By G. H. Hardy

This publication has been designed basically for using first 12 months scholars on the Universities whose skills achieve or process whatever like what's often defined as scholarship average. i am hoping that it can be beneficial to different sessions of readers, however it is that this category whose wishes i've got thought of first. It os at least ebook for mathematicians i've got nowhere made any try to meet the wishes of scholars of engineering or certainly any category of scholars whose pursuits aren't essentially mathematical.

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305 See f Squaring the Circle (Cambridge, 1913). , or the same writer s 26 REAL VARIABLES [l where n is an In this way it is possible to define a integer. number which is not rational nor yet belongs to any of the classes of irrational numbers which we have so far considered. And this number TT is no isolated or exceptional case. Any number of other examples can be constructed. In fact it is only special classes of irrational numbers which are roots of equations of this kind, just as it is only a still smaller class which can be expressed by means of surds.

If a 5. or = /3 < /3<y, or according as a=ft /3>a a>ft or a</3. then a=y. a<ft = /3 Prove that or j3<a, and/3 = y, ft = - 0. Prove that 1. that /3 = a, a /3 ^ y, -/3< then -a, or a<y. a, /3> according as a = /3, a<,3, a>/3. 6. Prove that 7. Prove that a 8. Prove that 9. Prove that, find an if a < a . 1< v/2 a>0 is positive, and a<0 if a is negative. | if a < v/3 < 2. and /3 are two different real numbers, we can always numbers lying between a arid /3.

This is so, for s/2 or (if we confine ourselves v/2 and x example, with the pair x 2 2 and x2 to positive numbers) with #2 Every rational number in possesses one or other of the properties, but not every real number, since Thus < N /2 > c > < * > < . either case v/2 escapes classification. There are now two member * I, or R possibilities!. has a least The discussion which member follows is in Either r, many ways L has a greatest Both of these events similar to that of 6.