SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface
SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface
SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface
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6 I. R. <strong>Shafarevich</strong>The rst two statements follow directly from the denition of even numbers: ifk =2l, then no matter whether m is even or odd, we always have km =2lm whichis an even number. However, the proof of the last statement requires Lemma 1.Let k 1 and k 2 be two oddnumbers. By Lemma 1 we can write them in the formk 1 = 2s 1 +1, k 2 = 2s 2 +1, where s 1 and s 2 are natural numbers or 0. Thenk 1 k 2 =(2s 1 +1)(2s 2 +1) = 4s 1 s 2 +2s 1 +2s 2 +1=2s+1 where s =2s 1 s 2 +s 1 +s 2 .As we know, any number of the form 2s + 1 is odd and so k 1 k 2 is odd.Notice the following particular case of Lemma 2: the square of an odd numberis odd.Now we can easily nish the proof of Theorem 2. Suppose that the equality(1) is true where m and n are positive integers, relatively prime. If n is odd, thenby Lemma 2, n 2 is also odd, whereas from (1) follows that n 2 is even. Hence, n iseven and can be written in the form n =2s. But m and n are relatively prime, andso m must be odd (otherwise they would have common factor 2). Substituting theexpression for n into (1) and cancelling by 2,we obtainm 2 =2s 2 i.e. the square of the odd number m is even, whichisincontradiction with Lemma 2.Theorem 2, and hence Theorem 1, are proved.Under the supposition that the result of measuring the length of a segment(with respect to a given unit segment) is a number and that the square root of apositive number is a number, we can look at Theorems 1 and 2 from a dierentpoint of view. These theorems assert that in the case of the diagonal of a square orin the case of p 2, these numbers are not rational, that is to say they are irrational.This is the simplest example of an irrational number. All the numbers, rational andirrational, comprise real numbers. In one of the following chapters we shall give amore precise logical approach to the concept of a real number, and we shall use itin accordance with the school teaching of mathematics, that is to say we shall notinsist too much on the logical foundations.Why didsuch a simple and at the same time important fact, the existence ofirrational numbers, have towait so long to be discovered? The answer is simple|because for all practical purposes, we can take, for instance, p 2tobea rationalnumber. In fact, we haveTHEOREM 3. No matter how small is a given number ", itispossible to ndarational number a = m n , such that a