How Do You Convert a Complex Number Between Cartesian, Polar, and Euler Form?

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Cartesian to Polar or Euler

For $z=a+bi$, calculate the modulus:

Then find the argument $\theta=\arg(z)$ from $\tan\theta=\frac{b}{a}$. This ratio gives a reference angle, but tangent alone does not identify the quadrant. Use the signs of $a$ and $b$ to select the correct angle, normally in radians unless degrees are specified.

Then write $z=r\operatorname{cis}\theta$ or $z=re^{i\theta}$.

Worked Example

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

The point $(-1,\sqrt{3})$ lies in quadrant II, so $\theta=\frac{2\pi}{3}$. Therefore,

Geometrically, Cartesian form gives horizontal and vertical components, while polar and Euler forms give distance from the origin and rotation from the positive real axis.

Polar or Euler to Cartesian

Use $a=r\cos\theta$ and $b=r\sin\theta$. Hence,

For example, $4e^{-i\pi/6}=2\sqrt{3}-2i$. Converting directly between polar and Euler form requires no new calculation because $e^{i\theta}=\cos\theta+i\sin\theta$.

A common misconception is using $\tan^{-1}(b/a)$ without checking the quadrant. For example, $(1,1)$ and $(-1,-1)$ both give $b/a=1$, but their arguments differ by $\pi$.

The conversion also explains why each form is useful. Cartesian form is usually best for addition and subtraction, while polar or Euler form is usually best for multiplication, division, powers, and roots because moduli multiply or divide and arguments add or subtract.

In an IB response, show $r$, justify $\theta$, and state the correct form. Use exact values on Paper 1; on Paper 2, a GDC may find the angle, but the quadrant must be justified. If an argument range is given, obey it; otherwise arguments differ by $2k\pi$.