Michael B. answered • 11/16/12

I can provide your 'A-HA' moment

Division by complex numbers is handled very much like division by radicals... when there is a radical in the denominator, we "rationalize" it by multiplying both the numerator and the denominator by the radical. This has the effect of "getting rid of" the radical in the denominator.

A very similar concept is used to "realize" the denominator when it has an imaginary component. Keep in mind that i * i = -1, and so if we multiply both the numerator and the denominator by i, then the denominator will simplify to -1. Thus:

` 4 i 4i 4i`

--- * --- = ---- = ---- = -4i

i i i^{2} -1

So the answer is **-4i**.

If the denominator was a complex number (instead of simply an imaginary number) then we would multiply by the complex conjugate of the complex number, which is just the same number but with the opposite sign on the imaginary part. For example, the complex conjugate of 1+2i is 1-2i. So if we had an expression that was something divided by 1+2i, we would multiply both the numerator and denominator by the complex conjugate 1-2i. The result of this is that the denominator will become a real number (no imaginary part.)