# Problems with Subtraction

Subtraction is the operation of “taking away”, and acts like the “opposite” of addition. For example, we can use a subtraction to “undo” an addition:

(*x* + *y*) − *y* = *x*

This is a fine idea, but unfortunately subtraction is not as well-behaved as addition, and I personally think it causes more problems than it solves.

## Subtraction is *Redundant*

We can define subtraction using addition and negation:

$$x - y \equiv x + \ngtv{y}$

Hence situations involving negatives don’t need subtraction: we can stick to additions, and sprinkle in some negatives as needed. Furthermore, if a situation *doesn’t* allow negatives, then it doesn’t allow subtraction either, since it gives negative answers when the second value is larger than the first (e.g. $5 - 7 = ).

So any time we might want subtraction, we could use negatives instead; and any time we can’t use negatives, we also can’t use subtraction. Thus subtraction isn’t *needed*. We might find it *useful* . Unf subtraction is just an abbreviation

We can’t swap around the values of a subtraction, since *x* − *y*/*n**e**q**y* − *x*. In fact, we get the negative result: $x - y = \ngtv{y - x}$ (we say that subtraction “anti-commutes”).

Subtraction becomes ambiguous when there are more than two values. For example *x* − *y* − *z* could mean (*x* − *y*) − *z* (“*x* take away *y*, then take away *z*”), *or* it could mean *x* − (*y* − *z*) ("*y* take away *z*, taken away from *x*). These are different things, e.g. $(1 - 2) - 3 = \ngtv{1} - 3 = \ngtv{4}$, whilst $1 - (2 - 3) = 1 - \ngtv{1} = 2$. In contrast can be done We can switch the values Subtraction is not commutative

$$x - y \equiv x + \ngtv{y}$$