Modulus

Introduction

In real numbers, the modulus is the magnitude, or distance of that number to the origin, $0$. For a complex number, it is the distance of that complex number to the origin, $0+0i$.

Given a complex number $a+bi$, the modulus is denoted as :

$$|z|=\sqrt{a^2+b^2}$$

Example

Properties

The following can be deduced, try prove them as an exercise. Given complex numbers $z,w$ and a real constant $a$ :

$$|az|=a|z|$$ $$|zw|=|z|\cdot|w|$$ $$|\overline{z}|=|z|$$

The last one is particularly important, taking the conjugate of a complex number $z$ simply flips it symmetrically over the real axis, but the distance to the origin remains the same.

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