Not sure quite how I’m going to say this but, everything you’ve been told about maths is a lie. Maybe I should be more specific…maths is not about being a human calculator or remembering facts about numbers. Oh heck, maybe the easiest thing to do is give you an example.
Example: group theory.
How to start. Well, picture the whole numbers …,-4, -3, -2, -1, 0, 1, 2, 3, 4,… . When adding them together, what sort of structure do we get? Well, we have
i) a number which ‘does nothing’ to every number (called the identity element)
Can you see which number satisfies property (i) in our case? The answer is here. Note that this number behaves in this way for all numbers. Also, no other number has this property. We’ll maybe come back to this in another post. Next, given any number,
ii) is there another number so that they add together to get to the identity? (called an inverse)
Let’s think about some examples. If we have the number 1, we need to add (-1) to it to get to zero (our identity element in this case). For the number 2, we need to add (-2). For 3, we need (-3). Can you spot the pattern? (Maths has a lot to do with pattern spotting which is why it’s used for forecasting things from the weather to finance to the path of an asteroid). So the pattern was…that we need to take the minus of the whole number. Now, does this work for negative numbers? Well, if we have (-5) then we need to add 5 to get to zero. Notice what happens when taking the inverse of the inverse of a number? Have a think. We’ll come back to this later. One way to remember the need to have an identity element and an inverse for each element is that they mean that we have
- from any element you can get to any other element.
What does this mean? Well, for these whole numbers with addition we have that for any two numbers, say 5 and 23, there is a number x such that
- 5 + x = 23.
And how can we get there? Well, I like to use a trick that comes up in maths now and then. First we are going to go from 5 to 0. We know we can do this because we have inverses.
- 5 + (-5) = 0
Then we’ll go from 0 to 23. That’s just adding 23. So this means we can put these together
- 5 + (-5) + 23 = 23 and so x is equal to (-5) + 23.
So in order to get from 5 to 23, we’ve gone from 5 to 0 to 23. In doing this I’ve actually used the final property needed for a group. Imagine you’re given three numbers
- a, b, and c
and you’re told to add them up in order. Do you do
- a + (b + c) or
- (a + b) +c?
The point is that it doesn’t matter the order you do these in. And so we require a group to satisfy one last rule, which is
iii) a + (b + c) = (a + b) +c (called associativity)
Any structure satisfying (i), (ii), and (iii) is called a group. The point is that, although I’ve used the whole numbers in my example, did I need to use the symbols 0, 1, 2, 3, etc? Mathematicians care about the structure that these form, not working out what the answer is. I used this example since the whole numbers are hopefully familiar to people. But groups can also be used to describe other systems in a less familiar way and tell us things we might not see so easily otherwise. I’m thinking of symmetries. More on that soon where I hope you’ll join me!
Edit: just to give a simple example which doesn’t satisfy (iii), the associativity rule, consider the whole numbers again but with subtraction. Under what conditions will this succeed and fail? Also, what goes wrong with (i) and (ii)? Find the answer in the comments. Thanks for reading.