Inspired by a question Mr Phillips posed…
Yes it’s true. The Maths teachers do just hang around and chat about maths in their spare time. The other day Mr Phillips and I were chatting about the cross product. It isn’t some sort of angry multiplication; it’s a way of combing two vectors to form another. It’s dead useful in physics and computer animation, but the question is where does it come from?
We couldn’t answer the question. The bell rang and we had to finish the conversation. Our conclusion was: it’s just useful and that’s enough. But I’ve been thinking: being useful is not enough. I don’t think maths is cool because it’s useful. Spoons are useful but there aren’t many cool spoons. On the other hand Glastonbury is cool and maths is more like that. I guess my analogy needs some explanation.

Glastonbury is cool because when you’re there you are free to do whatever you like just as long as you’re not nasty to anyone and don’t ruin whatever someone else is doing. If you don’t like Dizzee Rascle, your free not to see him, just don’t pull the plug on the amps. It would make Dizzies fans go Bonkers.
Maths works just the same. You are free to invent whatever you like. Just as long as it doesn’t ruin what came before. Here are some things that spring to mind. Numbers are maths most basic ingredient. Peoples first introduction to maths is from counting stuff 1, 2, 3, etc. Nice and friendly and no problems. We can add, subtract, multiply and divide happily all day long. But the numbers we use to count aren’t the objects themselves they are just ideas so we are free to do what we want to them. So let’s add a number we can’t see: zero.
The only rule is that when we start using zero it doesn’t ruin the maths we could do with the other numbers. Just like playing Lego with your friends. You can add to the construction but don’t bash it up. In order not to mess up our fun with the normal numbers we just have to leave dividing by zero to Chuck Norris. That means no dividing by zero for us. But that’s cool, these numbers are just ideas so we can do whatever we feel (as long as it doesn’t interfere with our previous constructions).


There are loads of examples of this extension of the number system. Like Jenga, add what you want. Just don’t topple the tower. Take negative numbers for example. If you stop and think about them on the surface they are pretty weird and don’t have much use in the real world. But, then how would you describe positive and negative charge? What about extending indices. The sequence 10^3, 10^2, 10^1 is fine. It’s just more exciting to continue 10^0, 10^-1 and see what comes next. Why not, we are free to imagine whatever we like.
What about x^2 = -1. Let’s imagine an answer to that. i can, and it’s OK just as long as it doesn’t interfere with what has come before. (It is interesting watching how each new thing combines to make something new and exciting – just like minecraft).
The same goes for the cross product. It’s there because maths gives use the freedom to imagine it. The brilliant thing is once we’ve come up with a new idea we can play with it. And use it to build a bunch of amazing new things on the edge of reality.
Awesome. Maths is cool because of its freedom not its utility.