If you ever bother to run a search for ‘basic maths’ on twitter - something you may not generally do, as you’re probably not the author of a book called ‘Basic Maths For Dummies’ - you’ll find the breakdown of tweets looks something like this:

  • 47%: Teenagers complaining that they have to learn basic maths
  • 31%: Adults complaining that they’ve forgotten basic maths
  • 10%: People criticising each other for not knowing basic maths
  • 7%: Adults complaining that kids can’t do basic maths((Like percentages.))
  • 4%: People arguing over artificial sums involving the order of operations
  • 2%: Sites illegally offering my book for download((My position on piracy is complicated. When it’s just my book, I don’t mind so much. Published books have to pay my editor and marketing people too.)).

It’s a bit of a downer how rarely basic maths gets a good press - and that the only bit of maths people ever argue over is ‘facebook sums’ like these:

\(4 \\times 4 + 4\\times 4 + 4\\times 4 + 4 \\times 0 + 4 \\times 4 = ?\) \(3 + 6 \\div 2(4-1) = ?\)

A little aside before I look at them, though: my student, Patrick, has made an awesome poster about the order of operations, which he’s made available under a Creative Commons BY-SA license.

The first one is easy enough - the rule is clear, the $\times$ is more important than the $+$s, so you work out each of the multiplications first, getting $16 + 16 + 16 + 0 + 16 = 64.$

The second, though, irritates me (and pretty much every mathematician). It’s ambiguously written, and no self-respecting mathematician would allow such a sum to escape his or her pencil.

Here’s the problem: applying the rules strictly, you work out the bracket to get 3, then do $6 \div 2$ to get $3 + 3 \times 3 = 3 + 9 = 12$. However, that’s not how a mathematican would instinctively read or write the sum: we would generally interpret $2(4-1)$ as a single element and work out $3 + 6 \div 6 = 4$. I could, and would, argue it both ways. Even calculators aren’t unanimous in how they work it out.

This is one of the reasons we don’t encourage use of the $\div$ sign. (Fractions are usually much clearer - except, possibly, if you have to divide them by each other.)

If a mathematician wanted to write the first version of it (to make 12), they’d write something like $3 + \frac { 6(4 - 1) }{2}$ - keeping together the things that need to be multiplied on top. If they meant the second, it’d be $3 + \frac{6}{2(4-1)}$, which was my immediate interpretation of the original question.

However, the correct answer to “What is $3 + 6 \div 2(4-1)$” is “That is ambiguously written, please rewrite in a clearer form.”