Subtraction

Subtraction is the inverse of addition: you take away one number from another to find out how much is left. The notation $a - b = c$ reads "a minus b equals c". The number $a$ is the minuend, $b$ is the subtrahend, and $c$ is the difference.

The names in subtraction

• minuend — the number you subtract from ($a$),
• subtrahend — the number you take away ($b$),
• difference — the result ($c$).

For example, in $8 - 5 = 3$ the number $8$ is the minuend, $5$ the subtrahend and $3$ the difference.

Subtraction and addition

Subtraction and addition are inverse operations — one "undoes" the other. That lets you check every subtraction with addition:

$a - b = c \iff b + c = a$

Since $15 - 6 = 9$, it must be that $6 + 9 = 15$ — and it is. This is the simplest way to be sure a result is correct.

Properties of subtraction

Unlike addition, subtraction is neither commutative nor associative — order and grouping matter:

$a - b \neq b - a$

For example $7 - 3 = 4$, but $3 - 7 = -4$. There are, however, two handy properties:

• subtracting zero leaves the number unchanged: $a - 0 = a$,
• a number minus itself is zero: $a - a = 0$.

Column subtraction

Larger numbers are subtracted in columns — digit under digit, right to left. When a digit of the minuend is smaller than the digit below it, we borrow one ten from the column to the left.

Take $52 - 27$. Ones: $2$ is smaller than $7$, so we borrow a ten — compute $12 - 7 = 5$. Tens: after the borrow $4$ is left, so $4 - 2 = 2$. The result is $25$.

Common mistakes

• Forgetting the borrow — if you don't reduce the neighbouring digit by 1 after borrowing, the result comes out too large.
• Subtracting the smaller digit from the larger "as a shortcut" — in a column you always subtract the subtrahend's digit from the minuend's, not the other way round; when you can't, you borrow.
• Mixing up the order — $a - b$ is not the same as $b - a$.