Friday, December 30, 2016

The last lesson (until the next one)

I’m taking a break from teaching mathematics for a while, so I thought I would just have one last gasp with some irritations from Facebook and try and convince people that these are just silly things that arise from playing fast and loose with the grammar of mathematics.

What is 5+3×2?

There are many people who believe that the answer to this problem is 16: 5+3=8 and 8×2=16. Yet, this is wrong and many a Facebook page has been curdled by disagreements about what the answer should be. Mathematics has its own grammar that comes from its understanding of structure. The structure of how numbers interact is properly called algebra. People may think that this is just using letters to stand for numbers, but the actual study of how the numbers combine and reduce is really what we mean by algebra.

What does a mathematician make of 5+3×2? Well, to see my point, let’s rephrase it in English. What are five and three lots of two? Still not sure? Try reading that sentence again, but leave a little pause after the five. What are five(,) and three lots of two? Well, three lots of two are six, and five more makes eleven. This is the correct answer. The mathematical grammar says that you must do multiplication before addition. We have to understand what three lots of two are before we add the five.

Now, let’s try reading that sentence another way. Put the pause between the “three” and the “lots”. What are five and three(,) lots of two? Ah! Now you’re asking a different question. Five and three are eight, and two lots of eight are sixteen. We’re doing the addition first. The way that a mathematician would write this is (5+3)×2, bracketing off the bit we do first.

I’ve seen some bizarre answers to things like: 1+1+1+1+1+1+1+1+1+1+1×0+1=? But apply the mathematical grammar, do the multiplication first: 1×0=0. So we now have:


I suspect that many people might have thought the answer to be 1, but that would be the answer to:


There are some genuine mathematical ambiguities out there. What about 1-1+1=?

Is that 1-(1+1)=-1?

No. Otherwise we’d put the bracket in to show that we are adding the 1 and 1 together first. We have to understand this as a credit of £1 plus a debt of £1 plus a credit of £1 is £1. This is why the old acronym of BIDMAS (or BODMAS or BEDMAS if you’re of an age) sometimes tricks us into getting the answer wrong. BIDMAS gives the order of operations: Brackets, Indices (Orders or Exponents), Division, Multiplication, Addition, Subtraction.

However, we see that this is a bit vague when it comes to the order of addition and subtraction. We have to see subtracting as adding a negative.

A truly ambiguous mathematical question is 6÷2(1+2). Clearly, as mathematicians, we should do the brackets first.

6÷2(1+2)= 6÷2×3.

Strict BIDMAS would now say do 6÷2=3, and thus 3×3=9.

However, what is not completely clear is whether it is intended for six scones to be shared between two families each of one and two people (Johnny and his parents, Dorothy and her parents), or if we’re trying to find out the number of eggs needed when bumping a recipe for an omelette for two people that requires six eggs up to an omelette for Johnny and his parents. If it is the former, then each person gets one scone each. If it is the latter, then nine eggs are needed.

So ambiguous statements do exist in mathematics if we are not careful. We could make that last calculation a little easier if we write:

(6÷2)(1+2)=9 or 6÷[2(1+2)]=1

Of course, strings of numbers and calculations are all rather fun to play about with, like word games which reveal much about our language. It’s making sure that we do the correct calculation when it counts.

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