e. p > 1, but =t=0. For p -- and for n = 1, (a) becomes an equality.For /> > 0, the validity of (a) follows immediately from the expansion of the left-hand side by the binomial theorem. The relation (a) is called Bernoulli's Inequality(James Bernoulli, Propositiones arithmeticae de seriebus, 1689, Prop. 4).

2. Sequences of rational numbers. 19Accordingly, we now have so that, however small c > may be, we haveI xn I I aU I < for every n > P8. In particular, besides the sequence f ) mentioned in 2., ( -), (-- J,//4\"\ . W \ 2n/ \ 3fi/( (?) )'i ui 9. A similar remark to that of 8, 1 may be apended to Definition 10: noessential modification is produced by reading "5* e" for "< e" there. In fact,if, for every n > w(), | xn \ < e, then a fortiori \ xn \ 5 c; conversely, if, given anye, ?2 can be so determined that | xn \ ' e for every n > w3, then choosing any posi-tive number et < c there is certainly an n1 such that | xn \ fg cl9 for every n > nand consequently | xn | < for every n > nt ;the conditions in their original form are thus also fulfilled. Precisely analogousconsiderations show that in Definition 10 "> HO" and " w " are practically inter-changeable alternatives.In any individual case, however, the distinction must of course be taken intoaccount.10. Although in a sequence every term stands entirely by itself, with a definitefixed value, and is not necessarily in any particular relation with the precedingor following terms, yet it is quite customary to ascribe "to the terms xn", or "tothe general term'1 any peculiarities in the sequence which may be observed onrunning through it. We might say, for instance, in 6, 1 the terms diminish; in6, 2 the terms increase; in 6, 4 or 6, 6 the terms oscillate; in 6, 1 the generalterm cannot be expressed by a formula, and so on. In this sense, the character-istic behaviour of a null sequence may be described by saying that the terms becomearbitrarily small, or infinitely small 8 ; by which neither more nor less is meant thanis contained in Definition 9 10, viz. that for every > however small the termsare ultimately (i. e. for all indices n > a suitable n ; or from and after, or beyond,a certain n(t) numericallv less than e.1. A null sequence is ipso facto bounded. For if we choose e I, then theremust be an integer n, such that, for every n > ni9 \ xn \ < 1. Among the finitenumber of values | .vt |, | x2 1, . . . , ! .vnl |, however, one (cf. 8, 5) is greatest, Msay. Then for K M -f 1, obviously | .vw | is akvays < K.12. To prove that a given sequence is a null sequence, it is indispensableto show that for a prescribed e > 0, the corresponding wy can actually be provedto exist (for instance, as in the examples that follow, by actually designating sucha number). Conversely, if a sequence (xn) is assumed to be a null sequence, it isthereby assumed that, for every t, the corresponding n may really be regarded asexistent. On the other hand, the student should make sure that he understandsclearly what is meant by a sequence not being a null sequence. The meaning isthis : it is not true that, for every positive number *, beyond a certain point | xn \6 This mode of expression is due to A. L. Caitchy (Analyse algebrique, p. 4and 2G).9 There need of course be no question here of the sequence being monotone.Also, in any case, some | xn | 's of index 5* w may already be < c.

20 Chapter I. Principles of the theory of real numbers.is always < e; there exists a special positive number e,,, such that | xn \ is not, beyondtiny // , always < c() ; after every // there is a larger index n (and therefore an in-finite number of such indices) for which | vn | ]> c .1 3. Finally we may indicate a means of interpreting geometrically the specialcharacter of a null sequence.Using the graphical representation 7, 5, the sequence is a nuii sequence ifits terms ultimately (for n > n) all belong to the interval 10 e . . . -f- . Letus call such an interval for brevity an e-neighbourhood of the origin; then we maystate (xn) is a null sequence if every c-neighbourhood of the origin (however small)contains all but a finite number, at most, of the terms of the sequence.Similarly, using the graphical representation 7, 6, we can state: (xn) is anull sequence if every *-stnp (however narrow) about the a\ts of absciae containsthe entire graph, with the exception, at most, of a finite initial portion, the e-stripbeing limited by parallels to the axis of abscissae through the two points (0, e).14. The concept of a null sequence, the "arbitrarily small given positivenumber c", to which we shall from now on have continually and indispensably toappeal, and which may thus be said to form a main support for the whole super-structure of analysis, appears to have been first used in 1055 by J. Walks (v. OperaI., p. 3S2/3). Substantially, however, it is already to be found in Euclid, Elements V.We are already in a better position to comprehend what is involvedin the idea, discussed above, of a meaning for V2 or T or log 5. Informing on the one hand (we keep to the instance of V2) the numbers*1 =l-4; *o=l-41; *a = 1-414; *4 == 1-4142; ...on the other, the numbersyi = I'O; y* - 1-42; - 1415; y, =-- 1-4143; . . .we are obviously constructing two sequences of (rational) numbers (xn)and (yn) according to a perfectly definite (though possibly very laborious)method of procedure. These two sequences are both monotone, (xn)increasing, (yn) decreasing. Furthermore xn is <yn for every //, but thedifferences, i. e. the numbers yn xn =- dnform, by 10, 8, a null sequence, since dn = n. These are clearly thefacts which convince us that we "know" V2, and can "calculate" it?and so on, although as we said before no one has yet had thevalue V2 completely within his view, so to speak. If we referagain to the more suggestive representation on the number-axis, then,obviously (cf. fig. 1, p. 25): the points xl and y determine an interval10 The word interval denotes a portion of the number-axis between a definitepair of its points. According as we reckon these points themselves as belongingto the interval or not, this is termed closed or open. Unless otherwise stated, theinterval will always in the sequel be regarded as closed. (For 10, 13 this is immaterial,by 10, 9.) Supposing a to be the left end point, b the right end point, of an interval,we call this for brevity the interval a ... b. 20 Chapter I. Principles of the theory of real numbers.is always < e; there exists a special positive number e,,, such that | xn \ is not, beyondtiny // , always < c() ; after every // there is a larger index n (and therefore an in-finite number of such indices) for which | vn | ]> c .1 3. Finally we may indicate a means of interpreting geometrically the specialcharacter of a null sequence.Using the graphical representation 7, 5, the sequence is a nuii sequence ifits terms ultimately (for n > n) all belong to the interval 10 e . . . -f- . Letus call such an interval for brevity an e-neighbourhood of the origin; then we maystate (xn) is a null sequence if every c-neighbourhood of the origin (however small)contains all but a finite number, at most, of the terms of the sequence.Similarly, using the graphical representation 7, 6, we can state: (xn) is anull sequence if every *-stnp (however narrow) about the a\ts of absciae containsthe entire graph, with the exception, at most, of a finite initial portion, the e-stripbeing limited by parallels to the axis of abscissae through the two points (0, e).14. The concept of a null sequence, the "arbitrarily small given positivenumber c", to which we shall from now on have continually and indispensably toappeal, and which may thus be said to form a main support for the whole super-structure of analysis, appears to have been first used in 1055 by J. Walks (v. OperaI., p. 3S2/3). Substantially, however, it is already to be found in Euclid, Elements V.We are already in a better position to comprehend what is involvedin the idea, discussed above, of a meaning for V2 or T or log 5. Informing on the one hand (we keep to the instance of V2) the numbers*1 =l-4; *o=l-41; *a = 1-414; *4 == 1-4142; ...on the other, the numbersyi = I'O; y* - 1-42; - 1415; y, =-- 1-4143; . . .we are obviously constructing two sequences of (rational) numbers (xn)and (yn) according to a perfectly definite (though possibly very laborious)method of procedure. These two sequences are both monotone, (xn)increasing, (yn) decreasing. Furthermore xn is <yn for every //, but thedifferences, i. e. the numbers yn xn =- dnform, by 10, 8, a null sequence, since dn = n. These are clearly thefacts which convince us that we "know" V2, and can "calculate" it?and so on, although as we said before no one has yet had thevalue V2 completely within his view, so to speak. If we referagain to the more suggestive representation on the number-axis, then,obviously (cf. fig. 1, p. 25): the points xl and y determine an interval10 The word interval denotes a portion of the number-axis between a definitepair of its points. According as we reckon these points themselves as belongingto the interval or not, this is termed closed or open. Unless otherwise stated, theinterval will always in the sequel be regarded as closed. (For 10, 13 this is immaterial,by 10, 9.) Supposing a to be the left end point, b the right end point, of an interval,we call this for brevity the interval a ... b.

2. Sequences of rational numbers. 21! of length dl ; the points x2 and jy2 similarly, an interval /2 of length. Sincethe second interval lies wholly within the first. Similarly, the points X3and V3 determine an interval of length d3, completely within /2, andgenerally, the points xn and yn determine an interval fn completelyinside Jn-V The lengths of these intervals form a null sequence; theintervals themselves shrink up, one surmises, about a definitenumber, contract to a quite definite point.It only remains to examine how near this surmise is to truth. Withthis purpose in view, we state, more generally, the following:Definition. To express the fact that a monotone ascending sequence 1.(xn) and a monotone descending sequence (yn) are given, whose terms forevery n satisfy the condition xn ynand for which the differences dn=yn - Xnform a null sequence, we say for brevity that we are given a nest ofintervals (Intervallschachtelung)*. TJie nth interval stretchesfrom xn to yn and has length dn. The nest itself will be denoted by ( /) orby (# | yn)-The conjecture which we made above now finds its first confirma-tion in the following:Theorem f. There is at most one (rational) point s belonging to all 12.the intervals of a given nest, that is to say satisfying, for every nt the in-equality *ns yn>Proof: If there were, besides \$, another number sf differing fromit, and also satisfying the inequalityfor every , then, for every , besidesxn <Ls< yn,* A set or series of similar objects is said to form a nest or to be nested (inein-ander geschachtelt) when each smaller one is enclosed or fits into that which is nextin size to it. The word nest is here used with the additional (ideal) characteristicimplied, that the sizes diminish to zero. When this is not implied, we shall use themore explicit phrase that each is contained in the preceding (or we might say thatthey are nested).f We note here for future reference that this theorem continues to hold un-altered when the numbers which occur are arbitrary real numbers.2 (051) 2. Sequences of rational numbers. 21! of length dl ; the points x2 and jy2 similarly, an interval /2 of length. Sincethe second interval lies wholly within the first. Similarly, the points X3and V3 determine an interval of length d3, completely within /2, andgenerally, the points xn and yn determine an interval fn completelyinside Jn-V The lengths of these intervals form a null sequence; theintervals themselves shrink up, one surmises, about a definitenumber, contract to a quite definite point.It only remains to examine how near this surmise is to truth. Withthis purpose in view, we state, more generally, the following:Definition. To express the fact that a monotone ascending sequence 1.(xn) and a monotone descending sequence (yn) are given, whose terms forevery n satisfy the condition xn ynand for which the differences dn=yn - Xnform a null sequence, we say for brevity that we are given a nest ofintervals (Intervallschachtelung)*. TJie nth interval stretchesfrom xn to yn and has length dn. The nest itself will be denoted by ( /) orby (# | yn)-The conjecture which we made above now finds its first confirma-tion in the following:Theorem f. There is at most one (rational) point s belonging to all 12.the intervals of a given nest, that is to say satisfying, for every nt the in-equality *ns yn>Proof: If there were, besides \$, another number sf differing fromit, and also satisfying the inequalityfor every , then, for every , besidesxn <Ls< yn,* A set or series of similar objects is said to form a nest or to be nested (inein-ander geschachtelt) when each smaller one is enclosed or fits into that which is nextin size to it. The word nest is here used with the additional (ideal) characteristicimplied, that the sizes diminish to zero. When this is not implied, we shall use themore explicit phrase that each is contained in the preceding (or we might say thatthey are nested).f We note here for future reference that this theorem continues to hold un-altered when the numbers which occur are arbitrary real numbers.2 (051)

2 Chapter I. Principles of the theory of real numbers.we should also have (v. 3, I, 4)by 3, I, 2 and 3, I, 5, the inequalitieswould therefore hold for every n. Choosing = | s sr |, dn would never(a fortiori not for every n beyond a certain // ) be < . This contradictsthe hypothesis that (dn) is a null sequence. The assumption that twodistinct points belong to all the intervals is therefore inadmissible 1.Q. E. D. Remarks and Examples.1. Let*n = "-"--, y = J; that is to say,/,, - 5J=J . . . "-J- 1, dn = ?We can at once verify that we actually have a nest of intervals here, since2xn xn+i " yn+i Vn or every nt an< since, for every n > , we have dn < thowever > be chosen.The number 5=1 here belongs to all the / 's, since n ~~-- < 1 < - ~* n n nfor every n. No number other than 1 can belong therefore to all the intervals.2. Let fn be defined as follows 12 : / is the interval ... 1; /l the left halfof A; Jz the right half ofy y3 the left half ofy2; and so on. These intervals areobviously each contained in the preceding; and sinceJn has length dn k>n , tmdthese numbers form a null sequence, we have a nest of intervals. A little considera-tion shows that the sequence of the xn's consists of the numbers0> 4' 4 10 ~~ 16' 4 + T6 ~*~ G4 "" 6T * * 'each taken twice running; and that the sequence of yn's begins with 1 and con-tinues with 1 ~" 2 = 2J l ~ 2 ~ 8 8' ~ 2 "" 8 ~~ 32 32* * ' *each taken twice running. Now1,1,1, , 1 1 A 1\ 14 16 Ci ' ' ' P = 3 ~ 4*- 3~ 4*-)1 From a graphical point of view, what the proof indicates is that if \$ and\$' belong to all the intervals, then each interval has a length at least equal to thedistance | s s' | between s and s' (v. 3, I, 6); these lengths cannot, therefore,form a null sequence.12 Here we let the index start from (cf. 7, 2).13 For any two numbers a and b, and every positive integer kt the formulaafc - bk = (a - b)(ak~l + ak~* b+ ... + a bk~2 + *fc~1)is known to hold. Whence, more particularly, for a =t= 1, the formulae1 + a + . . . + ak~* = ! ~ ** and a + a* + . . . + ak = \ " a* . a.l o 1 a 2 Chapter I. Principles of the theory of real numbers.we should also have (v. 3, I, 4)by 3, I, 2 and 3, I, 5, the inequalitieswould therefore hold for every n. Choosing = | s sr |, dn would never(a fortiori not for every n beyond a certain // ) be < . This contradictsthe hypothesis that (dn) is a null sequence. The assumption that twodistinct points belong to all the intervals is therefore inadmissible 1.Q. E. D. Remarks and Examples.1. Let*n = "-"--, y = J; that is to say,/,, - 5J=J . . . "-J- 1, dn = ?We can at once verify that we actually have a nest of intervals here, since2xn xn+i " yn+i Vn or every nt an< since, for every n > , we have dn < thowever > be chosen.The number 5=1 here belongs to all the / 's, since n ~~-- < 1 < - ~* n n nfor every n. No number other than 1 can belong therefore to all the intervals.2. Let fn be defined as follows 12 : / is the interval ... 1; /l the left halfof A; Jz the right half ofy y3 the left half ofy2; and so on. These intervals areobviously each contained in the preceding; and sinceJn has length dn k>n , tmdthese numbers form a null sequence, we have a nest of intervals. A little considera-tion shows that the sequence of the xn's consists of the numbers0> 4' 4 10 ~~ 16' 4 + T6 ~*~ G4 "" 6T * * 'each taken twice running; and that the sequence of yn's begins with 1 and con-tinues with 1 ~" 2 = 2J l ~ 2 ~ 8 8' ~ 2 "" 8 ~~ 32 32* * ' *each taken twice running. Now1,1,1, , 1 1 A 1\ 14 16 Ci ' ' ' P = 3 ~ 4*- 3~ 4*-)1 From a graphical point of view, what the proof indicates is that if \$ and\$' belong to all the intervals, then each interval has a length at least equal to thedistance | s s' | between s and s' (v. 3, I, 6); these lengths cannot, therefore,form a null sequence.12 Here we let the index start from (cf. 7, 2).13 For any two numbers a and b, and every positive integer kt the formulaafc - bk = (a - b)(ak~l + ak~* b+ ... + a bk~2 + *fc~1)is known to hold. Whence, more particularly, for a =t= 1, the formulae1 + a + . . . + ak~* = ! ~ ** and a + a* + . . . + ak = \ " a* . a.l o 1 a

3. Irrational numbers. 23Hence, for every nt xn < J < yn \ thus s J is the single number which belongsto all the intervals. Here, therefore, (/n) "defines" or "determines*' the number i,or (yn) shrinks up to the number J.3. vf we are given a nest of intervals (/n), and a number s has been recog-nised as belonging to all the /n's, then by our theorem, 5 is quite uniquely deter-mined by ( /n). We therefore say, more pointedly, that the nest (/n) "defines" or"encloses" the number s. We also say that 5 is the innermost point of all the intervals.4. If s is any given rational number and we put, for n 1, 2, . . . , xn ~ s1 nand yn s + -, then (xn \ yn) is evidently a nest of intervals determining the numbers itself. But this is also the case if we put, for every n, xn - s and yn s. Mani-festly, we can, in the most various ways, form nests of intervals defining a givennumber.This theorem, however, only confirms what we may regard as onehalf of our previously described impression; namely, that if a numbers belongs to all the intervals of a nest, then there is none other besideswith this property, s is uniquely determined by the nest.The other half of our impression, namely, that there must alsoalways be a (rational) number belonging to all the intervals of a nest,is erroneous and it is precisely this fact which will become our induce-ment for extending the system of rational numbers.This the following example shows. As on p. 20, let xl 14; x.> 1-41 ; . . .;yl 1 >; yz = 1-42; . . . Then there is no rational number s> for which xn !L A "? ynfor every n. In fact, if we putv ' v a v 7 v 2xn xn Vn ~ 3>nthen the intervals /n' xn' . . . yn' also form a nest 1. But xnf x < 2 for all n,and yn' -- yn2 > 2 for all n (because this was how xn and yn were chosen), i. e.xnf < 2 < yn'. On the other hand, if xn ;< s -_ yn we should have, by squaring(as we may, by 3, 1, 3), xn' ? s2 yn' for all n. By our theorem 12 this would in-volve s2 = 2, which is however impossible, by the proof given in footnote 17 onp. 12. Here, therefore, there is certainly no (rational) number belonging to all theintervals.In the following paragraphs, we will investigate what, in a case suchas this, should be done. 3. Irrational numbers.We must come to terms with the fact that there is no rationalnumber whose square is 2, that the system of rational numbers is toodefective, too incomplete, too full of gaps, to furnish a solution for the14 For it follows from xn xn l < yn+i yn since all the numbers arepositive, so that squaring (cf. 3, I, 3) is allowed that xn' *v'n+1 < y'nfi yn';further yn' xn' -- (yn + xn)(yn .vn); therefore, since .vr} and yn are certainly< 2 for every n, yn' xn' < --n, i. e. < s, provided J)n < ; and this, by 10, 8,is certainly the case for every n > a certain w .

24: Chapter I. Principles of the theory of real numbers.equation x2 2. Indeed, this is only one of many equations for whosesolution the material of the system of rational numbers proves insufficient.Almost all the numerical values which we are in the habit of denotingby \/nt log n, sin a, tan a and so on, are non-existent in the system ofrational numbers and can no more be immediately "obtained", or "deter-mined", or be "stated in figures", than can V2. The material is too coarsefor such finer purposes.The considerations brought forward in the preceding paragraphspoint to means for providing ourselves with more suitable material.We saw, on the one hand, that, behind the conviction that we doknow V2, there lay no more, substantially, than the fact that we possessa method by which a perfectly definite nest of intervals may beobtained ; for its construction, the solution of the equation x2 2 ofcourse gave the occasion lr>. We saw, on the other hand, that if anest (/n) encloses any number s capable of specification at all (this stillimplying that it is a rational number) then this number s is quite uniquelydefined by the nest ( /n), - so unambiguously, indeed, that it ia entirelyindifferent, whether I give (write down, indicate) the number directly,or give, instead, the nest (/) with the tacit addition that, by the latter,I mean precisely the number s which it uniquely encloses or defines. Inthis sense, the two data (the two symbols) are equivalent, and mayto a certain extent be considered equal 16, so that we may write in-deed: (/n) = * or (xn | yn) = s.15 The kernel of this procedure is in fact as follows: We ascertain thatI2 < 2, 2 > 2, and accordingly put # 1, y ~ 2. We then divide the intervalkJQ =- x . . . y into 10 equal parts, and taking the points of division, 1 + , fork -= 0, 1, 2, . . . , 9, 10, determine by trial whether their squares are > 2 or < 2.We find that the squares corresponding to k 0, 1, 2, 3, 4 are too small, thosecorresponding to k = 5y G, . . . , 10 too large, and accordingly we put Xi =1-4 andyt == 1-5. Next, we divide the interval /j. xl . . . yl into 10 equal parts, and gothrough a similar test with regard to the new points of division and so on. Theknown process for extracting the square root of 2 is intended mainly to make thesuccessive trials as mechanical as possible. The corresponding treatment of,for instance, the equation 10* = 2 (i. e. determination of the common logarithmof 2) involves the following nest of intervals: Since 10 < 2, 10l > 2, we here pu:XQ = 0, y = 1 and divide / = # . . . y into 10 equal parts. For the points ofdivision, lftt we next test whether 10*/10 < 2 or > 2, that is to say, whether 10fc< 210 or > 210. As a result of this trial, we shall have to put x ~ 0-3, y 0-4.The interval /l xl . . . yl is again divided into 10 equal parts, the same pro-3 kcedure instituted for the points of division -}- . and, in consequence, xz putequal to 30 and ya to 31 and so on. This obvious procedure is of coursemuch too laborious for practical calculations.16 The justification for this is provided by Theorems 14 to 19. 24: Chapter I. Principles of the theory of real numbers.equation x2 2. Indeed, this is only one of many equations for whosesolution the material of the system of rational numbers proves insufficient.Almost all the numerical values which we are in the habit of denotingby \/nt log n, sin a, tan a and so on, are non-existent in the system ofrational numbers and can no more be immediately "obtained", or "deter-mined", or be "stated in figures", than can V2. The material is too coarsefor such finer purposes.The considerations brought forward in the preceding paragraphspoint to means for providing ourselves with more suitable material.We saw, on the one hand, that, behind the conviction that we doknow V2, there lay no more, substantially, than the fact that we possessa method by which a perfectly definite nest of intervals may beobtained ; for its construction, the solution of the equation x2 2 ofcourse gave the occasion lr>. We saw, on the other hand, that if anest (/n) encloses any number s capable of specification at all (this stillimplying that it is a rational number) then this number s is quite uniquelydefined by the nest ( /n), - so unambiguously, indeed, that it ia entirelyindifferent, whether I give (write down, indicate) the number directly,or give, instead, the nest (/) with the tacit addition that, by the latter,I mean precisely the number s which it uniquely encloses or defines. Inthis sense, the two data (the two symbols) are equivalent, and mayto a certain extent be considered equal 16, so that we may write in-deed: (/n) = * or (xn | yn) = s.15 The kernel of this procedure is in fact as follows: We ascertain thatI2 < 2, 2 > 2, and accordingly put # 1, y ~ 2. We then divide the intervalkJQ =- x . . . y into 10 equal parts, and taking the points of division, 1 + , fork -= 0, 1, 2, . . . , 9, 10, determine by trial whether their squares are > 2 or < 2.We find that the squares corresponding to k 0, 1, 2, 3, 4 are too small, thosecorresponding to k = 5y G, . . . , 10 too large, and accordingly we put Xi =1-4 andyt == 1-5. Next, we divide the interval /j. xl . . . yl into 10 equal parts, and gothrough a similar test with regard to the new points of division and so on. Theknown process for extracting the square root of 2 is intended mainly to make thesuccessive trials as mechanical as possible. The corresponding treatment of,for instance, the equation 10* = 2 (i. e. determination of the common logarithmof 2) involves the following nest of intervals: Since 10 < 2, 10l > 2, we here pu:XQ = 0, y = 1 and divide / = # . . . y into 10 equal parts. For the points ofdivision, lftt we next test whether 10*/10 < 2 or > 2, that is to say, whether 10fc< 210 or > 210. As a result of this trial, we shall have to put x ~ 0-3, y 0-4.The interval /l xl . . . yl is again divided into 10 equal parts, the same pro-3 kcedure instituted for the points of division -}- . and, in consequence, xz putequal to 30 and ya to 31 and so on. This obvious procedure is of coursemuch too laborious for practical calculations.16 The justification for this is provided by Theorems 14 to 19.

3. Irrational numbers. 25Consequently, we will not say merely: "the nest (/n) defines the numbers" but rather "(/) is only another symbol for the number \$", or in fine,"(/n) is the number s" exactly as we are used to look upon the decimalfraction 0-3 ... as merely another symbol for the number , or as beingprecisely the number itself.It now becomes extremely natural to introduce tentatively ananalogous mode of expression with regard to those nests of intervalswhich contain no rational number. Thus if xn, yn denote the numbersconstructed previously in connection with the equation x2 = 2, onemight seeing that in the system of rational numbers there is nota single one whose square =2 decide to say that this nest (xn \ yn)determines the "true" "value of V2 " though one incapable of beingsymbolised by means of rational numbers, that it encloses this X U -J JFig. 1.value unambiguously in fine, "it is a newly created symbol for thisnumber", or, for brevity, "it is the number itself". And similarly in everyother case. If (/n) (xn \ yn) is any nest of intervals and no rationalnumber s belongs to all its intervals, we might finally resolve to say thatthis nest encloses a perfectly definite value, though one incapable ofbeing directly symbolised by means of rational numbers, it deter-mines a perfectly definite number, though one unfortunately non-existent in the system of rational numbers, it is a newly created symbolfor this number, or briefly: is the number itself; and this number, incontradistinction to the rational numbers, would then have to be calledan irrational number.Here certainly the question arises: Can this be done withoutfurther justification ? Is it allowable ? May we, without more ado,designate these new symbols, the nests (xn \ yn), as numbers? The fol-lowing considerations are intended to show that to this course there isno obstacle whatever.In the first instance, a simple graphical illustration of these factson the number-axis (see fig. 1) gives every appearance of justification toour resolution. If, by any construction, we have marked a point P onthe number-axis (e. g. by marking off to the right of O the length

26 Chapter I. Principles of the theory of real numbers.of the diagonal of a square of side O U) then we can in any numberof ways define a nest of intervals enclosing the point P. We maydo so in this way, for instance. First of all we imagine all integers marked on the axis. Of these, there will be exactly one, say p,such that our point P lies in the stretch from p inclusive to (/>+!)exclusive. Accordingly we put x -= p, y p + 1, and divide theinterval JQ = x . . . yQ into 10 equal parts 17. The points of divisionkare p + - (with k = 0, 1, 2, . . . , 10), and among them, there will againk kbe exactly one, say p + - J , such that P lies between xt p -[- *inclusive and y = p + * -y~ exclusive. The interval J xl . . . yis again divided into 10 equal parts, and so on. If we imagine this processcontinued indefinitely, we obtain a perfectly definite nest (Jn) all of whoseintervals Jn contain the point P. No other point P' besides P can lie in allthe intervals Jn. For, if that were so, all the intervals would have to con-tain the whole stretch P', which is impossible, as the lengths of theintervals (jn has length J form a null sequence.For every arbitrarily given point P on the number-axis (rational ornot) there are thus nests of intervals obviously, indeed, any numberof such nests which contain that point and no other. And in thepresent instance, i. e. in the graphical representation on the number-axis the converse appears most plausible; if we consider any nestof intervals, there seems to be always one point (and by the reasoningabove, only this one) belonging to all its intervals, which is thus deter-mined by it. We believe, at any rate, that we may infer this directly fromour conception of the continuity, or gaplessnessy of the straight line 18.Thus in this geometrical representation we should have completereciprocity: every point can be enclosed in a suitable nest of intervalsand every such nest invariably encloses one and only one point.This gives us a high degree of confidence in the adequacy of ourresolve to consider nests of intervals as numbers, which we now for-mulate more precisely as follows:13. Definition. We will say of every nest of intervals (Jn) or (xn \ yn),that it defines or, for brevity, it is, a determinate number. To represent17 Instead of 10 we may of course take any other integer 2. For furtheidetail, see 5.18 The proposition, by which the "continuity of the straight line" is expresslypostulated for a proof cannot be here expected, since it is essentially a descriptionof the form of our concept of the straight line which is involved is called theCantor-Dedekind axiom. 26 Chapter I. Principles of the theory of real numbers.of the diagonal of a square of side O U) then we can in any numberof ways define a nest of intervals enclosing the point P. We maydo so in this way, for instance. First of all we imagine all integers marked on the axis. Of these, there will be exactly one, say p,such that our point P lies in the stretch from p inclusive to (/>+!)exclusive. Accordingly we put x -= p, y p + 1, and divide theinterval JQ = x . . . yQ into 10 equal parts 17. The points of divisionkare p + - (with k = 0, 1, 2, . . . , 10), and among them, there will againk kbe exactly one, say p + - J , such that P lies between xt p -[- *inclusive and y = p + * -y~ exclusive. The interval J xl . . . yis again divided into 10 equal parts, and so on. If we imagine this processcontinued indefinitely, we obtain a perfectly definite nest (Jn) all of whoseintervals Jn contain the point P. No other point P' besides P can lie in allthe intervals Jn. For, if that were so, all the intervals would have to con-tain the whole stretch P', which is impossible, as the lengths of theintervals (jn has length J form a null sequence.For every arbitrarily given point P on the number-axis (rational ornot) there are thus nests of intervals obviously, indeed, any numberof such nests which contain that point and no other. And in thepresent instance, i. e. in the graphical representation on the number-axis the converse appears most plausible; if we consider any nestof intervals, there seems to be always one point (and by the reasoningabove, only this one) belonging to all its intervals, which is thus deter-mined by it. We believe, at any rate, that we may infer this directly fromour conception of the continuity, or gaplessnessy of the straight line 18.Thus in this geometrical representation we should have completereciprocity: every point can be enclosed in a suitable nest of intervalsand every such nest invariably encloses one and only one point.This gives us a high degree of confidence in the adequacy of ourresolve to consider nests of intervals as numbers, which we now for-mulate more precisely as follows:13. Definition. We will say of every nest of intervals (Jn) or (xn \ yn),that it defines or, for brevity, it is, a determinate number. To represent17 Instead of 10 we may of course take any other integer 2. For furtheidetail, see 5.18 The proposition, by which the "continuity of the straight line" is expresslypostulated for a proof cannot be here expected, since it is essentially a descriptionof the form of our concept of the straight line which is involved is called theCantor-Dedekind axiom.

3. Irrational numbers. 27ity we use the symbol denoting the nest of intervals itself, and only as an ab-breviation replace this by a small Greek letter, writing in this sense 19, e. g.(Jn) or (xn\yn) - a.Now, in spite of all we have said, this cannot but seem a very arbi-trary step, the question has to be repeated most insistently: will itpass without further justification? These purely ideal objects which wehave just defined these nests of intervals (or else that still extremelyquestionable 'something' which such a nest encloses or determines) canwe speak of these as numbers? Are they after all numbers in the samesense as the rational numbers, more precisely, in the sense in whichthe number concept was defined by our conditions 4?The answer can only consist in deciding, whether the totality oraggregate of all conceivable nests of intervals, or of the symbols (/n) or(xn \ yn) r <* introduced to denote them, forms a system of objects satis-fying these conditions 4 20; a system therefore to recapitulate theseconditions briefly whose elements are derived from the rational numbers,and 1. are capable of being ordered; 2. are capable of being combinedby the four processes (rules), obeying at the same time the fundamentallaws 1 and 2, I IV; 3. contain a sub-system similar and isomorphousto the system of rational numbers; and 4. satisfy the Postulate of Eud-oxus.If and only if the decision turns out to be favourable, all will bewell; our new symbols will then have vindicated their numerical char-acter, and we shall have established that they are numbers, whosetotality we shall then designate as the system or set of real numbers.Now the decision in question does not present the slightest diffi-culty, and we may accordingly be brief in expounding the details:Nests of intervals or our new symbols (xn \ yn) are certainlyconstructed by means of rational number-symbols alone; we have there-fore only to settle the points 4, 1 4. For this, we shall go to work inthe following way: Certain of the nests of intervals define a rationalnumber 21, something, therefore, for which both meaning and mode ofcombination have been previously established. We consider two suchrational-valued nests, say (xn \ yn) s and (xnf \ yn') = s'. With the tworational number-symbols s and s', we can immediately distinguish whetherthe first s is <, = or > the second s'; and we can combine the two bythe four processes of arithmetic. Essentially, what we have to do is toendeavour directly to recognise the former fact, and to carry out the latterprocesses, on the two nests of intervals themselves by which s and s' were19 <7 is an abbreviated notation for the nest of intervals ( /n) or (xn \ yn).20 The reader should here read these conditions through again.81 We will describe such nests for brevity as rational-valued. 3. Irrational numbers. 27ity we use the symbol denoting the nest of intervals itself, and only as an ab-breviation replace this by a small Greek letter, writing in this sense 19, e. g.(Jn) or (xn\yn) - a.Now, in spite of all we have said, this cannot but seem a very arbi-trary step, the question has to be repeated most insistently: will itpass without further justification? These purely ideal objects which wehave just defined these nests of intervals (or else that still extremelyquestionable 'something' which such a nest encloses or determines) canwe speak of these as numbers? Are they after all numbers in the samesense as the rational numbers, more precisely, in the sense in whichthe number concept was defined by our conditions 4?The answer can only consist in deciding, whether the totality oraggregate of all conceivable nests of intervals, or of the symbols (/n) or(xn \ yn) r <* introduced to denote them, forms a system of objects satis-fying these conditions 4 20; a system therefore to recapitulate theseconditions briefly whose elements are derived from the rational numbers,and 1. are capable of being ordered; 2. are capable of being combinedby the four processes (rules), obeying at the same time the fundamentallaws 1 and 2, I IV; 3. contain a sub-system similar and isomorphousto the system of rational numbers; and 4. satisfy the Postulate of Eud-oxus.If and only if the decision turns out to be favourable, all will bewell; our new symbols will then have vindicated their numerical char-acter, and we shall have established that they are numbers, whosetotality we shall then designate as the system or set of real numbers.Now the decision in question does not present the slightest diffi-culty, and we may accordingly be brief in expounding the details:Nests of intervals or our new symbols (xn \ yn) are certainlyconstructed by means of rational number-symbols alone; we have there-fore only to settle the points 4, 1 4. For this, we shall go to work inthe following way: Certain of the nests of intervals define a rationalnumber 21, something, therefore, for which both meaning and mode ofcombination have been previously established. We consider two suchrational-valued nests, say (xn \ yn) s and (xnf \ yn') = s'. With the tworational number-symbols s and s', we can immediately distinguish whetherthe first s is <, = or > the second s'; and we can combine the two bythe four processes of arithmetic. Essentially, what we have to do is toendeavour directly to recognise the former fact, and to carry out the latterprocesses, on the two nests of intervals themselves by which s and s' were19 <7 is an abbreviated notation for the nest of intervals ( /n) or (xn \ yn).20 The reader should here read these conditions through again.81 We will describe such nests for brevity as rational-valued.

28 Chapter 1. Principles of the theory of real numbers.given, and finally to extend the result to the aggregate of all nests ofintervals.Each provable proposition (A) relating to rational-valued nests will ac-cordingly give rise to a corresponding definition (B). We begin by settingdown concisely side by side these pairs of propositions (A) anddefinitions (B) 2.14. Equality: A. Theorem. If(xn \yn) = 5 and (xnf \yn') = s' are tworational-valued nests of intervals, then s = s' holds if, and only if,besides *n yn and xn' < yn'9we have 23for every n.On this theorem we now base the following:B. Definition. Two arbitrary nests of intervals cr (#n |j>n) anda .= (xnf | yn') are said to be equal if and only ifor every n. Remarks and Examples.1. The numbers xn and \n' on the one hand, yn and yn' on the other, needof course have nothing whatever to do with one another. This is no more sur-prising than that rational numbers so entirely different in appearance as , g'A,and 375 should be referred to as "equal". Equality is indeed something which2 The import of proposition and definition should in each case be interpretedin relation to the number-axis.23 Into the very simple proofs of the propositions 14 to 19 we do not proposeto enter, for the general reasons explained on p. 2. They will not present theslightest difficulty to the reader, once he has mastered the contents of Chapter I,whereas at this stage they would appear to him strange; moreover they will serveas exercises in that chapter. Merely as a specimen and example for the solutionof those problems, we will here prove Theorem 14:a) If s = s't then we have both xn \$ yn and xn' s yn'y whence atonce, xn < yn' and xn' y for every n.b) If conversely xn 5\$ yn' for every n, then s s' must hold. For if we hads > s', i. e. s s' > 0, then, since (yn xn) is a null sequence, we could so choosethe index p, that yp - xp < s s/ r XP - s' > y* - *As however s is certainly yp, this would imply xp s' > 0. We could thereforechoose a further index r for whichy/ - */ < * - s'.Since xr' \$', this would imply yr' < x Choosing an integer m exceed-ing both p and r, we could deduce, in view of the respective ascending and descend-ing monotony of our sequences of numbers, that a fortiori ym' < xm, which con-tradicts the hypothesis that xn y for every n. Thus s \$' is ensured.By interchanging throughout the above proof the accented and non-accentedletters, we deduce in the same manner that if xn' < yn for every n, then s' sIf then we have both xn' yn and xn yn' holding for every , then s ~ snecessarily follows. Q. E. D. 28 Chapter 1. Principles of the theory of real numbers.given, and finally to extend the result to the aggregate of all nests ofintervals.Each provable proposition (A) relating to rational-valued nests will ac-cordingly give rise to a corresponding definition (B). We begin by settingdown concisely side by side these pairs of propositions (A) anddefinitions (B) 2.14. Equality: A. Theorem. If(xn \yn) = 5 and (xnf \yn') = s' are tworational-valued nests of intervals, then s = s' holds if, and only if,besides *n yn and xn' < yn'9we have 23for every n.On this theorem we now base the following:B. Definition. Two arbitrary nests of intervals cr (#n |j>n) anda .= (xnf | yn') are said to be equal if and only ifor every n. Remarks and Examples.1. The numbers xn and \n' on the one hand, yn and yn' on the other, needof course have nothing whatever to do with one another. This is no more sur-prising than that rational numbers so entirely different in appearance as , g'A,and 375 should be referred to as "equal". Equality is indeed something which2 The import of proposition and definition should in each case be interpretedin relation to the number-axis.23 Into the very simple proofs of the propositions 14 to 19 we do not proposeto enter, for the general reasons explained on p. 2. They will not present theslightest difficulty to the reader, once he has mastered the contents of Chapter I,whereas at this stage they would appear to him strange; moreover they will serveas exercises in that chapter. Merely as a specimen and example for the solutionof those problems, we will here prove Theorem 14:a) If s = s't then we have both xn \$ yn and xn' s yn'y whence atonce, xn < yn' and xn' y for every n.b) If conversely xn 5\$ yn' for every n, then s s' must hold. For if we hads > s', i. e. s s' > 0, then, since (yn xn) is a null sequence, we could so choosethe index p, that yp - xp < s s/ r XP - s' > y* - *As however s is certainly yp, this would imply xp s' > 0. We could thereforechoose a further index r for whichy/ - */ < * - s'.Since xr' \$', this would imply yr' < x Choosing an integer m exceed-ing both p and r, we could deduce, in view of the respective ascending and descend-ing monotony of our sequences of numbers, that a fortiori ym' < xm, which con-tradicts the hypothesis that xn y for every n. Thus s \$' is ensured.By interchanging throughout the above proof the accented and non-accentedletters, we deduce in the same manner that if xn' < yn for every n, then s' sIf then we have both xn' yn and xn yn' holding for every , then s ~ snecessarily follows. Q. E. D.

3. Irrational numbers. 29is not fixed a priori, but needs to be established by some form of definition, andit i> perfectly compatible \vith marked dissimilarity in a purely external aspect.2. The two nests I 3 ) anc* * ~ are ctlual m accordance withour present definition3. By 14, we may write e. g. (s s -\- J = s --= (s \ s), the latter symboldenoting a nest all of whose intervals ha\e both their left and their right endpomtss. In particular, f - (0 | 0) = 0.w/4. It still remains to establish but the proof is so simple that vve will notgo into it further that (cf. Footnote 23), in consequence of our definition, wehave a) a a (Footnote 24), b) a -= a' always implies a' = a, and c) a a7, a' a"involve a = a".Inequality: A. Theorem. If (xn \ yn) = s and (xnf \ yn') s' are 15two rational-valued nests, then we have s < s', if and only ifxn yn' for every //, but not xnf 5 yn for every ;/,* e- y>n < xm for <** feast one M.B. Definition. Given any two nests of intervals a = (xn \ yn) anda (xnr | yn'), then we shall say a < o-', ifxn f yn' for every ;/, but not xn' yn for every n,i. e. for at least one m, ym -- xm'.Remarks and Examples.1. It is clear that by 14 and 15 the totality of all conceivable nests is ordered.For if a and a' are any two of them, either there is equality, a a7, or, for at leastone p, we have yv <* .Vj/, implying a < a7, or finally, for at least one r, yr' < .v|fimplying a' < a. The last two cases cannot occur simultaneously, since, for mgreater than r and />, we should then have, a fortiori, v?// <. v7/1', which is impossible.Thus between a ard a' one and only one of the three relationsalways holds, and the totality of these new symbols is thus ordered by 14 and 15.2. Here again it would have to be established in all detail that the laws oforder 1 continue to hold good with the adopted definitions of equality and in-equality. Taking as model the proof in the footnote to Theorem 14, this presentsso few essential difficulties that we will not enter into it further: The laic* of orderdo, effectually all remain valid.3. In consequence of 14 and 15 we now have, therefore, for every nAn < c yn.What does this mean r It means that each of the rational numbers xn is, in ac-cordance with 14 and 15, not greater than the nest a ~ (xn \ yn). Or: if we con-24 Here it may be clearly recognised that this "law" is by no means trivial:it has indeed to be proved that with the given definition of equality every nest ofintervals is effectually "equal" to itself, that is to say that the conditions of thatdefinition are fulfilled, when the same nest is taken for both of the nests of intervalswhich we are comparing. 3. Irrational numbers. 29is not fixed a priori, but needs to be established by some form of definition, andit i> perfectly compatible \vith marked dissimilarity in a purely external aspect.2. The two nests I 3 ) anc* * ~ are ctlual m accordance withour present definition3. By 14, we may write e. g. (s s -\- J = s --= (s \ s), the latter symboldenoting a nest all of whose intervals ha\e both their left and their right endpomtss. In particular, f - (0 | 0) = 0.w/4. It still remains to establish but the proof is so simple that vve will notgo into it further that (cf. Footnote 23), in consequence of our definition, wehave a) a a (Footnote 24), b) a -= a' always implies a' = a, and c) a a7, a' a"involve a = a".Inequality: A. Theorem. If (xn \ yn) = s and (xnf \ yn') s' are 15two rational-valued nests, then we have s < s', if and only ifxn yn' for every //, but not xnf 5 yn for every ;/,* e- y>n < xm for <** feast one M.B. Definition. Given any two nests of intervals a = (xn \ yn) anda (xnr | yn'), then we shall say a < o-', ifxn f yn' for every ;/, but not xn' yn for every n,i. e. for at least one m, ym -- xm'.Remarks and Examples.1. It is clear that by 14 and 15 the totality of all conceivable nests is ordered.For if a and a' are any two of them, either there is equality, a a7, or, for at leastone p, we have yv <* .Vj/, implying a < a7, or finally, for at least one r, yr' < .v|fimplying a' < a. The last two cases cannot occur simultaneously, since, for mgreater than r and />, we should then have, a fortiori, v?// <. v7/1', which is impossible.Thus between a ard a' one and only one of the three relationsalways holds, and the totality of these new symbols is thus ordered by 14 and 15.2. Here again it would have to be established in all detail that the laws oforder 1 continue to hold good with the adopted definitions of equality and in-equality. Taking as model the proof in the footnote to Theorem 14, this presentsso few essential difficulties that we will not enter into it further: The laic* of orderdo, effectually all remain valid.3. In consequence of 14 and 15 we now have, therefore, for every nAn < c yn.What does this mean r It means that each of the rational numbers xn is, in ac-cordance with 14 and 15, not greater than the nest a ~ (xn \ yn). Or: if we con-24 Here it may be clearly recognised that this "law" is by no means trivial:it has indeed to be proved that with the given definition of equality every nest ofintervals is effectually "equal" to itself, that is to say that the conditions of thatdefinition are fulfilled, when the same nest is taken for both of the nests of intervalswhich we are comparing.

30 Chapter I. Principles of the theory of real numbers.sider any particular one of the numbers xn> say xp, and denote it for brevity by x,then we may write (see 14, Rem. 3)(v;) -) x - x - - x + fj or - (x | x)and our statement takes the form(*!*) <.!*,).We may prove it as follows. If it were not true, then for at least one r,yr < x, i. e. yr < xand so afortiori, if m is greater than r and pyym < *m.which certainly cannot be the case. In the same way we see that a < yn. Accord-ingly, a is to be regarded as lyin between xn and yn for each n, in other word*, v con-tained within the intervalJn.The fact that no other number a', besides a, can possess the same propertyis now easily proved. If in fact there were a second nest of intervals a' - (\n' \ yn')such that for every definite index /> we also had xp a' < yp, then the left handinequality means, more precisely (cf 3), that (v | vp) r (vn' | yn') and so, by 14and 15, xp yn' for every n. Since this must hold in particular for // p, wededuce x9 1 yv' for every p, which signifies, by 14 and 15, that a a'. In thesame manner the right hand inequality is seen to imply that a' j <* Thus neces-sarily a a', which was what we set out to prove.4. By 15, a is > 0, i. e. "positive", if and only if (xn \ yn) > (0 | 0), that isto say, if for some suitable index p, xv > 0. But in this case, as the .vwfs increasewith n, we have a fortiori xn for every n > p. We may therefore* say : a(vn | yn) is positive if, and only if, all the endpomts ,vw, yn are positive from andafter a definite index. The exact analogue holds of course for a < 0.5. If or > 0, and, for every n p, xn > 0, let us form a new nest (xn' \ yn')= a' by putting x x\ . . . *V-i all equal to xp, but every other xn' andyn' equal to the corresponding xn and yn. By 14, obviously a a'; and we maysay: If a is positive, then there are always nests of intervals equal to it, for whichall the endpoints of intervals are positive. The exact analogue holds for a < 0.So far then, in respect of the possibility of ordering them, our nestsof intervals may be said to vindicate their character as numbers com-pletely. It is no more difficult to establish a similar conclusion with regardto the possibilities of combining them.16. Addition: A. Theorem 2r>. If(xn \yn) and (xn'\ynf) are any two nestsof intervals, then (xn + #n'> yn + yn') w also one, and if the former are bothrational-valued and respectively = s and = s\ then the latter is also rational-valued, and determines the number s + s''.B. Definition. If (xn \ yn) a and (xnf \ yn') ~ &' are any two nestsof intervals and a" denotes the nest (xn + xn', yn + yn') deduced from them,then we write a" = a + a'and a" ts called the sum of a and a'.18 With regard to the proof, cf. footnote 23. 30 Chapter I. Principles of the theory of real numbers.sider any particular one of the numbers xn> say xp, and denote it for brevity by x,then we may write (see 14, Rem. 3)(v;) -) x - x - - x + fj or - (x | x)and our statement takes the form(*!*) <.!*,).We may prove it as follows. If it were not true, then for at least one r,yr < x, i. e. yr < xand so afortiori, if m is greater than r and pyym < *m.which certainly cannot be the case. In the same way we see that a < yn. Accord-ingly, a is to be regarded as lyin between xn and yn for each n, in other word*, v con-tained within the intervalJn.The fact that no other number a', besides a, can possess the same propertyis now easily proved. If in fact there were a second nest of intervals a' - (\n' \ yn')such that for every definite index /> we also had xp a' < yp, then the left handinequality means, more precisely (cf 3), that (v | vp) r (vn' | yn') and so, by 14and 15, xp yn' for every n. Since this must hold in particular for // p, wededuce x9 1 yv' for every p, which signifies, by 14 and 15, that a a'. In thesame manner the right hand inequality is seen to imply that a' j <* Thus neces-sarily a a', which was what we set out to prove.4. By 15, a is > 0, i. e. "positive", if and only if (xn \ yn) > (0 | 0), that isto say, if for some suitable index p, xv > 0. But in this case, as the .vwfs increasewith n, we have a fortiori xn for every n > p. We may therefore* say : a(vn | yn) is positive if, and only if, all the endpomts ,vw, yn are positive from andafter a definite index. The exact analogue holds of course for a < 0.5. If or > 0, and, for every n p, xn > 0, let us form a new nest (xn' \ yn')= a' by putting x x\ . . . *V-i all equal to xp, but every other xn' andyn' equal to the corresponding xn and yn. By 14, obviously a a'; and we maysay: If a is positive, then there are always nests of intervals equal to it, for whichall the endpoints of intervals are positive. The exact analogue holds for a < 0.So far then, in respect of the possibility of ordering them, our nestsof intervals may be said to vindicate their character as numbers com-pletely. It is no more difficult to establish a similar conclusion with regardto the possibilities of combining them.16. Addition: A. Theorem 2r>. If(xn \yn) and (xn'\ynf) are any two nestsof intervals, then (xn + #n'> yn + yn') w also one, and if the former are bothrational-valued and respectively = s and = s\ then the latter is also rational-valued, and determines the number s + s''.B. Definition. If (xn \ yn) a and (xnf \ yn') ~ &' are any two nestsof intervals and a" denotes the nest (xn + xn', yn + yn') deduced from them,then we write a" = a + a'and a" ts called the sum of a and a'.18 With regard to the proof, cf. footnote 23.

3. Irrational numbers. 31Subtraction: A. Theorem. If (xn \ yn) is a nest of intervals, then so 17.is ( yn | xn); and if the former is rational-valued s, then the latteris also rational-valued, and determines the number s.B. Definition. If a = (xn \ yn) is any nest of intervals and a' de-note the nest of intervals ( yn \ xn)t we writea' = -aand say v is the opposite of cr. By the difference oftwo nests of inter-vals we then mean the sum of the first and of the opposite of the second.Multiplication: A. Theorem. If(xn \ yn) and (x \ yn') are any two 18.positive nests of intervals, replaced, if necessary, (in accordance with15, 5) by two nests of intervals equal to them, for which all the endpointsof intervals are positive (or at least non-negative), then (xn xnr \ynyn')is also a nest of intervals; and if the former are rational-valued and respec-tively s and = s', then the latter is also rational-valued, and determines thenumber s s'.B. Definition. If (xn \ yn) a and (xnr \ ynf) a are any twopositive nests of intervals for which all the endpoints of intervals are positivewhich is no restriction, by 15, 5 and a" denote the nest (xn xn' \ynyn')derived from them, then we writea" = <T- a'and call o-" the product of a and cr'.The slight modifications which have to be made in this definition ifone or both of a and or' are negative or zero, we leave to the reader, andhenceforth consider the product of any two nests of intervals as defined.Division: A. Theorem. // (xn \ yn) is any positive nest of intervals 19.for which all endpoints of intervals are positive, (cf. 15, 5) then so is ( J;Vn xn'and if the former is rational-valued, and = s, the latter is also rational-valued, and determines the number -.B. Definition. If (xn \ yn) = a is any positive nest of intervals forwhich all endpoints are positive, and a' denote the nest (-- ), then we\yn xjwriteand say a' is the reciprocal ofa. By the quotient of a first by a secondpositive nest ofintervals we then mean theproduct ofthefirst by the reciprocalof the second.The slight modifications necessary in this definition, if a (in the onecase) or the second of the two nests of intervals (in the other) is negative, 3. Irrational numbers. 31Subtraction: A. Theorem. If (xn \ yn) is a nest of intervals, then so 17.is ( yn | xn); and if the former is rational-valued s, then the latteris also rational-valued, and determines the number s.B. Definition. If a = (xn \ yn) is any nest of intervals and a' de-note the nest of intervals ( yn \ xn)t we writea' = -aand say v is the opposite of cr. By the difference oftwo nests of inter-vals we then mean the sum of the first and of the opposite of the second.Multiplication: A. Theorem. If(xn \ yn) and (x \ yn') are any two 18.positive nests of intervals, replaced, if necessary, (in accordance with15, 5) by two nests of intervals equal to them, for which all the endpointsof intervals are positive (or at least non-negative), then (xn xnr \ynyn')is also a nest of intervals; and if the former are rational-valued and respec-tively s and = s', then the latter is also rational-valued, and determines thenumber s s'.B. Definition. If (xn \ yn) a and (xnr \ ynf) a are any twopositive nests of intervals for which all the endpoints of intervals are positivewhich is no restriction, by 15, 5 and a" denote the nest (xn xn' \ynyn')derived from them, then we writea" = <T- a'and call o-" the product of a and cr'.The slight modifications which have to be made in this definition ifone or both of a and or' are negative or zero, we leave to the reader, andhenceforth consider the product of any two nests of intervals as defined.Division: A. Theorem. // (xn \ yn) is any positive nest of intervals 19.for which all endpoints of intervals are positive, (cf. 15, 5) then so is ( J;Vn xn'and if the former is rational-valued, and = s, the latter is also rational-valued, and determines the number -.B. Definition. If (xn \ yn) = a is any positive nest of intervals forwhich all endpoints are positive, and a' denote the nest (-- ), then we\yn xjwriteand say a' is the reciprocal ofa. By the quotient of a first by a secondpositive nest ofintervals we then mean theproduct ofthefirst by the reciprocalof the second.The slight modifications necessary in this definition, if a (in the onecase) or the second of the two nests of intervals (in the other) is negative,

32 Chapter I. Principles of the theory of real numbers.we may again leave to the reader, and henceforth consider the quotientof any two nests of intervals of which the second is different from 0, asdefined. If (xn \ yn) a = 0, then the above method fails to producea "reciprocal" nest: division by is here also impossible.The result of the preceding considerations is thus as follows: Bydefinitions 14 to 19, the system of all nests of intervals is ordered in thesense of 4, 1, and admits of having its elements combined by the fourprocesses in the sense of 4, 2. In consequence of the theorems 14 to 19,as stated in each case, this system possesses further, in the aggregate ofall rational-valued nests, a sub-system, similar and isomorphous to thesystem of rational numbers, in the sense of 4, 3. It remains to show thatthe system also fulfils the Postulate of Eudoxus. But if (xn \ yn) = a and(xn I yn) ~ v are any two positive nests for which all endpoints of in-tervals are positive (cf. 15, 5), let xm and ymf be a definite pair of theseendpoints; the theorem of Eudoxus ensures the existence of an integerp, for which p xm > ym', and the nest p a, or (p xn \ p yn), in accordancewith 15, is then effectually > a'.The next step should be to establish in all detail (cf. 14, 4 and 15,2) that the four processes defined in 16 to 19 for nests of intervals obeythe fundamental laws 2. This again offers not the slightest difficulty andwe will accordingly spare ourselves the trouble of setting it forth 26. TheFundamental Laws of Arithmetic, and thereby the entire body of rules validin calculations with rational numbers, effectually retain their validity in thenew system.By this, our nests of intervals have finally proved themselves inevery respect to be numbers in the sense of 4: The system of allnests of intervals is a number-system, the nests themselves are numbers 27.26 As regards addition, for instance, it should be shown that:a) Addition can always be carried out. (This follows at once from the defini-tion.) b) The result is unique; i. e. a a', T = T' (in the sense of 14) implya -f- r a1 \- r', if the sums are formed in accordance with 16 and the testfor equality carried out in accordance with 14. In the corresponding sense, it shouldbe shown further thatc) a + T = T -f- a always.d) fe + a) + T = g -|- (o- + T) always.e) a < a' implies a -\- T < a' 4* T always.And similarly for the other three processes of combination.27 Whether, as above, we regard nests of intervals as themselves numbers,or imagine some hypothetical entity introduced, which belongs to all the intervalsJn (cf. 15, 3) and thus appears to be in a special sense the number enclosed bythe nest of intervals and, consequently, the common element in all equal neststhis at bottom is a pure matter of taste and makes no essential difference. Theequality a -- (xn \ yn) we may, at any rate, from now on, (cf. 13, footnote 19) readindifferently either as "a is an abbreviated notation for the nest of intervals (xn \ yn)"9or as "a is the number defined by the nest of intervals (xn \ yn)". 32 Chapter I. Principles of the theory of real numbers.we may again leave to the reader, and henceforth consider the quotientof any two nests of intervals of which the second is different from 0, asdefined. If (xn \ yn) a = 0, then the above method fails to producea "reciprocal" nest: division by is here also impossible.The result of the preceding considerations is thus as follows: Bydefinitions 14 to 19, the system of all nests of intervals is ordered in thesense of 4, 1, and admits of having its elements combined by the fourprocesses in the sense of 4, 2. In consequence of the theorems 14 to 19,as stated in each case, this system possesses further, in the aggregate ofall rational-valued nests, a sub-system, similar and isomorphous to thesystem of rational numbers, in the sense of 4, 3. It remains to show thatthe system also fulfils the Postulate of Eudoxus. But if (xn \ yn) = a and(xn I yn) ~ v are any two positive nests for which all endpoints of in-tervals are positive (cf. 15, 5), let xm and ymf be a definite pair of theseendpoints; the theorem of Eudoxus ensures the existence of an integerp, for which p xm > ym', and the nest p a, or (p xn \ p yn), in accordancewith 15, is then effectually > a'.The next step should be to establish in all detail (cf. 14, 4 and 15,2) that the four processes defined in 16 to 19 for nests of intervals obeythe fundamental laws 2. This again offers not the slightest difficulty andwe will accordingly spare ourselves the trouble of setting it forth 26. TheFundamental Laws of Arithmetic, and thereby the entire body of rules validin calculations with rational numbers, effectually retain their validity in thenew system.By this, our nests of intervals have finally proved themselves inevery respect to be numbers in the sense of 4: The system of allnests of intervals is a number-system, the nests themselves are numbers 27.26 As regards addition, for instance, it should be shown that:a) Addition can always be carried out. (This follows at once from the defini-tion.) b) The result is unique; i. e. a a', T = T' (in the sense of 14) implya -f- r a1 \- r', if the sums are formed in accordance with 16 and the testfor equality carried out in accordance with 14. In the corresponding sense, it shouldbe shown further thatc) a + T = T -f- a always.d) fe + a) + T = g -|- (o- + T) always.e) a < a' implies a -\- T < a' 4* T always.And similarly for the other three processes of combination.27 Whether, as above, we regard nests of intervals as themselves numbers,or imagine some hypothetical entity introduced, which belongs to all the intervalsJn (cf. 15, 3) and thus appears to be in a special sense the number enclosed bythe nest of intervals and, consequently, the common element in all equal neststhis at bottom is a pure matter of taste and makes no essential difference. Theequality a -- (xn \ yn) we may, at any rate, from now on, (cf. 13, footnote 19) readindifferently either as "a is an abbreviated notation for the nest of intervals (xn \ yn)"9or as "a is the number defined by the nest of intervals (xn \ yn)".

4. Completeness and uniqueness of the system of real numbers. 33This system we shall henceforth designate as the system of real numbers.It is an extension of the system of rational numbers, in the sense inwhich the expression was used on p. 1, since there are not only rational-valued nests but also others besides.This system of real numbers is in one-one correspondence withthe whole aggregate of points of the number-axis. For, on the strengthof the considerations set forth on p. 24, 25, we can immediately assertthat to every nest of intervals a corresponds one and only one point,namely that common to all the intervals /n, which on account of the Cantor-Dedekind axiom is considered in each case as existing. Also two nests ofintervals a and cr' have, corresponding to them, one and the same point,if and only if they are equal, in the sense of 14. To each number cr (thatis to say, to all nests of intervals equal to each other) corresponds exactlyone point, and to each point exactly one number. The point correspondingin this manner to a particular number is called its image (or representative)point, and we may now assert that the system ofreal numbers can be uniquelyand reversibly represented by the points of a straight line.

4. Completeness and uniqueness of the system of realnumbers.Two last doubts remain to be dispelled 28 : Our starting point in3 was the fact that the system of rational numbers, by reason of its"gaps", could not satisfy all demands which would appear in the courseof the elementary processes of calculation. Our newly created number-system the system Z as we will call it for brevity is in this respectcertainly more efficient. E. g. it contains 29 a number a for which cr2 2.Yet the possibility is not excluded that the new system may still showgaps like the old, or that in some other way it may be susceptible of stillfurther extension.Accordingly, we raise the following question: Is it conceivable thata system Z, recognizable as a number-system in the sense of 4, and con-taining all the elements of the system Z, should also contain additionalelements distinct from these? * 28 Cf. the closing words of the Introduction (p. 2).29 For if CT = (xn | yn) denote the nest of intervals constructed on p. 20in connection with the equation A?3 = 2, then by 18 we have a (xn2 \ yn*). Since,however, #n2 < 2 and yn2 > 2, it follows that a2 = 2. Q. E. D.80 I. e. Z would have to represent an extension of Z in the same sense as Zitself represents an extension of the system of rational numbers.

(Parte 1 de 2)