The difference of two numbers $a$ and $b$ is a number $d$ such that $$b\!+\!d = a.$$ The difference of $a$ (the minuend) and $b$ (the subtrahend) is denoted by $a\!-\!b$ .

The definition is similar for the elements $a,\,b$ of any additive Abelian group (e.g. of a vector space).The difference of them is always unique.

Note 1. Forming the difference of numbers (resp. elements), i.e. subtraction, is in a certain sense converse to the addition operation: $$(x\!+\!y)\!-\!y \;=\; x$$

Note 2. As for real numbers, one may say that the difference between $a$ and $b$ is $|a\!-\!b|$ (which is the same as $|b\!-\!a|$ ); then it is always nonnegative. For all complex numbers, such a phrase would be nonsense.

Some identities

• $b\!+\!(a\!-\!b) = a$
• $a\!-\!b = a\!+\!(-b)$
• $-(a\!-\!b) = b\!-\!a$
• $n(a\!-\!b) = na\!-\!nb \quad (n\in\mathbb{Z})$
• $a\!-\!a = 0$