So far, we have met the idea of a rational number, treating them as chunks of apples, and how to add them together. Now we will discover how the idea of the anti-apple (by analogy with the integers' anti-cow) must work.
Just as we had an anti-cow, so we can have an anti-apple. If we combine an apple with an anti-apple, they both annihilate, leaving nothing behind. We write this as .
A very useful thing for you to ponder for thirty seconds (though I will give you the answer soon): given that means "divide an apple into equal pieces, then take copies of the resulting little-piece", what would mean? And what would mean?
would mean "divide an apple into equal pieces, then take copies of the resulting little-piece". That is, turn it into an anti-little-piece. This anti-little-piece will annihilate one little-piece of the same size: .
, on the other hand, would mean "divide an anti-apple into equal pieces, then take copy of the resulting little-anti-piece". But this is the same as : it doesn't matter whether we do "convert to anti, then divide up the apple" or "divide up the apple, then convert to anti". That is, "little-anti-piece" is the same as "anti-little-piece", which is very convenient.
What about chunks of apple? If we combine half an apple with half an anti-apple, they should also annihilate, leaving nothing behind. We write this as .
How about a bit more abstract? If we combine an apple with half an anti-apple, what should happen? Well, the apple can be made out of two half-chunks (that is, ); and we've just seen that half an apple will annihilate half an anti-apple; so we'll be left with just one of the two halves of the apple. More formally, ; or, writing out the calculation in full,
Let's go the other way round: if we combine an anti-apple with half an apple, what happens? It's pretty much the same as the opposite case except flipped around: the anti-apple is made of two anti-half-chunks, and the half apple will annihilate one of those chunks, leaving us with half an anti-apple: that is, .
We call all of these things subtraction: "subtracting" a quantity is defined to be the same as the addition of an anti-quantity.
Since we already know how to add, we might hope that subtraction will be easier (since subtraction is just a slightly different kind of adding).
In general, we write for "the -sized building block, but made by dividing an anti-apple (instead of an apple) into equal pieces". Remember from the pondering above that this is actually the same as , where we have divided an apple into equal pieces but then taken of the pieces.
Then in general, we can just use the instant addition rule that we've already seen. [1] In fact,
Why are we justified in just plugging these numbers into the formula, without justification? You're quite right if you are dubious: you should not be content merely to learn this formula%[2]%. In the rest of this page, we'll go through why it works, and how you might construct it yourself if you forgot it. I took the choice here to present the formula first, because it's a good advertisement for why we use the notation rather than talking about "-chunks" explicitly: it's a very compact and neat way of expressing all this talk of anti-apples, in the light of what we've already seen about addition.
Very well: what should be? We should first find a smaller chunk out of which we can build both the and chunks. We've seen already that will work as a smaller chunk-size.
Now, what is expressed in -chunks? Each -chunk is of the -chunks, so of the -chunks is lots of " lots of -chunks": that is, of them.
Similarly, is just lots of -chunks.
So, expressed in -chunks, we have lots of positive chunks, and lots of anti-chunks. Therefore, when we put them together, we'll get chunks (which might be negative or positive or even zero - after all the annihilation has taken place we might end up with either normal or anti-chunks or maybe no chunks at all - but it's still an integer, being a number of chunks).
So the total amount of apple we have is , just like we got out of the instant formula.
Recall: this was . Remember that the order of operations in the integers is such that in the numerator, we calculate both the products first; then we add them together.
This goes for all of maths! It's not simply a collection of arbitrary rules, but a proper process that we use to model our thoughts. Behind every pithy, unmemorable formula is a great edifice of motivation and reason, if you can only find it.