The Opposite of Multiplication

Following in from addition and subtraction, the opposite of multiplication is division. This is the process of taking a group of things and asking, if I have to separate these things into a certain number of equally sized groups, how many will be in each group. So if you have six apples and three hungry people, you would want to know how many apples each hungry person could have if they all had to get the same amount. This is the process of dividing six by thee. Each hungry person will get two apples.

Like subtraction and addition, we can treat the act of division as a special type of multiplication. But my feeling is that this gets hopelessly circular very quickly. The logic goes that if you divide a number by another number that is the same as multiplying the first number by the inverse of the second number.

Well, what the hell is an inverse? Good question. An inverse is the size of the individual pieces you get if you split the unit number (for us the number one remember) up into the same number of equal pieces as the original number. If you consider the apples and hungry people example, we are dividing six by three. The inverse of three is a third, because if you split the number one up into three equal pieces, each piece is a third in size. So instead of asking how many ones are in three, you ask the INVERSE of the question, which is how big is each piece if one is split up into three equal pieces.

What you then do, according to the multiplication theory of division, is multiply six by one third. Remember that multiplication is replicating one collection of things so that you end up with a second number of those collections. I did not say that you had to use WHOLE numbers.

So what is a whole number, and more importantly, what is NOT a whole number? As with negative numbers and addition, multiplying by an inverse forces us to consider another new type of number. A whole number is a number that is made up of ones and ones alone. It could have one one in it, or two, or three or fifteen billion, or negative twenty two, but it is only made up of ones. Another name for this type of number is an integer. If it is a positive integer (more than zero) then it is a natural number (but still an integer).

A number which is NOT a whole number is made up of PARTS OF ones. It may be made of some whole ones as well, but it will have parts of one in there too. So, the quantity two and a half is made up of two ones, and a half of one. The time a quarter past three is three hours (past midnight or midday) and a quarter part of an hour. The question is are these quantities or times really numbers? We quickly find ourselves back at the same kind of philosophical problem as we met at the negative number stage. Is it meaningless to take about half a kilogram of flour? No. Is it meaningless to talk about half an egg? Maybe - after all you would never go into a shop and try to buy three and a half eggs.

Well the answer, as you may have expected, is that we are quite comfortable now with the idea of these parts of one being numbers in their own right. They are just not whole numbers. They are also much less abstract than negative numbers. If you have half a cow, you can at least see and smell it, albeit the experience may not be a pleasant one.

So this part, or fraction, of one is a number is its own right. You can then multiply another number by that number. What does it mean to multiply a number by a fraction? Well remember that multiplying a group of things by a number means you end up with that number of groups of things. So all that happens if you multiply by a fraction is that you end up with a fraction of the original group. So if you have a group of twenty things, and you multiply that by a half, you end up with ten things. It still does not matter what order you do this in, so if you gather up twenty halves, and put them all together, each one finds a friend and you end up with ten whole things. In that example, of course, one half is the INVERSE of two, because there are two equal halves in one, rather than two ones in two.

So why is all this circular? Because we represent fractions as one number divided by another. So we show a half as one divided by two. So we are back to division already. Brilliant. So to be able to multiply by an inverse instead of dividing the first thing you have to do is divide to get the value of the inverse. Seemingly pointless. However it will be surprisingly useful when we come to try to multiply fractions together. Lets have a look at that kind of thing before we look at our last opposite operation.