Associated Topics || Dr. Math Home || Search Dr. Math

### 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.

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

---- = ------------ = ------------------
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/
```
Associated Topics:
High School Imaginary/Complex Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search