Complex Numbers: Subtraction, Division

Date: 08/12/2001 at 23:04:42
From: April 
Subject: Subtraction and division for imaginary numbers 

I know that when adding imaginary numbers the formula is

(a+bi)+(c+di) = (a+c)+(b+d)i 

but for subtraction, is the formula (a+c)-(b+d)i ?

And how do you divide imaginary numbers - like for instance: 

a+bi/a-bi?

Date: 08/12/2001 at 23:39:04
From: Doctor Peterson
Subject: Re: Subtraction and division for imaginary numbers 

Hi, April.

The formula for addition works because complex numbers follow the 
associative and commutative properties just as real numbers do. We can 
do the same for subtraction, and get

    (a+bi) - (c+di) = (a+bi) + -(c+di)
                    = (a+bi) + (-c + -di)
                    = a + bi + -c + -di
                    = a + -c + bi + -di
                    = (a-c) + (b-d)i

That is, you subtract the real parts and the imaginary parts.

For division, we generally think of it as simplifying a fraction, much 
like the way you rationalize the denominator of a fraction. In this 
case, you can multiply numerator and denominator by the complex 
conjugate of the denominator. So in general, we get

    a+bi   (a+bi)(c-di)   (ac+bd) + (bc-ad)i
    ---- = ------------ = ------------------
    c+di   (c+di)(c-di)        c^2 + d^2

Since the denominator is now a real number, you can just divide the 
real and imaginary parts of the numerator by it to get the answer.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/