Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How would I go about proving the following identity:

$$\frac{1}{\left|z\right|} = \left|\frac{1}{z}\right|$$

I keep finding myself going in circles. I've tried using this identity: $|z|^2 = z^*z$ conjugate.

share|improve this question
    
Exponential form of complex number. –  Gina Jul 8 at 4:09
    
@Gina is that the "e^(i*theta)" form? –  Jackson Jul 8 at 4:11
    
The left side is $\frac{1}{\sqrt{a^2+b^2}}$. To compute the right side, try writing $z=a+bi$ and then computing $1/z$. Hint: multiply the numerator and denominator by $a-bi$. (If you have polar coordinates available, however, this problem is very simple.) –  Ian Jul 8 at 4:12
    
$z=re^{i\theta}$ –  Gina Jul 8 at 4:12

1 Answer 1

up vote 6 down vote accepted

Hint: Remember that for complex numbers $a$ and $b$, $|a\cdot b| = |a|\cdot|b|$. Based on this, what can we say about $\left|z\cdot \frac{1}{z} \right|$?

share|improve this answer
    
Great hint! Figured it out. –  Jackson Jul 8 at 5:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.