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 $x^ax^b~=~x^{a+b}.$

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