Put on your shoes and socks: commutation

"Put on your shoes and socks - but not in that order", I say to my daughter, who sighs at my frequently repeated joke. The order you put your clothes on matters. Putting on your shoes then your socks is quite different from putting on your socks first and then your shoes.

On the other hand, it doesn't matter which sock you put on first, your left or your right.

In mathematics, when the order you do something is not important we say that operation is commutative. It commutes. The actions 'putting on your left sock' and 'putting on your right sock' commute. It doesn't matter which you do first, you get the same result. However, whether you put on your shoes or your socks first does matter. These tasks do not commute.

So because is some cases the order is important, the operation "putting on" is not commutative. However, if we restrict this and only deal with the operation of "putting on a sock", then this operation does commute, because it doesn't matter which foot you put your sock on first.^*

In mathematics some operations commute and others do not.

Addition commutes, meaning that:

a + b = b + a

For example:

3 + 2 = 5
2 + 3 = 5

We get the same answer, no matter the order.

Multiplication also commutes:

ab = ba

3 X 2 = 6
2 X 3 = 6

So what does not commute? For some reason mathematicians seem to like jumping to more exotic examples like matrices, rotations or quaternions. But how about subtraction? Subtraction does not commute.

a - b ≠ b - a

For example:

3 - 2 = 1
2 - 3 = -1

Division does not commute either.

a / b ≠ b / a

For example:

3 / 2 = 1.5
2 / 3 = 0.666...

And 'raising to the power of' does not commute also:

a^b ≠ b^a

3² = 9
2³ = 8

In future posts I'll look at other properties of these simple operations.

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*I am assuming that you do not put more than one sock on each foot - then the order would matter (if you can tell your socks apart) as it would effect which sock was underneath and which sock on top.