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

Convert a complex number from Cartesian form by finding its modulus and argument, then substitute them into polar or exponential form. To convert back, expand the expression using cosine and sine.

This is AHL 1.13, Complex numbers continued, so it is HL-only. The explanation assumes an HL Paper 1 short-response style, although this skill may appear on any GDC-active AI HL paper.

Conversion Method

For a complex number

$$z=a+bi,$$

calculate

$$r=|z|=\sqrt{a^2+b^2}, \qquad \theta=\arg(z).$$

Use your GDC's argument function, or calculate the reference angle and check the quadrant carefully. Euler's formula, $e^{i\theta}=\cos\theta+i\sin\theta$, connects the polar and exponential representations.

Arguments are not unique: $\theta+2k\pi$, where $k\in\mathbb{Z}$, represents the same direction. Unless otherwise requested, give the principal argument displayed by your GDC.

Worked Example

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

First, find the modulus:

$$r=\sqrt{(-1)^2+(\sqrt{3})^2}=2.$$

The point $(-1,\sqrt{3})$ lies in quadrant II, so

$$\theta=\frac{2\pi}{3}.$$

Therefore,

$$z=2\left(\cos\left(\frac{2\pi}{3}\right)+i\sin\left(\frac{2\pi}{3}\right)\right)=2e^{\frac{2\pi i}{3}}.$$

To convert back, use

$$a=r\cos\theta, \qquad b=r\sin\theta.$$

This gives $a=-1$ and $b=\sqrt{3}$, recovering $$.

These forms are equivalent but useful for different operations. Cartesian form is usually most convenient for addition and subtraction, while polar or exponential form is more efficient for multiplication, division, and integer powers. In polar form, moduli are multiplied or divided, while arguments are added or subtracted. This connection is important when applying De Moivre's theorem later in AHL 1.13.

When using a GDC, enter the Cartesian coordinates carefully and confirm that the calculator is in radian mode. If the GDC returns a negative principal argument, it is still valid; an equivalent positive argument can be found by adding $2\pi$. Do not round the argument before converting back, because premature rounding can change the Cartesian components.

A common misconception is using $\tan^{-1}(b/a)$ without checking the quadrant. This may produce an argument differing by $\pi$. In an IB response, show both $r$ and $\theta$, use radians unless degrees are explicitly indicated, and retain exact values where possible. Final decimal answers should normally be given to three significant figures.