Wikipedia article is at the bottom

One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)

Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.

One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.

Books

Some of the most prolific popularisers of mathematics include Keith Devlin, Martin Gardner and Ian Stewart. Titles by these 3 authors can be found on their respective pages.

On zero

• Charles Seife (2000). Zero: The Biography of a Dangerous Idea. Souvenir Press. ISBN 978-0-285-63594-4.
• Robert Kaplan (2000). The Nothing That Is: A Natural History of Zero. Oxford. ISBN 978-0-19-514237-2.

On infinity

• Brian Clegg (2003). Brief History of Infinity: The Quest to Think the Unthinkable. Constable and Robinson. ISBN 978-1-84119-650-3.
• Robert Kaplan & Ellen Kaplan (2004). The Art of the Infinite: Our Lost Language of Numbers. Penguin. ISBN 978-0-14-100886-8.

On constants

• Petr Beckmann (1976). A History of Pi. Saint Martin's Press. ISBN 978-0-312-38185-1.
• Eli Maor (1998). "e", The Story of a Number. Princeton. ISBN 978-0-691-05854-2.
• Mario Livio (2003). The Golden Ratio. Headline Review. ISBN 978-0-7472-4988-7.
• Julian Havil (2003). Gamma. Princeton. ISBN 978-0-691-09983-5.

On complex numbers

• Paul J. Nahin (1998). An Imaginary Tale: The Story of . Princeton. ISBN 978-0691027951.
• Paul J. Nahin (2006). Dr. Euler's Fabulous Formula. Princeton. ISBN 978-0-691-11822-2.

On the Riemann hypothesis

• John Derbyshire (2004). Prime Obsession. Plume Books. ISBN 978-0-452-28525-5.
• Marcus Du Sautoy (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. ISBN 0-06-093558-8.
• Dan Rockmore (2006). Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers. Vintage. ISBN 0-375-72772-8.
• Karl Sabbagh (2002). Dr Riemann's Zeros. Atlantic Books. ISBN 1-84354-100-9.

On recently solved problems

• Robin J. Wilson (2003). Four Colours Suffice. Penguin. ISBN 978-0-14-100908-7.
• Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1-84115-791-0.
• Donal O'Shea (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
• George G. Szpiro (2003). Kepler's Conjecture. Wiley. ISBN 0-471-08601-0.
• George G. Szpiro (2007). Poincaré's Prize.

On classification of finite simple groups

• Mario Livio (2006). The Equation That Couldn't Be Solved. Souvenir Press. ISBN 978-0-285-63743-6.
• Mark Ronan (2006). Symmetry and the Monster. Oxford. ISBN 0-19-280722-6.

Biographies

• Paul Hoffman (1998). The Man Who Loved Only Numbers. Fourth Estate. ISBN 1-85702-811-2.
• Bruce Schechter (2000). My Brain is Open: The Mathematical Journeys of Paul Erdos. Simon & Schuster. ISBN 0-684-85980-7.
• Robert Kanigel (1991). The Man Who Knew Infinity: A Life of the Genius Ramanujan. Washington Square Press. ISBN 0-671-75061-5.

Magazines and journals

• Popular science magazines such as New Scientist and Scientific American sometimes carry articles on mathematics.
• Plus Magazine is a free online magazine run under the Millennium Mathematics Project at the University of Cambridge.

The journals listed below can usually be found at your local university library.

• American Mathematical Monthly is designed to be accessible to a wide audience.

• The Mathematical Gazette contains letters, book reviews and expositions of attractive areas of mathematics.

• Mathematics Magazine offers lively, readable, and appealing exposition on a wide range of mathematical topics.

• The Mathematical Intelligencer is a mathematical journal that aims at a conversational and scholarly tone.

• Notices of the AMS - Each issue contains one or two expository articles that describe current developments in mathematical research, written by professional mathematicians. The Notices also carries articles on the history of mathematics, mathematics education, and professional issues facing mathematicians, as well as reviews of books, plays, movies, and other artistic and cultural works involving mathematics.

Audio and video

• Simon Singh's Fermat's Last Theorem is available in audio and there is also a Horizon television program.