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 \(\sqrt{z~\overline{z}}\).
If \(z = 1+2i\), then \(\lvert z \rvert = \sqrt{1^2 + 2^2} = \sqrt{5}\).
If \(z = 3-4i\), then \(\lvert z \rvert = \sqrt{3^2 + (-4)^2} = \sqrt{9+16} = \sqrt{25} = 5\).
If \(z = 2i\), then \(\lvert z \rvert = \sqrt{0^2 + 2^2} = \sqrt{4} = 2\).
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.
There are a number of properties of the modulus that are worth knowing. Let \(z, w\in \mathbb{C}\). Then,
\(\lvert \overline{z} \rvert = \lvert z \rvert\).
\(\lvert -z\rvert = |~z~\rvert\).
If \(\lvert z \rvert = 0\), then \(z = 0\).
\(\lvert z w \rvert = \lvert z \rvert \lvert w \rvert\).
\(\displaystyle\left|\frac{z}{w}\right| = \frac{\lvert z \rvert}{\lvert w \rvert}\).
\(\lvert z+w \rvert \leq \lvert z \rvert + \lvert w \rvert\).
Let \(z = 2i\) and \(w = -2+i\). Give the answer to each of the following.
\(\lvert \overline{z} \rvert\)
\(\lvert zw\rvert\)
\(\lvert z^4 \rvert\)
\(\displaystyle\left|\frac{1}{z}\right|\)