# factors with minus sign

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

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

derived in the parent entry (http://planetmath.org/productofnegativenumbers) and concerning numbers and elements $a$, $b$ of an arbitrary ring, may be generalised to the following

Theorem.  If the sign of one factor (http://planetmath.org/Product) 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:

 $(-a)^{2n+1}=-a^{2n+1},\quad(-a)^{2n}=a^{2n}\qquad(n\in\mathbb{N})$
Title factors with minus sign FactorsWithMinusSign 2015-02-04 12:30:11 2015-02-04 12:30:11 pahio (2872) pahio (2872) 7 pahio (2872) Topic msc 97D40 msc 13A99 sign rules for products GeneralAssociativity Multiplication DoublyEvenNumber