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| Title of object: |
product of negative numbers |
| Canonical Name: |
ProductOfNegativeNumbers |
| Type: |
Derivation |
| Created on: |
2007-10-20 10:16:23 |
| Modified on: |
2007-10-20 10:33:25 |
| Classification: |
msc:13A99, msc:97D40 |
| Synonyms: |
product of negative numbers=product of two negative numbers |
Revision comment (for changes between this and next version):
| Changes for correction #13181 ('grammar, slight confusion'). |
Preamble:
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\usepackage{amssymb}
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Content:
\PMlinkescapeword{factor} \PMlinkescapeword{order}
\textbf{\PMlinkescapetext{Why the product of two negative numbers is positive?}}\\
The following properties of multiplication are well known:
\begin{itemize}
\item The product of two numbers equals to zero always when one of the numbers equals to zero.
\item Commutative law:\, $xy = yx$
\item Distributive law:\, $x(y\!+\!z) = xy\!+\!xz$
\end{itemize}
It is natural to require these properties also when one or both of the numbers is negative.
We need in the following calculation only the first and the last property:
$$0 = (+a)\!\cdot\!0\, = \,(+a)[(+b)\!+\!(-b)]\, = \,(+a)(+b)\!+\!(+a)(-b)\, = \,ab\!+\!(+a)(-b)$$
Because the value of the sum in the end is zero, the latter summand $(+a)(-b)$ must be the opposite number of the former addend $ab$. Accordingly we may write:
\begin{itemize}
\item $(+a)(-b) = -(ab)$
\end{itemize}
This result means that as the sign of the second factor of the product $(+a)(+b)$ is changed, the sign of the whole product changes. The same concerns of course also the first factor of the product, since by the commutative law, the order of the factors can be changed.
But if one changes in the product $(+a)(+b)$ the signs of both factors, first one and then the other, the sign of the product changes twice, i.e. it remains unchanged. Thus we obtain the final result
\begin{itemize}
\item $(-a)(-b) = ab.$
\end{itemize}
This says that {\em we have to define the product of two negative numbers positive}.
\begin{thebibliography}{9}
\bibitem{VA}{\sc K. V\"ais\"al\"a:} {\em Algebran oppi- ja esimerkkikirja I}. Fifth edition. Werner S\"oderstr\"om osakeyhti\"o, Porvoo \& Helsinki (1952).
\end{thebibliography}
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