Negatives iii – plus and minus

Rough Maths 4

This series of posts started when someone said on Twitter that he’d asked three maths teachers to give a real world example of dividing a negative by another negative: and none of them had been able to answer.

As an example, we can use one of the earliest known application of negative numbers – to represent debts. The ancient Chinese represented debts in black ink and credits in red, the opposite of modern conventions.

Suppose that I owe $36,000, and that this is represented in my accounts by


I’m tightening my budget and want to reduce my fortnightly repayments to $400. I’d like see how long it will take me to clear the debt. It’s an interest-free loan, to keep things simple.

In other words, if I have a single debt of $36,000, and divide it into smaller debts of $400 that I can afford to pay off once a fortnight, how many fortnights will it take me?

-36,000 / -400 = 90

Which is about three and a half years. (More on that later.)

This might seem arbitrary: couldn’t I have expressed the debts as positive numbers and arrived at the same result? We could, but in a truly real-world example, our convention would have to match our accounting software, the banks’, the tax office, and so on.

In an important sense, the signs with which we represent credits and debits – positive and negative, or black and red, or red and black – are labels for a particular logical structure. This structure has two states (A and B, say) and maps combinations of those states onto the states themselves, so that AA and BB can be represented by A, and AB or BA by B. In other words, the idea of “sameness” is associated with A, and “difference” with B.


There is a more technical reason for two negatives to make a positive, which is the internal consistency of our arithmetic. Let’s start with a couple of results which follow from last week’s rule that a positive times a negative gives us another negative:

-1 * 36,000 = -36,000 and -1 * 400 = -400

Now let’s suppose that my neg / neg = pos example is wrong, and that the right answer is -90:

-36,000 / -400 = -90

This implies:

(-1 * 36,000 ) / (-1 * 400) = -90

The -1s in the left-hand side cancel out, and we have

36,000 / 400 = -90

Dividing the two positives gives

90 = -90

Which is not true. If we agree on every step of the way that led us here, it’s safe to say that the starting point was false.

This kind of reasoning – called proof by contradiction or the more splendid term reductio ad absurdum – is very common in maths. It’s based on the idea that a system of mathematical reasoning must be internally consistent. The fact that arithmetic and other mathematical structures are self-consistent is not important just because it makes mathematicians happy: it also makes physical science possible, because it turns out that this type of formal self-consistency is also a characteristic of the natural world.

About the three-and-a-half year payment plan: given what we’ve covered so far, we can’t give an answer to that yet, because so far, we only have whole numbers. Next post: fractions, decimals, and why your computer is crap at arithmetic.


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