# Determining Variables Involving Complex Numbers

1. Sep 18, 2011

### Freye

1. The problem statement, all variables and given/known data

Let a, b in R, not both zero. Find c, d in R such that (a+bi)^-1 = c+di

2. Relevant equations

i^2=-1

R is the set of all real numbers

3. The attempt at a solution
I have a feeling I'm approaching this problem incorrectly but:

1 = (a + bi)(c + di)
=ac + adi + cbi + bdi^2 but i^2=-1
so 1 = ac - bd + (ad + bc)i^2

This is as far as I've attempted becuase I realised that my solution really isn't going anywhere. Maybe someone could just give me a hint to start me off on the right track.

2. Sep 18, 2011

### LCKurtz

You are on the right track, but that last equation should have i instead of i2. It should read:

Set the real and imaginary parts equal to each other, giving two equations in the two unknowns c and d.

3. Sep 18, 2011

### Freye

Oops, I had i written down but I just misstyped it as i^2 here.

So I do:

but I dont see how that gives me equations to solve for c and d.

4. Sep 18, 2011

### LCKurtz

What I meant is to set the real parts equal to each other and ditto the imaginary parts.

5. Sep 18, 2011

### Freye

so from:

Am I allowed to say that:

If so, is this because the imaginary numbers of the equation cannot affect the real numbers and vice versa?

6. Sep 18, 2011

### LCKurtz

So ad + bc = 0, you don't need the i in that equation. But yes, two equations in the unknowns c and d.

7. Sep 18, 2011

### Freye

Ok, thank you very much