difference of squaresDifferenceOfSquares2008-01-20 18:36:462008-03-27 14:05:09Topicdifference% this is the default PlanetMath preamble. as your knowledge
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One of the most known and used \PMlinkname{formulas}{Equation} of mathematics is the one concerning the product of sum and difference:
\begin{align}
(a+b)(a-b) = a^2-b^2
\end{align}
This form may be used for multiplying any sum of two numbers (terms) by the difference of the same numbers (terms).\\
In the form
\begin{align}
a^2-b^2 = (a+b)(a-b)
\end{align}
the formula is used for factoring binomials which are the difference of two squares.\\
(1) is sometimes called the {\em conjugate rule}, especially in articles written in Sweden (in Swedish: {\em konjugatregel}).\\
(1) is an identic equation for all numbers $a,\,b$ and, more generally, for arbitrary elements $a,\,b$ of any commutative ring.\, Conversely, it is easy to justify that if (1) is true for all elements $a,\,b$ of a ring, then the ring is commutative.\, By the way, $a\!+\!b$ and $a\!-\!b$ also commute with each other in a non-commutative ring.