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Zero

The number that鈥檚 not a number

YOU could be forgiven for thinking that zero is not a proper number. After all, numbers are the things we use to count, and you can鈥檛 count nothing.

We have evidence for counting going back five millennia, but the history of zero only began with the Babylonians in about 1800 BC. Even then, it was not a fully fledged number. The point of zero for them was like the zero in our modern representation of a number like 3601 鈥� it鈥檚 a position-setting symbol that distinguishes the number from 361.

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The Babylonians鈥� symbol was two diagonal arrows; the familiar squashed egg shape only came into being around AD 800, still as an accounting symbol. It was the work of Indian mathematicians that sparked the genesis of zero as a number, when they first appreciated that numbers can have an abstract existence distinct from counting physical objects. The astronomer Brahmagupta, for example, laid out a number line that included positive and negative numbers and zero.

This line of thought wasn鈥檛 embraced in the West until much later, partly because zero was considered a gateway to the negative numbers, where debt and fraud lay. By the late 19th century, however, mathematicians had become interested in establishing rules of mathematical logic. When the Italian mathematician Giuseppe Peano set out a list of rules for arithmetic, his first axiom insisted that zero must be a number. Otherwise how would you perform a calculation that traverses the boundary between positive and negative, like 7 鈥� 9?

Zero鈥檚 number status was secured, but it had an even greater role still to come, in defining what numbers really are. Our best answer to that question involves set theory. The set, first conceptualised by Georg Cantor in 1874, is an abstract mathematical container; it might be the set of dwarfs in Snow White or the set of days in a week. If you could define a unique set that had intuitively 7 elements, that would help explain what we mean by the number 7. It turned out the best way involves a totally unique set: the empty set, which has zero members (see diagram).

That鈥檚 not to say everything is wrapped up. But that鈥檚 another story, for which we must visit the other end of the number line (see 鈥淚nfinity鈥�).

Michael Brooks

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Infinity

The concept that makes and breaks maths

A BILLION! A squillion! Infinity! Infinity plus one! Children of a certain age delight in outdoing each other with the biggest number. It鈥檚 just a shame that we grow up into a humdrum world where nothing as exotic as infinity exists.

Or does it? Despite our innate discomfort with the idea, infinity is baked into the way we understand the world. Take Einstein鈥檚 theory of gravity, general relativity. It predicts black holes where space-time is infinitely curved. Then there鈥檚 calculus, a crucial technique used to describe processes of continual change that relies on the fact that space and time are infinitely divisible.

That鈥檚 not to say physicists are comfortable with infinity. When equations in cosmology give us infinity, they go to great lengths to rewrite the theories.

But there鈥檚 no avoiding it in another bedrock of science 鈥� set theory. The set, an abstract mathematical container, can help explain what a number is if defined rigorously enough. Our first attempt at that definition involved the empty set (see 鈥淶ero鈥�). It worked well enough until infinity got involved.

Try imagining the set of all whole numbers. How large is it? You can keep counting forever, so the answer seems to be 鈥渋nfinitely large鈥�. But then consider a head-scratcher posed by the German mathematician David Hilbert in 1924. He imagined a hotel with an infinite number of rooms, all occupied. Then an infinite busload of new guests turns up. Can the hotel cope?

Intuition says no, but infinity says yes. Move the guest in room 1 to room 2, the guest in room 2 into room 4 and so on; to put it another way, every guest in room n winds up in room 2n. Now all of the odd-numbered rooms are free and your hotel, which was infinitely full, has infinite vacancies.

In other words, sets of whole numbers that have apparently different sizes can both, on close inspection, be infinite. Set theory has never managed to deal with such contradictions.

Even worse, some infinities are larger than others. Think of the set of all whole numbers 鈥� it is infinitely large. But now think of all the decimals between each of those whole numbers 鈥� it鈥檚 infinitely larger again. How鈥檚 that for a way to win a playground argument?

Stuart Clark

This article appeared in print under the headline 鈥淲onders of听numberland鈥�