Maths in a minute: negative numbers

Negative numbers are easy to imagine if you think of the number line as a giant thermometer which includes sub-zero temperatures. This makes addition and subtraction easy, as you just move up or down the number line by the according amount.

But what about those tricky multiplication rules? Why does positive times negative give negative, and negative times negative give positive? Here the number line can help us too.

Suppose you're standing at the point 0, facing in the positive direction of the number line. You take two steps backwards and you do this 4 times. You end up at the point -8, showing that -2 steps times 4 is -8, ie (-2)x4=-8.

Now suppose you're back at 0, this time facing in the negative direction. You take 2 steps forwards and you do this 4 times. You also end up at point -8, showing that 2 steps times -4 is -8, ie 2x(-4)=-8.

Again, go back to 0, looking in the negative direction. Take 2 steps backwards and do this 4 times. You end up at the point 8. Stepping backwards gives you a -2. Facing in the negative direction gives you a -4. Putting all this together gives (-2)x(-4)=8.


How do you know to face the negative direction instead of the positive direction?

Negative direction is just the where the numbers get smaller or more negative.

I find it easier to visualise a number as consisting of the number's value plus an infinite number of +1, -1 pairs. Then, for me, subtracting negative numbers is just a matter of removing the appropriate number of -1's from the pairs and seeing what's left.
So 5 - -1=(5+1-1)- -1=5+1=6
(-2)x4=subtract 2 lots of +4 from the +1-1 pairs=-8
(-2)x(-4)=subtract 2 lots of -4 from the +1-1 pairs=+8