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.