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.

The anti-apple

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 $1+(-1)=0$.

A very useful thing for you to ponder for thirty seconds (though I will give you the answer soon): given that $\frac{a}{n}$ means “divide an apple into $n$ equal pieces, then take $a$ copies of the resulting little-piece”, what would $\frac{-1}{n}$ mean? And what would $-\frac{1}{n}$ mean?

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 $\frac{1}{2}+(-\frac{1}{2})=0$.

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, $1=\frac{1}{2}+\frac{1}{2}$); 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, $1+(-\frac{1}{2})=\frac{1}{2}$; or, writing out the calculation in full, $$1+(-\frac{1}{2})=\frac{1}{2}+\frac{1}{2}+(-\frac{1}{2})=\frac{1}{2}$$

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, $(-1)+\frac{1}{2}=-\frac{1}{2}$.

We call all of these things subtraction: “subtracting” a quantity is defined to be the same as the addition of an anti-quantity.

General procedure

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 $-\frac{1}{n}$ for “the $\frac{1}{n}$-sized building block, but made by dividing an anti-apple (instead of an apple) into $n$ equal pieces”. Remember from the pondering above that this is actually the same as $\frac{-1}{n}$, where we have divided an apple into $n$ equal pieces but then taken $-1$ of the pieces.

Then in general, we can just use the instant addition rule that we’ve already seen. ^[1] In fact, $$\frac{a}{m}-\frac{b}{n}=\frac{a}{m}+\left(\frac{-b}{n}\right)=\frac{a\times n+(-b)\times m}{m\times n}=\frac{a\times n-b\times m}{m\times n}$$

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 $\frac{a}{n}$ notation rather than talking about “$\frac{1}{n}$-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 $\frac{a}{m}-\frac{b}{n}$ be? We should first find a smaller chunk out of which we can build both the $\frac{1}{m}$ and $\frac{1}{n}$ chunks. We’ve seen already that $\frac{1}{m\times n}$ will work as a smaller chunk-size.

Now, what is $\frac{a}{m}$ expressed in $\frac{1}{m\times n}$-chunks? Each $\frac{1}{m}$-chunk is $n$ of the $\frac{1}{m\times n}$-chunks, so $a$ of the $\frac{1}{m}$-chunks is $a$ lots of ”$n$ lots of $\frac{1}{m\times n}$-chunks”: that is, $a\times n$ of them.

Similarly, $\frac{b}{n}$ is just $b\times m$ lots of $\frac{1}{m\times n}$-chunks.

So, expressed in $\frac{1}{m\times n}$-chunks, we have $a\times n$ lots of positive chunks, and $b\times m$ lots of anti-chunks. Therefore, when we put them together, we’ll get $a\times n-b\times m$ 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 $\frac{a\times n-b\times m}{m\times n}$, just like we got out of the instant formula.

1. ^︎

   Recall: this was $\frac{a}{m}+\frac{b}{n}=\frac{a\times n+b\times m}{m\times n}$. 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.

2. ^︎

   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.