factors with minus sign

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

(+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 valuesMathworldPlanetmathPlanetmathPlanetmath 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=-a2n+1,(-a)2n=a2n  (n)
Title factors with minus sign
Canonical name FactorsWithMinusSign
Date of creation 2015-02-04 12:30:11
Last modified on 2015-02-04 12:30:11
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Topic
Classification msc 97D40
Classification msc 13A99
Synonym sign rules for products
Related topic GeneralAssociativity
Related topic Multiplication
Related topic DoublyEvenNumber