The Order Relation in is defined by

P1) OEN

3. The Order Relation in is defined by

ab ifandonlyif b—aEN.

Theorem. The ring Z of integers is linearly (completely) ordered by the relation <. For all a, b, c E Z the relation a 6 implies a + c 6 + c and,

when c> 0, a.c<bc as well.

The natural numbers other than zero are thus the integers > 0, the so- called positive integers. A number a is said to be negative whenever —a is positive.

Remarks. Every commutative ring R expressible as a disjoint union R =

—P U (0) U P where P is additively and multiplicatively closed, can be totally ordered by the relation a 6 if 6 — a PU {0).

Historically, it was also DEDEKIND who introduced the idea of defining integers by pairs from N xN. In a letter from the 82-year-old mathematician written in 1913 to a former student, DEDEKIND ([10], p. 490) describes an

extension of the domain N of natural numbers to the domain G of the

integers. LANDAU [18] first constructs the rational numbers 0 from

and then extends this set by means of the negative rational numbers, to the field Q (see §4) obtaining Z as a subring of Q.

THE RATIONAL NUMBERS

1. Historical. Division, as the inverse of multiplication, cannot be done without restriction in the domain of integers. Fractions, which make division always possible, were already considered in early times. They were never surrounded by such mystery as were the negative numbers, which were thought of as being in some never-never land below "nothing," or the irrational and imaginary numbers, which we still have to discuss. The first systematic treatment of rationals is found in Book VII of EUCLID'S Elements, which deals with the ratios of natural numbers. The idea, which is so familiar to us, of interpreting ratios as fractions and of extending in this way the domain of whole numbers first arises in comparatively modern times. The first theoretical investigations stem from the nineteenth century.

BOLZANO [5J in a posthumously published paper entitled "Reine Zahien- lehre" developed a theory of rational numbers, and in fact a theory of those sets of numbers that are closed with respect to the four elementary arithmetic operations. One also finds, in a paper by OHM [22] (the brother of the famous physicist) an intention to define the rational numbers "solely through the basic truths relating to addition, substxaction, multiplication and division."

Their foremost consideration was therefore the investigation of certain arithmetical relationships, and not a philosophical question about the na- ture of number. Finally, with HANKEL ([13], p. 2), in his Theorie ^{dei' corn-}

plezen Zahiensysieme of 1867, it comes down to this: The laws of these operations determine "the system of conditions ... ^which are necessary and sufficient to define the operation formally." Apart from the rational numbers, the notion of a field (as a concept, even if not yet under this name) had also been discussed in the writings of ABE!.. and GAL0Is, where, for example, a root of an equation is adjoined to the rationals and an investigation is made of all possible expressions that can be formed from it by means of the four operations, addition, subtraction, multiplication and division. KRONECKER in 1853 speaks in his theory of algebraic quantities of

"domains of rationality" (KRONECKER [17], §1), and DEDEKIND, at first of

"rational domains" and finally of "fields" in the case of real and complex numbers (DEDEKIND (121, p. 224). Number fields were also investigated by WEBER [30] and HILBERT [141. In 1910 STEINITZ [261 gave an abstract def- inition of this fundamental algebraic concept. STEINITZ also brought out clearly the fact that behind this extension of the integers to the rational numbers there lies a general algebraic construction, namely, that of the

embedding of an integral domain in a field by the formation of fractions.

2. The Field Q. Following the example of WEBER in his Lehrbuch der Al- gebr'ti of 1895, we shall introduce fractions as equivalence classes of integers, and guided by the relation

ifandonlyif ad=bc,

we start from the equivalence relation defined on Z x (Z \ {O)) by (a, 6) (c, d) if and only if ad = ^bc.

These definitions are independent of the particular choice of representa-

tives. In LANDAU [18], Chapter 2, is given a detailed proof of the Theorem. The set Q of rational numbers, with the addition and multiplication defined above, constitutes a field.

Z is mapped isomorphically on the subring i(Z) C Q by the mapping

Q, a

Z is usually identified with e(Z). The field Q is the smallest field containing Z as a subring.

3. The Ordering of Q. A fraction a/b is said to be positive if a, b are

both positive or both negative. The set P of positive fractions is closed

with respect to the operations + and

.. Q is expressible as a union of disjoint sets —P U (0) U P. As in the remark in 3.3 a total order relation on Q can be defined by r s if and only ifS— r E Pu(0) which coincides with the order on Z defined in 3.3.

The order relation in Q is Archimed eon, that is, for all positive rational numbers r,s E Q there exists a natural number n with .c < n r. To prove

this, we write a =

p/h

and r =

q/h as fractions whose numerators and denominators are natural numbers and with a common denominator h.

The truth of the statement then follows as soon as it has been proved that

p < n .

^q for natural numbers > 0. The latter can be demonstrated for a

fixed q 1 by induction over p =

1,2,... A noteworthy property, which distinguishes the field Q from the ring of integers Z is its density: for all

r,s EQ with r < s, a t E Q can always be found such that r < t <8.

One

can, for example, choose the arithmetic mean I := +s).

REFERENCES

[1] ARISTOTELES: Metaphysik, Aristotelis Opera, ed. I. Bekker, Berlin 1831, repr. Darmstadt 1960

[2J ARISTOTELES: Phy8ik, Aristotelis Opera, ed. 1. Bekker, Berlin 1831, repr. Darmstadt 1960

[3] BECKER, 0.: Grundlagen der Mathem. in geschichtlicher Entwick- lung, Freiburg/Munchen 1954, 21964, Frankfurt 1975

[41 BOLZANO, B.: Paradoxien des Unendlichen, ed. F. Leipzig 1851, ^Berlin 21889, ed. A. Höfler, Leipzig 1920, mit Einl., ^{Anm., Reg.}

u. Bibliographie ed. B. van Rootselaar, Hamburg 1955, ²¹⁹⁷⁵ [5) BOLZANO, B.: Reine Zahlenlehre, in: B. Boizano —Gesamtausgabe

(eds. E. Winter, J. Berg, F. Kambartel, J. B. v. Rootselaar), Reihe II NachIall, A. Nachgelassene Schriften, Bd. 8 Grö8enlehre I!, Reine Zahlenlehre, Stuttgart/Bad Cannstatt 1976

[6] BOURBAKI, N.: Elements de mathématique, Paris, 1939 and later [7] BRUINS, E.M., RUTrEN, M.: Textes de Suse, Mém-

oires de Ia Mission Archéologique en Iran, Tome 34, Paris 1961 [8] CANTOR, G.: Gesam. Abh. mathem. u. philos. Inhalts, Berlin 1932,

repr. Berlin 1980

[9] DEDEKIND, R.: Was sind und was sollen die Zahlen? Braunschweig 1888, 101965, repr. 1969

[10] DEDEKIND, R.: Mathem. Werke Bd. 3, Braunschweig 1932, repr.New York 1969

[11] FREGE, G.: Die Grundlagen der Arithmetik. Elne logisch mathema- tische Untersuchung über den Begriff der Zahi, Breslau 1884, repr.

Darmstadt/Hildesheim 1961

[12] GAUSS, C.F.: Werke Bd. 3, Göttingen 1876

[13] HANKEL, H.: Theorie der complexen Zahlensysteme, Leipzig 1867 [14] HILBERT, D.: Uber den Zahibegriff, in: Jahreaber. d. Deut8chen Math.

Verein. 1900, 180—184

[15] JUSCHKEWITSCH, A.P.: Geschichte der Mathematik im Mittelalter, dt. Leipzig 1964

[16] KRONECKER, L.: Uber den Zahlbegriff, in: Journ. f. d. reine u. angew.

Mathem. 101 1887, 339, in: Math. Werke Bd. 3, Leipzig 1899/1931, repr. New York 1968, 249—274

[17] KRONECKER, L.: Grundzüge einer arithmetischen Theorie der alge- braischen Grö6en, in: J. f. d. reine u. angew. Mathem. 1882, 1—122, in: Math. Werke Sd. 2, Leipzig 1897, repr. New York 1968, 237—287 118] LANDAU, E.: Grundlagen der Analysis, Leipzig 1930, repr. Darmstadt

1963

[19] LEpsius, R.: Uber eine Hieroglyphiache Inschrift am Tempel in Edfu, in: Abh. d. Kgl. Akad. d. Wiss., Berlin 1855, 69—111

[20] NEUOEBAUER, 0.: Mathem. Keilschrifttexte, Quellen u. Studien A3, Berlin I, 111935, III 1937

[21] NEUMANN, J.v.: Zur Einführung der transfiniten Zahlen, in: Acta Szeged 11923, 199—202, repr. in: A.H. Taub (ed.), Collected Works, Oxford/London/Paris Bd. 11961, 24—33

[22] OHM, M.: Die reine Elementarmathematik Bd. 1, Berlin 21834 [23] PAPYRUS RIHND, (Hrsg. A. Eisenlohr) Leipzig 1877; A.B. Chace, The

Rhind Mathem. Papyrus, Oberlin 11927, 111929

(24] PEANO, G.: Arithmetices principia nova exposita, in: Opere scelte Bd. II, Horn 1958, 20—55

[25] RUSSELL, B.: The principles of mathematics, London 1903,

[26] STE!NITZ, E.: Algebraische Theorie der Körper, in: 3. f. d. reine u.

angew. Math. 137 1910, 167—309

[27] STIFEL, M.: Arithmetica integra, Nurnberg 1544

[28] STRUWE, W.W.: Papyrus des staatl. Museums der schönen Künste in Moskau, Quellen u. Studien Al 1930

[29) TROPFKE, J.: Geschichte der Elementarmathematik, Bd. I Arith- metik und Algebra, vollst. neu bearb. von H. Gericke, K. Reich u. K.

Vogel, Berlin

[30] WEBER, H.: Lehrbuch der Algebra Bd. 11895, repr. der 3. AufI. New York 1961

Real Numbers

K. Mainzer

&' EIVCLt cYUVEXèc ötUV wore ICaL re

olç icul aiijw(vct

(ARISTOTLE, Physics 227a, 11—12).

[1 call it holding together if it is the same and a single thing that becomes the boundary for each of the parts to which they cling and, as the word signifies, it is kept together.]

Continuum est totum cuius duse quaevis partes cointegrantes (sen quac simul sumtae toti coincidunt) habent aliquid com- mune, ...saltem habent communem terminum)

(G.W. LEtBNIz, Mathem. Schr. VII, 284).

(A continuum is a whole when any two component parts thereof (or more precisely any two parts which together make

up the whole) have something in common, ... ^at the very least they have a common boundary.]

Zerfallen alle Punkte der Geraden in zwei Klassen von dci Art, dafi jeder Punkt dci ersten Klasse links von jedem Punkt dci sweiten Klasse liegt, so existiert em und nur em Punkt, weicher diese Einteilung slier Punkte in zwei Kiassen, diese Zerschnci- dung dci Geraden in zwei Stücke, hervorbringt (R. DEDEKIND, Stetigkeit und irrationale Zahien, Braunschweig 1872, 10).

(If the points of a line are divided into two classes, in such a way that each point of the first class lies to the left of every point of the second class, then there exists one and only one point of division which produces this particular subdivision into two classes, this cutting of the line into two parts.]

§1. HISTORICAL

1. HIPPASUS and the Pentagon. When today we define the real numbers as elements of a completely ordered field, we tend to forget the mag- nitude of the intellectual and philosophical crisis brought about by the discovery that there were things outside the grasp of the rational numbers.

Indeed, if we can trust later legends, the discoverer incurred the wrath of the Gods. We mean of course the discovery ascribed to the 5th century

B.C. Pythagorean, HIPPASUS of METAPONT, that there are line segments whose ratios are incommensurable. The discovery is said to have caused a great shock in Pythagorean circles because it finally called into question one of the basic tenets of their philosophy, that everything was expressible in terms of whole numbers.

To understand the effects of thi8 crisis, one has to remember that the Pythagoreans were not only active as a highly influential mathematical school, who were the first to raise the requirement for exact mathematical science and who insisted on a strict education in arithmetic, geometry, astronomy and music for their members, but that in addition to all this they pledged themselves to an orderly way of life. Until the uprising of 445 BC, they had been a dominant force throughout Southern Italy. In this political turmoil, HIPPASUS is presumed to have played an important role (see IAMB[,ICHUS [14), p. 77, 6f; also FRITz [10], HEu.ER [11)).

The treatment of ratios of line-segments had come out of traditionally employed practices in measurement. A segment a of a line had traditionally been measured by laying along the line unit, measures e, one after the other, along the line, as many times as were necessary:

a=e+1-e=me.

m times

Two segments a0 and a1 are said to be commensurable if they can both be measured, in this sense, with the same unit of measurement e, so that

00 =

m e and = n .

^e with in, ⁿ being two natural ^numbers. In this case the ratio 00 ^: a1 of the line segments is equal to the ratio m: n of two natural numbers.

The method of finding a common measure of two line segments ao, 01 ^had already been practiced, before the days of Greek philosophy and 8CieflCe, by craftsmen, by a process of alternate "taking away." EucLiD described the process in his Elements which now goes by the name of the Euclidean algorithm. The smaller segment 01 is taken away from the larger segment

00 ^as many times as possible, until the residue left is smaller than a1, so that, if 02 is this residue, then

ao=nlal+02

^with ^02<01.

One then continues in the same way:

01 = fl202 + with 03 <02, 02 = fl303+ ⁰⁴ ^with⁰⁴ ^<03,

If00 and have a common measure, the process comes to an end after a finite number of steps, so that there is a k with ak—i = fl*ak, ^and is a common measure of GO and

At first, it was probably felt intuitively that thi8 process would always terminate, and that therefore there would always be a common measure.

In modern language, however, all that this procedure shows 18 that every ratio of line segments can be developed as a continued fraction

at = ni + ^{: 01}

1 1

=nl+ _01:02 =nl+

1 1

=nl+ =...=flt+

which is finite when 00 and a1 is commensurable.

The badge or symbol of their order used by the Pythagoreans was the Penta gram, which still retained its magical potency in mediaeval astrology and according to legend was used by Faust to exorcize Mephistopheles.

There is good reason to believe that 1-IIPPASUS by working from this symbol found that two of the lines therein were incommensurable (see IAMBL.ICHUS

[151, p. 132, 11—12; for references to the sources see FRITZ [10], HELLER [11), TROPFKE [23)).

To see this, we begin with the regular pentagon ABCDE in which all five diagonals have been drawn. The diagonals intersect to form a smaller regular pentagon A'JJ'C'D'E' in the middle. Because of symmetry, each side of a regular pentagon is parallel to one of the diagonals. Thus, the triangle AED and BE'C have their corresponding sides parallel and are therefore similar, so that AD: AE =

BC: BE'. Now BE' =

BD— BC,

since BC =

AE = DE',

as EA is parallel to DB, and DE is parallel to

AC. Consequently for any regular pentagon

diagonal: side = side:(diagonal — side).

If we denote the diagonal by a0, the side by and their difference by a2 = 00—01, then 00 : a1 = 01 : a2 and in particular <a1. If we now form the difference 03 =a1 —a2, we obtain the same equation between the ratios : a2 = : 03, and in particular 03 < 02. The process can dearly be continued indefinitely:

02=00—01,

a3=al—a2,

ao:aj=a1 :a2=a2:aa=a3:04=•

The Euclidean algorithm for 00 and namely a0 = ¹ ^.al + 02, Cj = I . + 03, 02 = 1.03+ 04

never terminates, thus the side ai and diagonal 00 of the pentagon are not commensurable.

We obtain for the ratio, the continued fraction

a001l+

¹¹

1+

1+1+...

It follows from aO a1 = 01 (00 —ai) that ao 01 = + ^This ratio is known as the goldensection. Thefact that the Euclidean algorithm never terminates can be seen at once from the diagram, which shows that each pentagon always has a smaller one within it so that there is an infinity of pentagons, whose sides are of length 01, a3, a5,... and diagonals of length a2, a4, 06,... respectively.

2. EUDOXUS and the Theory of Proportion. The Babylonians

Dalam dokumen Numbers (Halaman 40-49)