How can I break up $z = \frac{5}{9+3i}$

Question

Into its real and imaginary components? Wolfram tells me it's equivalent to $\frac{1}{2}+\frac{i}{6}$, but I don't know how to arrive there myself.

Thank you!

What have you tried? You might also want to consider doing something about your accept rate.
You might want to extract obvious factors to give $z = \cfrac 5 3 .\cfrac 1 {3+i} .\cfrac {3-i}{3-i}$
Going with Sean's hint, think of what happens when you multiply a complex number by its conjugate.
The general procedure is to multiply the numerator and denominator of the given complex fraction by the complex conjugate of the denomnator when that denominator is not real.

Makoto Kato · Answer 1 · 2012-10-07 21:10:39Z

up vote 5 down vote

$z = \frac{5}{9+3i} = \frac{5(9-3i)}{(9+3i)(9-3i)} = \frac{45-15i}{9^2 + 3^2} = \frac{45}{90} - \frac{15i}{90} = \frac{1}{2} - \frac{i}{6}$

answered Oct 7 '12 at 21:10

Makoto Kato
11.3k1946

Xuan Huang · Answer 2 · 2012-10-08 17:17:38Z

Here, we must use our knowledge of complex numbers and their conjugates as well as our knowledge of the difference of squares:

$z = \frac{5}{9+3i}$

$ = \frac{5(9-3i)}{(9+3i)(9-3i)}$

$ = \frac{45-15i}{9^2 - (i^2)(3^2)}$

$ = \frac{45-15i}{90}$

$ = \frac{1}{2} - \frac{1}{6}i$

2 Answers