Modulus of a complex number

In the plane, \(a+bi\) is the point \((a,b)\), and the Pythagorean theorem gives its distance to the origin: \(|z|=\sqrt{a^2+b^2}\). For example, \(|3+4i|=5\). The modulus is always a real number \(\ge0\), and equals \(0\) only for \(z=0\). For a real number it agrees with the usual absolute value.

The relation with the conjugate, \(z\bar z=|z|^2\), explains division: \(\frac1z=\frac{\bar z}{|z|^2}\) for \(z\neq0\). It also shows that the modulus is multiplicative: \(|zw|=|z|\,|w|\) and \(\left|\frac zw\right|=\frac{|z|}{|w|}\). The sum, in contrast, does not preserve the modulus; it only satisfies the triangle inequality \(|z+w|\le|z|+|w|\) and its variant \(|z+w|\ge\big||z|-|w|\big|\).

The modulus measures distances: \(|z-w|\) is the distance between the points \(z\) and \(w\). Thus \(|z-w|=r\) is the circle with center \(w\) and radius \(r\), and \(|z-w|<r\) is the corresponding open disk. These disks will be the basis of limits and continuity.

Common mistakes: writing \(|a+bi|=a+b\) or \(\sqrt{a^2+(bi)^2}\) (with the \(i\) inside the root), and believing \(|z|^2=z^2\), which holds only when \(z\) is real. The next step is to combine this modulus with the angle of \(z\): the polar form.