قراءات إضافية

Two other books in the OUP VSI series that complement and expand on the current one are Mathematics by the Field’s medallist Timothy Gowers and Cryptography by Fred Piper and Sean Murphy. Probability and statistics, fields that were neglected here in Numbers, are the subject of the VSI Statistics by David J. Hand.

An insight into the nature of numbers can be read in David Flannery’s book, The Square Root of 2: A Dialogue Concerning a Number and a Sequence (Copernicus Books, 2006). This leisurely account is in the Socratic mode of a conversation between a teacher and pupil. One to Nine: The Inner Life of Numbers by Andrew Hodges (Short Books, 2007) analyses the significance of the first nine digits in order. Actually it uses each number as an umbrella for examining certain fundamental aspects of the world and introduces the reader to all manner of deep ideas. This contrasts with Tony Crilly’s 50 Mathematical Ideas You Really Need To Know (Quercus Publishing, 2007), which does as it says, digesting each of 50 notions into a four-page description in as straightforward a manner as possible. The explanations are mainly through example with a modest amount of algebraic manipulations involved, rounded off with historical details and timelines surrounding the commentary. A particularly nice account on matters concerned with binomial coefficients is the paperback of Martin Griffiths, The Backbone of Pascal’s Triangle (UK Mathematics Trust, 2007), in which you will read proofs of Bertrand’s Postulate and Chebyshev’s Theorem, giving bounds for the number of primes less than .

Elementary Number Theory by G. and J. Jones (Springer-Verlag, 1998) gives a gentle but rigorous introduction and goes as far as aspects of the famous Riemann Zeta Function and Fermat’s Last Theorem. The classic book An Introduction to the Theory of Numbers, by G. H. Hardy and E. M. Wright, 6th edn (Oxford University Press, 2008) assumes little particular mathematical knowledge but hits the ground running. The author’s book Number Story: From Counting to Cryptography (Copernicus Books, 2008) has more in the way of the history of numbers than this VSI and includes mathematical details in the final chapter. The Book of Numbers by John Conway and Richard Guy (Springer-Verlag, 1996) is full of history, vivid pictures, and all manner of facts about numbers. Quite a lot of the history and mystery surrounding complex numbers is to be found in An Imaginary Tale: The Story of (Princeton University Press, 1998) by Paul J. Nahin. Paul Halmos’s Naive Set Theory (Springer-Verlag, 1974) gives a quick mathematical introduction to infinite cardinal and ordinal numbers, which were not introduced here.

A popular account of the Riemann Zeta Function is the book by Marcus du Sautoy, The Music of the Primes, Why an Unsolved Problem in Mathematics Matters (HarperCollins, 2004), while Carl Sabbagh’s, Dr Riemann’s Zeros (Atlantic Books, 2003) treats essentially the same topic.

There are two accounts of the solution to Fermat’s Last Theorem, those being Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem by Amir D. Aczel (Penguin, 1996) and Fermat’s Last Theorem by Simon Singh (Fourth Estate, 1999). The best popular book on the history of coding up to the RSA cipher is also an effort of Simon Singh: The Code Book (Fourth Estate, 2000). The unsolvability of the quintic (fifth-degree polynomial equations) was not explained in our text here but is the subject of an historical account: Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (MIT Press, 2003) by Peter Pesic.

مواقع ويب

A very high-quality web page that allows you to dip into any mathematical topic, and is especially rich in number matters, is Eric Wolfram’s MathWorld: mathworld.wolfram.com. For mathematical history topics, try The MacTutor History of Mathematics archive at St Andrews University, Scotland: https://www-history.mcs.st-andrews.ac.uk/history.index.html. Web pages accessed 8 October 2010. Wikipedia’s treatment of mathematics by topic is generally serious and of good quality, although the degree of difficulty of the treatments is a little variable. For example, Wikipedia gives a good quick overview of important topics such as matrices and linear algebra.