difference

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 for the elements $a,b$ of any 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

• $b+(a-b)=a$

• $a-b=a+(-b)$

• $-(a-b)=b-a$

• $n(a-b)=na-nb\mathit{\hspace{1em}}(n\in \mathbb{Z})$

• $a-a=\mathrm{\hspace{0.33em}0}$