# Modulus of a complex number

Definition. The modulus (or absolute value) of a complex number $$z$$ is defined as $$|z|=\sqrt{x^2+y^2}$$ where $$x = \text{Re}(z)$$ and $$y=\text{Im}(z)$$. In other words, it is the distance from the origin on the complex plane.

Notice that when $$z$$ is a real number, this definition coincides with the definition of the absolute value.

Example. The modulus of $$3+4i$$ is $$\sqrt{3^2+4^2} = 5$$.

## Modulus properties

The modulus is non-negative, that is, for all complex $$z$$, we have $$|z| \ge 0.$$

The modulus is multiplicative, that is, for all complex $$z$$, $$w$$, we have $$|zw| = |z||w|.$$

The implication of this is that the modulus preserves division: $$\left| \frac{z}{w} \right| = \frac{|z|}{|w|}.$$

It also implies that, for positive integer $$n$$, we have $$\left|z^n\right| = |z|^n.$$

We have the following identity which links the conjugate to the modulus: $$z \overline{z} = |z|^2.$$

Let us prove the multiplicativity of the modulus.

Theorem. For complex $$z,w$$, we have $$|zw| = |z||w|$$.
Proof. Let $$z = a+bi$$ and $$w = c+di$$. Then by expanding, we have \begin{align*} |zw|^2 &= |(a+bi)(c+di)|^2 \\ &= |(ac-bd)+(ad+bc)i|^2 \\ &= (ac-bd)^2 + (ad+bc)^2 \\ &= a^2 c^2 + b^2 d^2 + a^2 d^2 + b^2 c^2. \end{align*} Similarly, we have
\begin{align*} |z|^2 |w|^2 &= (a^2 + b^2) (c^2+d^2) \\ &= a^2 c^2 + b^2 d^2 + a^2 d^2 + b^2 c^2. \end{align*} Hence, $$|zw|^2 = |z|^2 |w|^2$$. Since the modulus is non-negative, we take the positive square root of both sides to see that $$|zw| = |z||w|$$.

## Problems

1. For complex number $$z=x+yi$$, use Pythagoras’ theorem to find the distance between $$0$$ and $$z$$.
2. Find the modulus of:
1. $$1-i$$
2. $$-12-5i$$
3. $$1+\sqrt{3} i$$
4. $$8$$
5. $$-2i$$
3. Prove the following for complex $$z,w$$:
1. $$\left| \frac{z}{w} \right| = \frac{|z|}{|w|}$$
2. $$\left|z^n\right| = |z|^n$$
3. $$z \overline{z} = |z|^2$$