factors with minus sign
The sign (cf. plus sign, opposite number) rule
derived in the parent entry (http://planetmath.org/productofnegativenumbers) and concerning numbers and elements , 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:
|Title||factors with minus sign|
|Date of creation||2015-02-04 12:30:11|
|Last modified on||2015-02-04 12:30:11|
|Last modified by||pahio (2872)|
|Synonym||sign rules for products|