by Jon Dunning (May 2020)
Jon set an imaginative task during school closure to his Year 7 class: “Ask me two questions about what we’ve studied in this topic. If you feel confident you know everything, think of two imaginative questions you could ask a mathematician about negative numbers.”
Below some brilliantly imaginative questions, and Jon’s answers to his class. Will this work with any other topic?
Everything you ever wanted to know about negative numbers and weren’t afraid to ask!
Disclaimer: I’m not a historian or a scientist; I’m your maths teacher. But I’m 90% sure I’m remembering the historical and physics stuff correctly.
Who created negative numbers?
We don’t know!
Chinese texts mention negative numbers in the 2nd century BCE – many hundreds of years before they (or 0) appear in Europe. When they did arrive in Europe, even the smartest people got their knickers in a twist about them – ‘how could there be less than nothing?’ they asked – and it was some time before they were accepted as being as real as positive numbers. It wasn’t really until the 16th or 17th century that we started to feel like they weren’t a silly idea.
If you ever struggle with negative numbers, remind yourself that there was a time when some of the cleverest people around were convinced that they were nonsense.
Should we call them negative numbers or minus numbers?
Well, some maths teachers are very uptight about this and others don’t care.
There is a clear difference between the technical meanings of ‘negative’ and ‘minus’, though. Negative refers to a type of number: those below 0 on the number line. Minus is a word connected with subtraction.
When I was learning maths, I never bothered about the difference. But when I started teaching, I noticed that some kids got a bit muddled when working with negative numbers. That’s understandable because it sounds confusing if you say it as ‘minus four minus minus three’.
So I always try to use the distinction when I’m talking to you: I think it helps us to understand each other and say exactly what we mean.
Is 0 negative or positive?
It’s just 0.
Can you have negative fractions or decimals?
They’re exactly the same as positive fractions or decimals, just on the other side of 0.
Is the priority of operations the same with negative numbers?
For example, multiplication and division always come before addition and subtraction, even with negative numbers.
Is there negative infinity?
If you could walk up number line, you’d be approaching infinity; if you could walk down the number line, you’d be approaching negative infinity.
What is a negative number squared or cubed?
Multiplying a number by a negative number will change the original number’s direction.
So negative x negative = positive
But then (negative x negative) x negative = (positive) x negative = negative
But then (negative x negative x negative) x negative = (negative) x negative = positive
Each time you multiply by another negative number, the direction of the answer changes.
So squaring (raising to a powers of 2) gives positive but cubing (raising to a power of 3) gives negative.
Can you see what pattern would result from raising a negative number to other powers?
Why does a negative multiplied by a negative always give a positive?
I would argue that it’s because that’s what we need it to do if we want all numbers to follow the same rules of arithmetic. (This raises an interesting point: are the rules of arithmetic created by people, or were they true even before we thought of them?)
The rule of arithmetic that’s particularly relevant here is the distributive property of multiplication over addition. That’s the same rule that we use when expanding brackets.
First, we need to show that positive x negative always gives negative. We’ll start with the fact that any number and its negative counterpart (I’ve chosen 3 and -3) sum to 0, and then try multiplying that sum by something (I’ve chosen 5)
Now we can apply the same logic to negative x negative…
Can you have a negative length?
You can’t have negative length but you can have negative displacement.
Displacement is the word we use for distance in a particular direction. Say I walk 3 metres forward: my distance from where I started is 3m and I could say that my displacement from where I started is +3m. If I’d walked backwards 3m instead, I’d still have travelled a distance of 3m but my displacement would be -3m
Another interesting perspective comes from algebra. What would happen if was negative in the diagram below? What would the shape look like? Can you show that the formula for the area would still work?
Can you have negative speeds?
You can’t have negative speed but you can have negative velocity. Just like displacement is distance with a direction, we use velocity as speed with a direction. I refer you here to They Might Be Giants for more information.
Can you have negative time?
I’m a bit out of my depth here but I think it depends on whether you consider time to have a direction, which is less straightforward than you’d imagine.
Most laws of physics would still work perfectly well mathematically if time had no direction but there is an important exception which is (very roughly) to do with the movement of heat.
In school maths, we do sometimes get negative values for time, for example when working with equations that describe the motion of an object through the air. You’ll see these in Year 12 and learn how to interpret them. (They don’t really mean that there is negative time, just a time before the events described by the equations.)
Can you have a negative area?
Area can’t be negative but we there is a sense in which consider the areas beneath the axis of a graph to be negative. We do this because it fits nicely with another Year 12 topic, the calculus. This is the very important part of maths for which Isaac Newton is so famous. (He didn’t really discover gravity, as in the story about the apple bumping him on the head. But he did help to describe mathematically how things move under gravity by inventing the calculus.)
Can you have the square root of a negative number?
No! But yes! But only kind of!
First the ‘no’ part:
When you square a negative number, you get a positive number. So both positive numbers and negative numbers square to make positive numbers. None of the numbers on the number line squares to make a negative number, so negative numbers don’t have a square root on the number line.
Also, the square root is the length of a square of given area. For example, if a square has an area of 9cm2, it has a length of 3cm – that’s why the square root of 9 is 3. If we accept that areas are positive (see above), then squares have positive area, so we’re not square rooting a negative.
Now the ‘but yes’ part:
Mathematicians have a number we call i, which when you square it gives you -1. So you often hear people say that ‘the square root of -1 is i’.
One interesting thing about i is that it’s a number but it’s nowhere on the number line! In fact, there are infinitely many numbers that aren’t on the number line. Just think about that: there are numbers that aren’t on the number line!
The simplest versions of them are called imaginary numbers. This was in fact initially an insult – the famous mathematician Descartes mocked the idea that such numbers could exist – but it stuck as a name.
Most modern mathematicians would consider imaginary numbers as real as any other number, and they are a vital part of mathematics, which you can learn about in Year 12. (Modern electronics and engineering depend on imaginary numbers so, far from being imaginary, they are actually vital for making stuff in the real world.)
Now the ‘but only kind of’ part:
Actually, I’m just going to leave it at ‘only kind of’ because I don’t want to confuse what you already know about square roots. But it’s not exactly true to say that the square root of -1 is i. This is one we really will have to leave for Year 12!