# difference

The 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

• $b\!+\!(a\!-\!b)\;=\;a$

• $a\!-\!b\;=\;a\!+\!(-b)$

• $-(a\!-\!b)\;=\;b\!-\!a$

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

• $a\!-\!a\;=\;0$

 Title difference Canonical name Difference Date of creation 2013-03-22 17:33:35 Last modified on 2013-03-22 17:33:35 Owner pahio (2872) Last modified by pahio (2872) Numerical id 16 Author pahio (2872) Entry type Definition Classification msc 20K99 Classification msc 00A05 Classification msc 11B25 Related topic VectorDifference Related topic SetDifference Related topic Multiple Related topic GeneralAssociativity Related topic Quotient Related topic DifferenceOfVectors Defines minuend Defines subtrahend