A Mathematician's ApologyG. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940 as his mathematical powers were declining, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his aphorisms and idiosyncrasies, and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times. 
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Hardy's book sets out to present the mathematical career to non mathematicians. He was what a mathematician would call a "purist". Someone who creates mathematics for the advancement of the subject and not to specifically use it to solve real world problems. It is sad in parts because he laments being too old to continue doing what he loves. Euclid's proof of the infinity of primes is a beautiful example of what mathematicians really do. Hardy actually proved half of the Riemann Hypothesis in 1914, basically that there are an infinite number of zeros on the critical line 1/2. I suppose that was too much mathematics in the last sentence but nonetheless, the book is great. Read it.
A mathematician's apology is a book about soul of mathematics, written by a pure number theorist and the book is intended for a lay man. I mean that it has no technicality of mathematics. But if you are a truly lay man at mathematics then you can not enjoy the charm that is contained in this book. Hardy has a very clear cut view for "usefulness" of mathematics but his meaning of "usefulness" may be unaccessible to a layman. It is also self justification of his mathematical career.