So far in our study of the arithmetic of rational numbers, we've had addition ("putting apples and chunks of apples side by side and counting what you've got"), subtraction ("the same, but you're allowed anti-apples too"), and multiplication ("make a rational number, but instead of starting from apple, start from some other number").
Division is what really sets the rational numbers apart from the integers, and it is the mathematician's answer to the question "if I have some apples, how do I share them among my friends?".
If you have an integer number of apples (that is, some number of apples and anti-apples - no chunks allowed, just whole apples and anti-apples), and you want to share them with friends, sometimes you'll get lucky. If you have four apples, for instance, then you can share them out between yourself and one friend, giving each person two apples.
But sometimes (often, in fact) you'll get unlucky. If you want to share four apples between yourself and two others, then you can give each person one apple, but there's this pesky single apple left over which you just can't share.
The trick, obvious to anyone who has ever eaten a cake, is to cut the leftover apple into three equally-sized pieces and give each person a piece. Now we have shared out all four apples equally.
But in order to do so, we've left the world of the integers, and in getting out our knife, we started working in the rationals. How much apple has everyone received, when we shared four apples among three people (that is, myself and two friends as recipients of apple)?
Everyone got .
Indeed, everyone got one whole apple; and then we chopped the remaining apple into three -chunks and gave everyone one chunk. So everyone ended up with one apple and one -chunk.
By our instant addition rule [1],
There's another way to see this, if (laudably!) you don't like just applying rules. We could cut every apple into three pieces at the beginning, so we're left with four collections of three -sized chunks. But now it's easy to share this among three people: just give everyone one of the -chunks from each apple. We gave everyone four chunks in total, so this is .
The rationals provide the natural answer to all "sharing" questions about apples.
If you've forgotten it, check out the addition page again; it came from working out a chunk size out of which we can make both the -chunk and the -chunk.