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