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/neqy − 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


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