factors with minus signFactorsWithMinusSign2007-11-03 14:23:182008-06-24 06:11:20Topicproduct of negative numbersproductpower% this is the default PlanetMath preamble. as your knowledge
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\PMlinkescapeword{factor} \PMlinkescapeword{factors} \PMlinkescapeword{base}
\PMlinkescapeword{power} \PMlinkescapeword{powers}
The sign (cf. plus sign, opposite number) rule
\begin{align}
(+a)(-b) = -(ab),
\end{align}
derived in the \PMlinkname{parent entry}{ProductOfTwoNegativeNumbers} and concerning numbers and elements $a,\,b$ of an arbitrary ring, may be generalised to the following
\textbf{Theorem.} If the sign of one \PMlinkname{factor}{Product} in a ring product is changed, the sign of the product changes.
\textbf{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.
\textbf{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:
$$(-a)^{2n+1} = -a^{2n+1}, \quad (-a)^{2n} = a^{2n} \qquad (n \in \mathbb{N})$$