opposite number

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

$$-a+a=\mathrm{\hspace{0.33em}0}.$$

Some properties:

• $-a=(-1)\cdot a$

• $-0=\mathrm{\hspace{0.33em}0}$

• $-(-a)=a$

• $-(a+b)=(-a)+(-b)$

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

• $-(a-b)=b-a$

• $-{\sum }_{j=1}^n{a}_j={\sum }_{j=1}^n(-a_j)$

• $-{\int }_a^{b}f(x)𝑑x={\int }_b^{a}f(x)𝑑x$

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

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