Determining Variables Involving Complex Numbers

[Freye]

Homework Statement

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

Homework Equations

i^2=-1

R is the set of all real numbers

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 realized that my solution really isn't going anywhere. Maybe someone could just give me a hint to start me off on the right track.

 

[LCKurtz]

Homework Statement

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

Homework Equations

i^2=-1

R is the set of all real numbers

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 realized that my solution really isn't going anywhere. Maybe someone could just give me a hint to start me off on the right track.

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

1 + 0i = (ac-bd)+(bc+ad)i

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

 

[Freye]

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

1 + 0i = (ac-bd)+(bc+ad)i

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

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

So I do:

1-ac+bd=(bc+ad)i + 0i

but I don't see how that gives me equations to solve for c and d.

 

[LCKurtz]

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

So I do:

1-ac+bd=(bc+ad)i + 0i

but I don't see how that gives me equations to solve for c and d.

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

 

[Freye]

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

so from:

1+0i=ac-bd+(ad+bc)i

Am I allowed to say that:

ac-bd=1 and (ad+bc)i =0i ?

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

 

[LCKurtz]

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

 

[Freye]

Ok, thank you very much