Editing DifferenceOfTwoSquares
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[[[>60 This second version forms the basis of a relatively quick way to square integers, and that in turn gives us a technique for multiplying integers in general. To square a two digit number like $53,$ think of that as $a$ in the above version. [[[>40 We had a free choice for $b,$ and by using $b=3$ we get $(53-3)$ which is $50$ - a nice number to work with. ]]] By using $b=3$ we get: * $53^2~=~(53+3)(53-3)~+~3^2$ So that simplifies to $53^2~=~56{\times}50~+~9.$ Now $56{\times}50$ is just $28{\times}100,$ which is $2800.$ So that means $53^2~=~2800~+~9$ which is $2809.$ With a little practice and knowledge of your times tables and number bonds, this can be done quickly and easily in your head. ]]] An identity that turns up repeatedly in algebra, and which needs to be recognised in all its many disguises. * $a^2-b^2~=~(a+b)(a-b)$ Note that this works in any commutative algebra, which means it applies not just to integers, but also to real numbers, and complex numbers. It does /not/ apply to matrices or quaternions, where multiplication is non-commutative. It can also be re-written as: * $a^2~=~(a+b)(a-b)~+~b^2$ ---- Related: * Square numbers