A reader asks:

“When you’re working out an expression, why do you sometimes divide after you multiply, when the BIDMAS rules say D comes before M?”

This is exactly the reason I don’t like BIDMAS - because it suggests something that simply isn’t true (that division is before multiplication and addition before subtraction).

What is BIDMAS?

BIDMAS is a well-intentioned memory-aid to help work out the order you work things out in: it stands for Brackets, Indices, Divide, Multiply, Add, Subtract, and it’s very nearly very useful.

How it works: you compute anything in a bracket before you do anything else. You then work out any indices (things like $3^2$) you can. After that, it gets messy (and this is the problem): divide and multiply are really the same thing in reverse, so they have the same precedence - you do them from left to right, after you’ve done the indices, but before you do the add/subtracts. Although BIDMAS looks like D comes before M, they could just as easily be the other way around.

Similarly, add and subtract happen from left to right as well.

If you’re going to use BIDMAS, I recommend writing it like this: B I (DM) (AS)

Some examples

Let’s start with an easy(ish) one:

$3 - 4 + 7$

If you do that the way BIDMAS seems to suggest, you get the wrong answer: you think “A comes before S, so I work out $4+7 =11$ and then take it away from $3$ to get $-8$.”

Our survey said nut-uh.

You start by doing $3-4$, because the minus is to the left of the plus. You get $-1 + 7 = 6$, the correct answer.

Now, I’m writing this first thing on a Monday morning, and my coffee hasn’t kicked in properly yet, so I’ve not been able to find an example where dividing before multiplying gives the wrong answer. The order you do things in can make a difference, though. Consider:

$6 \div 3 \times 2$

The right way is to say “$6 \div 3 = 2$, then multiply that by $2$ to get $4$.”

The wrong way - our poor American cousins who remember PEMDAS might fall into this trap - would be to say “Multiply first! $3 \times 2 = 6$ - then $6 \div 6 = 1$.” Wrong answer.

But WHY?!

Add and subtract are opposites - but they’re the same sort of thing: counting up and counting down, if you like. You can even see subtraction as ‘adding a negative number’ - which sounds more complicated, but is actually easier once you get your head around it.

Similarly, multiply and divide are opposites of the same sort: adding repeatedly and taking away repeatedly. Dividing by 3 (say) is just the same as multiplying by $\frac{1}{3}$ - in fact, it’s very unusual for a mathematician to use the $\div$ symbol; we generally prefer fractions.

So, add and subtract are on the same BIDMAS level because they’re really the same thing! Similarly, a divide is really a multiply in disguise.