On Zero
In documentaries on the history of mathematics, there is often a scene where the presenter discusses the invention of the number zero. If there is enough money they might even get to fly out to India to do it. It always seemed strange to me that "zero" was a concept that needed discovering. Surely if you asked someone in Rome what happens if you take 3 denari from a person with 3 denari, they would have told you that the person now has no denari, or "non denarium habet"?
So what do we mean by the discovery of the number zero? It became clear when I was reading The Joy of X (Bookshop, Amazon) by Steven Strogatz. In Rome, they would obviously have known that:
But this isn't actually very useful. Maths is about what you do with numbers that exist, not numbers that don't. So why do we care so much about zero? To realise why, we have to stop thinking about the the number line. Counting in Roman numerals is like counting with tally marks. You just keep adding lines. There are some shortcuts. The symbol for a 5 is V, you can put a lesser symbol before a greater symbol to indicate that it is one unit less instead of writing 4 or 9 units etc.
This is a problem when you do calculations. You need to parse out the whole string of numerals because each one can affect the value of numerals to its left and right. In the Hindu-Arabic number system numbers revolve around a base of 10 yet there is no numeral, or digit, for 10. Instead the number 0 acts as a placeholder for 10. The 0 itself still means nothing. But its presence changes the value of the digit to its left. It raises it to a higher power of 10.
This way of laying out numbers is so much easier. You can read them in one direction and work out the value of digits by wich column they are in, rather than just by the shape of the digit itself. That's why zero is important. Not because it means we have discovered the concept of nothingness, but because it means we can use placeholders in mathematics, and use the position of digits in numbers tell us about their value.