Editing MultiplyingComplexNumbers
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Given two complex numbers, $a+bi$ and $c+di,$ we can ask about the product: * $(a+bi)(c+di)$ There are two ways to proceed. The easiest is simply to use the usual rules of algebra. We expand brackets: * $(a+bi)(c+di)~=~ac+adi+bci+bdi^2$ We remember that $i^2=-1,$ so this is: * $(a+bi)(c+di)~=~ac+adi+bci-bd$ We rearrange: * $ac+adi+bci-bd~=~ac-bd+adi+bci$ Then we can take out a common factor of $i$ to get the complex number in the standard form: * $(a+bi)(c+di)~=~(ac-bd)+(ad+bc)i$ Alternatively, we can express the numbers using the polar representation of a complex number: |>> $z_1=re^{i\theta},\quad\quad~z_2=se^{i\phi}$ <<| Then the product is: |>> $z_1z_2~=~rse^{i(\theta+\phi)}$ <<| This uses the index laws that $x^ax^b~=~x^{a+b}.$