## Modulus of a complex number

Let $$z = a+bi$$ be a complex number. The modulus of $$z$$, denoted by $$\lvert z \rvert$$, is the real number given by $$\sqrt{a^2+b^2}$$. Note that this quantity can also be written as $$\sqrt{z~\overline{z}}$$.

### Examples

1. If $$z = 1+2i$$, then $$\lvert z \rvert = \sqrt{1^2 + 2^2} = \sqrt{5}$$.

2. If $$z = 3-4i$$, then $$\lvert z \rvert = \sqrt{3^2 + (-4)^2} = \sqrt{9+16} = \sqrt{25} = 5$$.

3. If $$z = 2i$$, then $$\lvert z \rvert = \sqrt{0^2 + 2^2} = \sqrt{4} = 2$$.

4. If $$z = 3$$, then $$\lvert z \rvert = \sqrt{3^2 + 0^2} = \sqrt{9} = 3$$.

The notion of the modulus of a complex number extends the notion of the absolute value of a real number because if $$z$$ is a real number, the modulus of $$z$$ is simply the absolute value of $$z$$. For example, the modulus of $$-2$$ is $$2$$.

Just as the absolute value of a real number represents its distance from $$0$$ on the number line, the modulus represents the distance between a complex number and $$0$$ on the complex plane.

## Properties

There are a number of properties of the modulus that are worth knowing. Let $$z, w\in \mathbb{C}$$. Then,

1. $$\lvert \overline{z} \rvert = \lvert z \rvert$$.

2. $$\lvert -z\rvert = |~z~\rvert$$.

3. If $$\lvert z \rvert = 0$$, then $$z = 0$$.

4. $$\lvert z w \rvert = \lvert z \rvert \lvert w \rvert$$.

5. $$\displaystyle\left|\frac{z}{w}\right| = \frac{\lvert z \rvert}{\lvert w \rvert}$$.

6. $$\lvert z+w \rvert \leq \lvert z \rvert + \lvert w \rvert$$.

## Exercise

Let $$z = 2i$$ and $$w = -2+i$$. Give the answer to each of the following.

1. $$\lvert \overline{z} \rvert$$

2. $$\lvert zw\rvert$$

3. $$\lvert z^4 \rvert$$

4. $$\displaystyle\left|\frac{1}{z}\right|$$