PureMath.pdf

# It is natural to regard length as a magnitude capable

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it is natural to regard length as a magnitude capable of sign, positive if the length is measured in one direction (that of A 0 A 1 ), and negative if measured in the other, so that AB = - BA ; and to take as the point representing r the point A - s such that A 0 A - s = - A - s A 0 = - A 0 A s .

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[I : 3] REAL VARIABLES 3 We thus obtain a point A r on the line corresponding to every rational value of r , positive or negative, and such that A 0 A r = r · A 0 A 1 ; and if, as is natural, we take A 0 A 1 as our unit of length, and write A 0 A 1 = 1, then we have A 0 A r = r. We shall call the points A r the rational points of the line. 3. Irrational numbers. If the reader will mark off on the line all the points corresponding to the rational numbers whose denominators are 1, 2, 3 , . . . in succession, he will readily convince himself that he can cover the line with rational points as closely as he likes. We can state this more precisely as follows: if we take any segment BC on Λ , we can find as many rational points as we please on BC . Suppose, for example, that BC falls within the segment A 1 A 2 . It is evident that if we choose a positive integer k so that k · BC > 1 , * (1) and divide A 1 A 2 into k equal parts, then at least one of the points of division (say P ) must fall inside BC , without coinciding with either B or C . For if this were not so, BC would be entirely included in one of the k parts into which A 1 A 2 has been divided, which contradicts the supposition (1). But P obviously corresponds to a rational number whose denominator is k . Thus at least one rational point P lies between B and C . But then we can find another such point Q between B and P , another between B and Q , and so on indefinitely; i.e. , as we asserted above, we can find as many as we please. We may express this by saying that BC includes infinitely many rational points. * The assumption that this is possible is equivalent to the assumption of what is known as the Axiom of Archimedes.
[I : 3] REAL VARIABLES 4 The meaning of such phrases as ‘ infinitely many ’ or ‘ an infinity of ’, in such sentences as ‘ BC includes infinitely many rational points’ or ‘there are an infinity of rational points on BC ’ or ‘there are an infinity of positive integers’, will be considered more closely in Ch. IV . The assertion ‘there are an infinity of positive integers’ means ‘given any positive integer n , however large, we can find more than n positive integers’. This is plainly true whatever n may be, e.g. for n = 100 , 000 or 100 , 000 , 000. The assertion means exactly the same as ‘we can find as many positive integers as we please ’. The reader will easily convince himself of the truth of the following assertion, which is substantially equivalent to what was proved in the second paragraph of this section: given any rational number r , and any positive integer n , we can find another rational number lying on either side of r and differing from r by less than 1 /n . It is merely to express this differently to say that we can find a rational number lying on either side of r and differing from r by as little as

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• Fall '14

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