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The sign (cf. plus sign, opposite number) rule

$\displaystyle(+a)(-b)=-(ab),$ | (1) |

derived in the parent entry and concerning numbers and elements $a$, $b$ of an arbitrary ring, may be generalised to the following

Theorem. If the sign of one factor in a ring product is changed, the sign of the product changes.

Corollary 1. The product of real numbers is equal to the product of their absolute values^{} equipped with the “$-$” sign if the number of negative factors is odd and with “$+$” sign if it is even. Especially, any odd power of a negative real number is negative and any even power of it is positive.

Corollary 2. Let us consider natural powers of a ring element. If one changes the sign of the base, then an odd power changes its sign but an even power remains unchanged:

$(-a)^{{2n+1}}=-a^{{2n+1}},\quad(-a)^{{2n}}=a^{{2n}}\qquad(n\in\mathbb{N})$ |

## Mathematics Subject Classification

97D40*no label found*13A99

*no label found*

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