A student asked: is zero a number? This is my answer.
Yes, zero is a number different from all others. You can’t divide by it, and it is neither positive nor negative.
The idea of “number” in the abstract (as distinct from a heap of particular numbers) and the idea of “human being” in the abstract (as distinct from a heap of particular humans)
But why shouldn’t numbers include one different from all others? If you think of all the people in the world, there is one different from all others, i.e. yourself. You are the only person the back of whose head or down the back of whose throat you can’t see without a mirror, the only person you can’t tickle, etc.
Zero is a number different from all others for roughly the same reason that yourself is a different person from all others. Numbers represent a movement (along a line, or, for complex numbers, in a plane). Not moving at all is a special sort of movement. You identify people by their difference from you, and zero difference from yourself is a special sort of difference.
For thousands of years people couldn’t conceptualise 0 as a number. Less than 540 years ago, very recently in human evolution, the first-ever printed maths book, the Treviso Arithmetic, did not consider 1 to be a number, let alone 0 (or fractions).
It took many tens of thousands of years to develop the idea of “all the people in the world”, and it takes some years for every toddler to develop the idea of a human society of which she or he is part. Similarly with developing the idea of a “universe” of numbers and so developing arithmetic beyond rules-of-thumb for calculating into a science.
Literacy and abstraction: the idea of “beauty” in the abstract (as distinct from a heap of beautiful things)
Maybe it took specially long with arithmetic, because until only a few hundred years ago arithmetic was a manual skill (with fingers, abacuses, etc.) rather than a written-down logical system. The results of arithmetic were written down (in accounts, and so on), and some rules of arithmetic were written down by way of giving examples of how to do certain sums, but no general theory. General theories about geometry got written down and developed logically with Euclid; but the Treviso Arithmetic, contains no proofs like Euclid’s, only rules of thumb.
Logic developed only where and when reading and writing developed. Plato wrote:
“The multitude [he meant people who did not read or write, the great majority in his day] cannot accept the idea of beauty in itself rather than many beautiful things, nor anything conceived in its essence instead of the many specific things. Thus the multitude cannot be philosophic”.
Once the idea of “number in itself” rather than “many things which are numbers” is developed, and so we get the idea of a system of numbers whose laws we can investigate and write down, then the reasons for considering 0 a number become strong.
Groups: systems of symmetries, as distinct from a heap of things being more or less symmetrical
A group, in mathematics, is the simplest algebraic structure. It is a set of elements and an operation which from any two elements makes another element in the set. The operation is associative: a*(b*c)=(a*b)*c. There is an “identity” element, e, with a*e=e*a=a for all a. For every element a there is an inverse (−a) with a*(−a)=e
The whole numbers, with addition, make a group with identity zero. Leave out zero from the numbers, and there is no identity. Also, leave out zero, and some addition sums give us answers which are not numbers, like 2+(−2). Take the two addition sums 4+(2−2) and (4+2)−2. They are just two ways of writing the same sum. The second way, there is no problem doing the sum while remaining all the time within the universe of numbers. But if we write the same sum the first way, then we have to go outside the numbers to do the calculation.
Leave out zero from the numbers, and you can’t conceptualise numbers as a system with its own laws to investigate.
The idea of a group came from thinking about symmetry not in terms of “many symmetric objects” but in terms of systems of symmetries. For example, an equilateral triangle has the symmetries “rotate by 120 degrees”, “rotate by 240 degrees”, “reflect in the axis through A”, “reflect in the axis through B”, “reflect in the axis through C”… and: do nothing.
To be able to study systems of symmetries scientifically, mathematicians had to make the same move with the symmetry “do nothing” as with the number zero in number theory.
The distributive law, rings, and fields
Still, you might ask, why can’t we redefine multiplication so that we no longer have the oddity with numbers that you can divide by every number except zero?
Why not? Because of the distributive law:
The sort of algebraic structure formed by the rational numbers, or the real numbers, or the complex numbers, is called a field.
It is a group with one operation, which we usually think of as adding, and almost a group, with the other one, which we call multiplying. There is an “identity” for multiplying (like 1 for numbers, or I for matrices), and an “identity” for adding (a zero).
And the two are linked by the distributive law.
But if the distributive law holds,
a×(b+0)=a×b+a×0 for all a, therefore a×0=0 for all a, therefore you can’t divide by 0. And therefore the field isn’t quite a group with the multiplication operation.
There are less developed structures with two operations, called rings. For example, 2×2 matrices form a ring. They are equivalent to linear transformations in two dimensions. Polynomials in x with real coefficients form a ring. With rings, there may be lots of elements you can’t divide by. (There are lots of singular 2×2 matrices). With the most developed structures, fields, you reduce the “can’t-divide-by” bit to the minimum: zero. But you can’t reduce it below that minimum.
The rational numbers, the real numbers, and the complex numbers, with our usual addition and multiplication, all form fields. There are many other sorts of fields. The numbers 0, 1, 2, 3…. (p−1), if p is a prime, with clock-arithmetic addition and multiplication, form a field. All the elements (polynomial in x)/(another polynomial in x), with real coefficients, form a field.
Every field has that property, that 0 is an element, but a strange one.