Any Ideas?

He loved his parents, The Queen and Superman

Why BODMAS is the ambiguous grammar of maths.


"Let's see who's dumb" said the provocative post that went viral recently.

60 + 60 x 0 + 1 = ?

Those who said the answer is 1 are apparently dumb; smart people (it claimed) are supposed to say 61.

This is the latest in a long-running saga of maths questions on social media about the order in which you do calculations, known (depending on where you grew up) as BODMAS, BIDMAS, BEDMAS, PEDMAS or PEMDAS.  It probably has other names too.

The initials stand for:

B - Brackets (or Parentheses). Always calculate what's in brackets first.

O - 'Order' (or Index, or Exponential), bit of a fudge this, but it means 'things raised to a power'.

DM - Division/Multiplication

AS - Addition/Subtraction.

BODMAS is part of the 'grammar' of maths, and the point of grammar is to make things easy to read and unambiguous.  And grammar rules have to be treated carefully.  For example, we're taught that in lists you should never put a comma before the final 'and'.  Which is fine, until you get something like:

HE LOVED HIS PARENTS, THE QUEEN AND SUPERMAN.

To avoid ambiguity, this is crying out for an extra comma after 'Queen'.  

But if instead I'd written HE LOVED THE QUEEN, SUPERMAN AND HIS PARENTS, there wouldn't have been any ambiguity.  To ensure there is no confusion, the responsibility lies with the writer, not the reader.

And so it is with BODMAS.  

For 60 + 60x0 + 1, BODMAS says you should do the multiplication part before doing the additions, so according to this rule, the answer is 60 + 0 + 1 = 61.  If you have a basic calculator, however, it will tell you the answer is 1, because it does the calculations in sequence, starting at the left.  Is the basic calculator 'wrong' or 'dumb'? No, but it's probably not doing what the person who posed the question meant it to be doing.  If anyone was 'dumb' it was the person who asked the question and expected everyone to come up with the same answer.

Where BODMAS gets really contentious is when you have a calculation like this next one (which went viral on Twitter about three years ago):

6÷2(2+1)

Here, "2(2+1)" is shorthand for 2 multiplied by (2+1) - there's a convention not to use a multiplication sign before brackets because it looks like an 'x' and might be mistaken for one.

For this question, there is far more argument about the answer.  Some people say the answer is '1', but the majority say it is '9'.  

Why the disagreement?  Some of those who say 9 do so because they think BODMAS says you should do division before multiplication. But in this case BODMAS is misleading, there's no official convention in maths that division should come before multiplication, the two have equal status.

The acceptable argument for '9' as the answer is that when deciding between division and multiplication (or between addition and subtraction) you should calculate in the order in which they appear.  Hence you'd calculate the above, starting with the brackets, as:

(2+1) = 3

6/2 = 3

3 x 3 = 9

That's the convention, and I agree with it in most cases. However, if I was forced to state what 6÷2(2+1) equals, I would say  '1'.  Why? Because I would interpret that as a shorthand way of writing this fraction:

   6     

2(2+1)

Can a modern calculator resolve this dispute?

Fat chance.  I have a new-ish Casio.  When I ask it to calculate 6÷2(2+1)  it tells me the answer is '1'.

Hooray, I win!

But wait. If I now insert that missing multiplication symbol and ask it for: 6÷2x(2+1) it thinks the answer is now 9.

If a calculator can't agree with itself on the right answer, that's a sure sign that BODMAS is a guideline to be handled with extreme care.