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Without a convention for the order in which operations should be applied, many mathematical
expressions are quite ambiguous. Five plus four times three: should I do the plus first,
or the times? Eighteen p minus two: well, that’s eighteen times p minus two, but should
I do the times first, or the minus? If we didn’t have a suitable convention for this,
we would have to use brackets all the time. If we didn’t, we could never be sure which
operation was intended to be done first.
The last example here is one I’ve taken straight from a popular post on Facebook.
As a mathematics teacher, I am horrified—and disappointed—by how many people simply don’t
know, or don’t remember how to apply, the correct mathematical convention for the order
of operations.
Many people just do the operations from left to right, in the order they’re written,
as if it’s an English sentence to be read and performed in simple linear fashion. This
is incorrect. Others vaguely remember something called “BODMAS” or “BEMDAS”, and do
all additions before any subtractions. That’s also incorrect.
Here is the correct mathematical convention for order of operations. These rules apply
every time you read or write mathematics, every time you construct or evaluate an expression.
They always apply. Always.
Brackets come first. Work out the inner-most brackets first, then go outwards, one pair
of brackets at a time.
Exponents come next: squares, roots, powers.
Then multiplication and division, in order from left to right. In the old “BOMDAS”
or however you learnt it, the M and the D were meant to be interchangeable: You do them
in the order they’re written in the expression.
And finally, addition and subtraction, again in order from left to right, as they’re
written in the expression.
I’m going to say it again: This convention for order of operations always applies.
So, in the first example here, five plus four times three, we do the times first, not the
add: five, plus four-times-three. Four times three is twelve, so it’s five plus twelve,
which is seventeen.
Eighteen p minus two. Well, I can’t actually evaluate this one, because I don’t have
a value for p. But I do know that the multiplication gets done first: it’s eighteen times p,
minus two.
Five plus two squared times eight plus four. Notice how I already know to interpret the
second two as “squared”, because of the way it’s written, higher up and small. It’s
an exponent, so we have to do that first, and it’s written on the right of the thing
that’s being squared: two squared, which is four. Next, the multiply: four times eight,
which is thirty-two. Now I’ve just got two additions left: five plus thirty-two is thirty-seven,
plus four is forty-one.
Finally, seven minus one times zero plus three divided by three. Okay, so first we have to
do the multiplication and division: one times zero is zero, and three divided by three is
one. Now I have a minus and a plus. What does my convention say? In order, from left to
right. So in this case the minus comes before the plus. Seven minus zero is seven, plus
one is eight.
That’s how this convention works. Always. Every time. There are no exceptions.