How Do You Find the Modulus and Argument of a Complex Number?

For a complex number $z=a+bi$, its modulus is $|z|=\sqrt{a^2+b^2}$, while its argument is the angle measured anticlockwise from the positive real axis to $z$ on an Argand diagram. The quadrant must be used to select the correct angle.

The Method

The complex number $z=a+bi$ corresponds to the point ((a,b)). Its modulus is the distance from the origin, so Pythagoras gives

$$|z|=\sqrt{a^2+b^2}.$$

To find the argument, first calculate a reference angle using

$$\alpha=\tan^{-1}\left(\left|\frac{b}{a}\right|\right).$$

Then adjust this angle according to the quadrant containing ((a,b)). The principal argument, written $\arg(z)$, is normally taken in the interval $-\pi<\arg(z)\leq\pi$.

The argument is not simply an acute triangle angle. It records direction, so signs of both real and imaginary parts matter. In calculator work, an atan2-style function can account for these signs automatically.

If $a=0$, do not divide by zero. The argument is $\frac{\pi}{2}$ when $b>0$, or when . If , the argument is for , or for .

For example, let

$$z=-1+\sqrt{3}i.$$

Its modulus is

$$|z|=\sqrt{(-1)^2+(\sqrt{3})^2}=\sqrt{4}=2.$$

The point $(-1,\sqrt{3})$ lies in Quadrant II. Its reference angle is

$$\alpha=\tan^{-1}(\sqrt{3})=\frac{\pi}{3},$$

so

$$\arg(z)=\pi-\frac{\pi}{3}=\frac{2\pi}{3}.$$

Therefore, $|z|=2$ and $\arg(z)=\frac{2\pi}{3}$. This gives the polar form , linking modulus and argument to multiplication, division, powers, and roots.

Arguments differing by $2\pi$ describe the same direction, but when asked for the principal argument, report only the representative in the stated interval.

A common misconception is to accept $\tan^{-1}(b/a)$ without checking the quadrant. This can produce an angle differing by $\pi$. Also, $z=0$ has modulus $0$, but its argument is undefined. A GDC may select the quadrant automatically, but written working should still identify it.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 1.12. On Paper 1, show the modulus calculation, identify the quadrant, and give an exact argument where possible; on Paper 2, ensure your GDC is in radian mode unless degrees are specified.