IBMathsDPApplications & Interpretation (AI)HLRevision Notes1. Number & AlgebraFurther Complex NumbersConversion between Forms of Complex Numbers

Conversion between Forms of Complex Numbers (DP IB Applications & Interpretation (AI): HL): Revision Note

Written by: Amber

Reviewed by: Mark Curtis

Updated on 26 November 2025

Conversion of forms

How do I convert out of Cartesian form?

To convert $z = x + i y$ from Cartesian form to modulus-argument (polar) form or exponential (Euler) form, you first need to find
- the modulus
  - $r = | z | = \sqrt{x^{2} + y^{2}}$
- and the argument
  - $θ = \arg z$
- then use
  - $z = r (\cos θ + isin θ) = r e^{i θ} = r cis θ$

Examiner Tips and Tricks

The relationship $z = r (\cos θ + isin θ) = r e^{i θ} = r cis θ$ is given in the formula booklet.

How do I convert into Cartesian form?

To convert from modulus-argument (polar) form or exponential (Euler) form to Cartesian form
- you need to work out the values of
  - $\cos θ$
  - $\sin θ$
  - which may be exact
- then expand $z = r (\cos θ + isin θ)$ into the components $z = x + i y$

Examiner Tips and Tricks

It is not possible to go directly from exponential (Euler) to Cartesian form without having to first convert to modulus-argument (polar) form.

Examiner Tips and Tricks

Your GDC may be able to convert between complex number forms, as some have the option to 'convert to polar' or 'convert to rectangular' (Cartesian).

Worked Example

Two complex numbers are given by $z_{1} = 2 + 2 i$ and $z_{2} = 3 e^{\frac{2 π}{3} i}$ .

(a) Write $z_{1}$ in the form $r e^{i θ}$ where $r > 0$ and $0 \leq θ < 2 π$ .

Answer:

1-9-2-ib-aa-hl-forms-of-cn-we-solution-3-a

(b) Write $z_{2}$ in Cartesian form.

Answer:

1-9-2-ib-aa-hl-forms-of-cn-we-solution-3-b

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Conversion between Forms of Complex Numbers (DP IB Applications & Interpretation (AI): HL): Revision Note

Conversion of forms

How do I convert out of Cartesian form?

Examiner Tips and Tricks

How do I convert into Cartesian form?

Examiner Tips and Tricks

Examiner Tips and Tricks

Worked Example

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Author: Amber

Reviewer: Mark Curtis