# opposite number

The opposite number of a number $a$ is such a number  $-a$  that

 $-a\!+\!a\;=\;0.$

Some properties:

• $-a\;=\;(-1)\!\cdot\!a$

• $-0\;=\;0$

• $-(-a)\;=\;a$

• $-(a\!+\!b)\;=\;(-a)\!+\!(-b)$

• $-(a\!\cdot\!b)\;=\;a\!\cdot\!(-b)\;=\;(-a)\!\cdot\!b$

• $-(a\!-\!b)\;=\;b\!-\!a$

• $-\sum_{j=1}^{n}a_{j}\;=\;\sum_{j=1}^{n}(-a_{j})$

• $-\int_{a}^{b}f(x)\,dx\;=\;\int_{b}^{a}f(x)\,dx$

Exactly similar properties (except the last) are valid in every ring.  The fifth property implies the

If one changes the sign of one factor of a ring product, then the sign of the whole product changes.

 Title opposite number Canonical name OppositeNumber Date of creation 2013-03-22 15:03:25 Last modified on 2013-03-22 15:03:25 Owner pahio (2872) Last modified by pahio (2872) Numerical id 10 Author pahio (2872) Entry type Definition Classification msc 97D99 Classification msc 12D99 Synonym negative [as a noun] Related topic Ring Related topic OppositePolynomial Related topic ConditionOfOrthogonality Related topic Automorphism4 Related topic ProductOfNegativeNumbers Related topic PlusSign