[Rough Maths 1 – a series of posts inspired by Sean M Elliott’s Rough Science. Note: I am not a mathematician. At best I’m a keen amateur.]
What is a number? This seems like a dumb question, as numbers are something we all learn about in pre-school. However, it’s a question which has had different answers over the last couple of thousand years, and looking at the different answers which mathematicians have given to it is a good way to introduce both the history of mathematics and at the same time, explain some fundamental ways in which mathematics works.
The history of mathematics is in itself a strange idea, as maths is thought of as the one subject where you can be definitely right or wrong. How can such a discipline have a history? When mathematicians – and maths teachers – are so certain of the truth of their results, what can change? The invention of zero is one of the earliest examples of this process.
This guy’s wall is an example of the simplest way of recording numbers: tally marks. It’s not the most basic – he’s scratching every fifth mark across the preceding four, making bundles which are easier to count – but the idea is: one mark for each thing you’re counting. This method has two problems: you run out of room quickly, and there’s no convenient way to do arithmetic. There is no way to add two tallies apart from counting all the marks in both of them.
Unfortunately for school students ever since, Roman numerals were a way of solving the first problem which made the second problem much worse. It’s thought that they were based on shepherds’ tally sticks, with the addition of new signs – V, X, L, C, D, M – which behave like tally marks carrying higher values – 5, 10, 50, 100, 500 and 1000. They save space – the prisoner could write his sentence as CVI – but they make arithmetic extremely difficult.
The number systems which replaced systems like the Roman are called place-value systems. They were invented independently in many parts of the ancient world, but the version we use came from India to the Arabic world and then to Europe in the 12th century. In this place-value systems, the digits 1-9 represent different values according to the position in which they are found in the number. The leftmost column represents ones, the second represents tens, and so on.
This requires two big steps forward in our ideas of number. The first is that a symbol, such as 1, can represent different numeric values depending on where it appears.
If we break the prisoner’s tally into powers of ten, it can be written with two digits: 1, representing 100, and 6. It turns out that every round number can be broken into powers of ten in this way.
The problem with such a system us that if we write 1 hundreds and 6 ones as 16, it is impossible to then distinguish it from 1 tens and 6 ones. What we need is a placeholder to show that there are no tens.
That placeholder is zero. It might seem like it’s just a workaround to make place-value notation work, but it is the first of our additions to the numbers.
Numbers as tally marks are always counts of something: zero was a disquieting idea for philosophers because it seems to be making a something out of nothing. Zero is not “no sheep” or “no years” or “no bushels of grain”. Or, if it is, it is also none of an impossible encyclopaedic list: “no eels, no galaxies, no words, no hailstorms, …” This is the beginning of a process of abstraction which will continue for centuries.