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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.

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Exponential form of complex number. – Gina Jul 8 '14 at 4:09
@Gina is that the "e^(i*theta)" form? – Jackson Jul 8 '14 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 '14 at 4:12
$z=re^{i\theta}$ – Gina Jul 8 '14 at 4:12
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 $\left|z\cdot \frac{1}{z} \right|$?

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Great hint! Figured it out. – Jackson Jul 8 '14 at 5:09

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