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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 \(\sqrt{z~\overline{z}}\).


  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.


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. \(\lvert z+w \rvert \leq \lvert z \rvert + \lvert w \rvert\).

Quick Quiz


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. \(\displaystyle\left|\frac{1}{z}\right|\)