What are the Real Numbers c and d for Inverting Complex Numbers?

[TheDoorsOfMe]

Homework Statement

Suppose a and b are real numbers, not both 0. Find real numbers c and d, such that

\frac{1}{a + bi} = c + di

Homework Equations

I said that:

1 = (c + di) (a + bi)

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

so if b = 0 then
\frac{1}{a} = c + di

The rationale holds for if a = 0. Since this is equal to a real number 1/a can I just say that c and d are real numbers?

 

[rootX]

You need to understand that if
x + yj = a + bj
then
x = a
and
y = b

2)
Multiply left side by a-bj / a-bj