The Opposite of Addition

Last time we looked at three mathematical operations, addition (combining two groups of things into one big thing and counting how many things you have in the big thing), multiplication (a special type of addition where you add together a certain number of equal groups of things) and raising to the power (a special type of multiplication where you multiply a number by itself a certain number of times).

Each of these operations has its opposite, and I want to say a few words about those. These do not arise in the formula that we are looking at, but they do crop up in some of the explanations of the individual components of the formula.

Firstly the opposite of addition is subtraction. You start with some number of things, remove a certain amount of them, and count how many are left. You have subtracted the certain amount from the number of things.

What you can also do is just consider this to be a special type of addition where you add a negative number to your starting number. At this point we encounter our first real philosophical problem. What is a negative number? We have been imagining lots of "things" in our discussions of addition, subtraction, and raising to the power. These things could have been anything - carrots, tractors, stamps, people, bacteria - it doesn't really matter. But you could picture the thing in your minds eye - you could touch it, look at it - it was real. We call these numbers of things, NATURAL numbers. But what can it possibly mean to talk about a negative tractor? What colour would it be?

Well, this is a thorny problem that ancient mathematicians dealt with by ignoring it and hoping it would go away. For instance the ancient Egyptian mathematicians said there was no sensible answer to the question "what number, if you multiply it by four and then add twenty, equals zero". Today we would say that the answer is negative five, because four lots of negative five make negative twenty, and twenty added to negative twenty makes zero. In the middle ages, Hindu and Muslim mathematicians worked out how to use negative numbers, primarily as a reflection of a debt. Meanwhile the western world was trying to work out how to stop the barbarians from clubbing them over the head and being eaten. As recently as the Eighteenth century (when America got its independence) mathematicians would ignore negative answers to problems because they were meaningless.

Nowadays we are relatively comfortable with the idea of negative numbers. If you gift me your only tractor for my birthday, then you have zero tractors. If, however, I borrow your only tractor you have a negative tractor because I am due to return your tractor to you.

So if you have five of something, and you add negative three somethings to it you end up with two of the somethings left.

There is an important difference between ACTUAL subtraction and adding a negative number. Remember that with addition, it does not matter which way round you do it. So, if you have four and you add negative three to that you get one. If you have negative three and add four to it you get one again. Just as we would have expected. BUT if you have four and SUBTRACT three you get one, WHILE if you have three and SUBTRACT four you get negative one. So watch with this - if subtracting you need to get the numbers in the right order, but if you just add using negative numbers it does not matter.

We use the symbol \(-\) to show the action of subtracting, and we put the same symbol to the left of a number to show that it is negative: \(-5\).