The Irrationality of and Its Continued Fraction Expansion

ARGAND

6. The Irrationality of and Its Continued Fraction Expansion

The statement that the circumference and diameter of a circle are incom- mensurable had already been asserted by ARISTOTLE, but the first proof of the irrationality of was given in 1766 by Johann Heinrich LAMBERT (1728-1777) in his VorlIufige Kenninisse fur' die, so die Quadratur and Rectification dci C*rculs suchen (Werke 1, 194—212), a sort of manual for would-be circle squarers, written in highly original language. The proof was based on the theory of continued fractions. He found the infinite continued fraction

tanz =

z 1—

7—...

and deduced from it the irrationality of tan(z) for all real rational argu- ments z 0, and in particular the result that Qsince tan = 1. How- ever LAMBER.T'S proof is not completely rigorous because it lacks a lemma on the irrationality of certain infinite continued fractions (having particularly good convergence). This lemma was proved in 1806 by Adrien-Marie

LEGENDRE (1752-1833) in the 6th edition of his Elements de geometric,

Note JV. LECENDRE also shows there that is irrational.

LAMBERT'S continued fraction for is, in fact, irrational, by virtue of LEGENDRE'S lemma, for all q E Q, q 0, and therefore R

q E Q ^is impossible, because tan 11 = 0.

0

LAMBERT's and LEGENDRE'S work on the subject is readily accessible

in the article by RuDlo mentioned in the introduction to this chapter.

Perhaps the simplest modern proof of the irrationality of runs as follows.

We introduce the polynomial pn(Z) := ^— xv', n 1, and begin by noting that

1) 0 <p,,(z) <

for 0 < x < 1; E Z for all ii.

2) For := ^—

+

^— +

(—1) (2n)(a,)) we have

— = b IL

The inequalities in 1) are trivial; the statement that is an integer follows, by induction on n, from the equation

=

⁽¹ the statement 2) is easily proved by first noting that the derivative on the left

is simply + ^sin

Ifnow were rational, say .r2 = a/bwith a, 6 1 natural numbers, then the values of and formed with this 6 would, by 1), be rational integers. Consequently we should have, by 2), since =

j pn(x)

^sin ^{dx =} ^— ^con

P,(1)

On the other hand, since 0< sin irx < 1 for 0< x < 1, we deduce from 1) that:

p,(x)sinTzdx<w—1<1

forlargen,

o Ti.

since = ⁰ for every a IR, in view of the convergence of the exponential series.

Thus, for all large enough n, we should have

0 < + < I in contradiction to + E Z.

0

This proof is based on an idea of I. NIvEN: A simple proof that is irrational, in Bull. Amer. Math. Soc., 53, 1947, 509. The extension to is due to Y. IWAMOTO: A proofthat x2 is irrational, in I. Osaka Inst. Sci.

Tech., 1, 1949, 147—148. The reader should also compare the proof given in the book by G.H. HARDY and E.M. WRIGHT An introduction to the theory of numbers, ^3rdedn., Oxford, Clarendon Press, 1954, especially p. 47. Also compare a paper by J. "A simple proof of the irrationality of ir4,"

Amer. Math. Monthly 93 (1986), 374—375.

0

The following two continued fractions among others exist for the number

it I

4 ¹²

7+

2+

₅₂ 15+

2+272

The expansion on the Left was found in 1656 by Lord BROUNCKER (1620—

1684), the first President of the Royal Society, by transforming WALLIS'S product. EULER, in §369 of his used the LEIBNIZ series instead.

The expansion on the right is the so-called regular continued fraction for

Every positive real number has a unique representation in the form of a regular continued fraction, in which only positive integers appear, and in which all the "numerators in the denominators" are 1. There is no known law governing the successive terms in the regular continued fraction for Thesuccessive integers 3, 7, 15, 1, 292,.. ^.shown are simply calculated from the decimal representation of w by using the continued fraction algorithm.

BROUNCKER'S continued fraction has a very poor convergence. Regu- lar continued fractions on the other hand have excellent convergence. The first few convergents to w, for example, give the approximations 3,

the approximation in 1.3 given by Zu CH0NO-Zm is thus the fourth convergent. For further details on the relation between w and continued fractions we refer the reader to the two volume work by 0. PERRON:

Die Ldire von den KeUenbriichen, Stuttgart, Teubner Verlag, 3rd edn., 1954—1957. The approximation of and ,r2 by rational numbers p/q has some fundamental limitations. For example, a result of M. MIGNOTTE, Ap- ralionelles de quelgucs aufres nombres, Bull. Soc. Math.

France, Mem. 37, 121—132 shows that:

p ¹ 2 p ¹

Wq >q206

^for

q>1,

^—— ^for

7. Transcendence of

^The problem of constructing a square equal in area to a given circle by means of a ruler and compass construction had already engaged the attention of the ancient Greeks. This is the problem usually referred to as "squaring the circle." It is shown in Algebra that a real number can be constructed by these means if and only if it lies in a finite extension of the field Qformed by successive adjunction of square roots. In particular therefore the numbers constructible by ruler and compass are at most those which are algebraic (over Q), thatis to say which annihilate a polynomial p E Q[X] \ {O}.

The problem of squaring the circle is equivalent to the question of whether is constructible by ruler and compass. In view of the foregoing remarks, would then have to be an algebraic number. EULER, LAMBERT and LEGEN- DRE were already of the opinion that this is not so. Thus LEGENDRE at the end of his paper on the irrationality of says quite clearly (see RUDIO,

p. 59) "It is probable that i is not even contained among the algebraic

irrationals, in other words it cannot be the root of an algebraic equation with a finite number of terms, and rational coefficients. However it seems difficult to prove this theorem rigorously."

Numbers which are not algebraic are called "transcendental" (omnem rationem transcendunt). Thus LEGENDRE in 1806 conjectured that is

transcendental. This was an extraordinarily bold conjecture, because at

that time, no one even knew that there were such things as transcendental numbers (in contrast to irrational numbers, such as, for example, whose existence had been known to the Greeks). It was not until 1844 that Joseph LIOUVILL.E (1809—1882) first proved that all (irrational) numbers having

"very good" rational approximations, such as for example the number 10.hI +

+

^10_3! = 0.1100010000...

are transcendental. In 1874 Georg CANTOR (1845—1918) gave his sensa- tional proof, using an enumerative argument, that there are only countably many algebraic numbers, but uncountably many transcendental numbers (see on this, for example, 0. PERRON: Irrntionalzahlen, Berlin, de Gruyter,

1960, 174—181).

The great breakthrough in the theory of transcendental numbers came in 1873 when the French mathematician Charles HERMI'rE (1822—1901) devel- oped methods by which he was able to prove that the number e is transcendental. By a refinement of HERMITE'S argument the German mathematician Carl Louis Ferdinand von LINDEMANN (1852—1939) who had taught and HURWITZ in Königsberg, and subsequently went to Munich in 1893, proved in 1882 in a short paper "Uber die Zahi ^published in

Math. Ann., 20, 213—225, his famous theorem that:

is transcendental.

In this way the thousand-year-old question about the quadrature of the circle was finallyanswered in the negative. Oblivious to this fact, amateur mathematicians still try to tackle this problem as they did before; they often find good approximation processes, and in most cases it is difficult to convince them that their "solution" does not contradict the transcenden-

talityofir. 0

LINDEMANN himselfseems to have been quite surprised at having been able to solve a thousand-year-old problem. Thus we read in the introduction to his paper (p. 213): "Man wird sonach die Unmöglichkeit der Quadratur des Kreises darthun, wenn man nachweist, dass die ZahI iiberhaupt nicht Wurzel einer algebraischen Gleichung irgend welchen Grades mit rutionalen Coefficienten scm kann. Den dafür nothigen Beweis zu erbringen, 1st im Folgenden versucht worden." [The impossibility of the quadrature of the circle will thus have been established when one has proved that the number

can never be the root of any algebraic equation of any degree with rational coefficients. We seek to prove this in the following pages.] The propositions of HERMITE and LINDEMANN are included in the following general theorem.

The LINDEMANN-WEIERSTRASS Theorem (see WEIERSTRASS:

Zu Lindemann's Abhandlung: "Uber die Ludolph'sche Zahi," in Math.

Werke, 2, 341—462, particularly 360—361). Let c1,. .. ,c,, E C be pairwise distinct algebraic numbers belonging to C. Then there exists no equation

aiedl + ^•

= 0 in

which are algebraic numbers and are not all equal to zero.

If, in this theorem, we put n := ^2, := c, c2 :=0, we obtain the result:

for every algebraic number c E C* the number a := ec is transcendental.

The case c :=

¹ proves the transcendence of e, and since 1 = e2ti the

transcendence of w follows as well.

0

Meanwhile it is also known that ew = is transcendental (GELFOND, 1929). As for the number w nothing is known for certain, and on the whole our knowledge about transcendental numbers is still extremely limited. As e is transcendental, the numbers ew and e + w cannot both be algebraic;

but it is still not known whether ew or e + w is rational.

6 The p-Adic Numbers

Dalam dokumen Numbers (Halaman 168-174)