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a course of pure mathematics

by g h godfrey harold hardy

Author: g h godfrey harold hardy

Science » Mathematics

Added by: Sue (Flushing...

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This book is a perfect example of classic scientific masterpiece. The textbook is crystal clear, well organized, and thorough. "A Course of Pure Mathematics" is great for teaching yourself mathematics. Hardy is a sincerely elegant, fine, incisive thinker, and his unbounded enthusiasm for his subject, duly controlled by British understatement, shines through every page. He conveys the irresistible, almost addictive quality of math. This book is a little quaint. The terminology used by the author is a bit out of date, e.g. ‘sequences’ are ‘functions of a positive integral variable’. But that’s the thing that makes "A Course of Pure Mathematics" so charming. The book contains a lot of examples. This is a great book for teachers, tutors, and certainly students.

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...n. Hence n=\q 2, where A. Is an integer. But this involves m = Xp 2 : and as m and n have no common factor, A must be unity. Thus m =p 2, n = q 2, as was to be proved. In particular it follows, by taking n â– â€” 1, that an integer cannot be the square of a rational number, unless that rational number is itself integral. It appears then that our requirements involve the existence of a number x and a point P, not one of the rational points already constructed, such that A P = cc, x 2 = 2; and (as the reader will remember from elementary algebra) we write x = *J2. The following alternative proof that no rational number can have its square equal to 2 is interesting. Suppose, if possible, that p/q is a positive fraction, in its lowest terms, such that (p/q) 2 = 2 or p 2 = 2q 2 . It is easy to see that this involves (2qâ€”p) 2 = 2(p-q) 2 ; and so (Zq-p)/(p â€” q) is another fraction having the same property. But clearly q x- or x x > x. Thus there are larger members of L than x; and as x is...