Complex numbers fundamentals

IB Mathematics: Analysis and Approaches HL· IB AA HL 1.11· 15 min read

1. Core Definitions and Rectangular Form★★☆☆☆⏱ 5 min

📘 Definition

Complex Number (Rectangular Form)

$z = a + bi$

A number consisting of a real part $a = Re (z)$ and imaginary part $b = Im (z)$ , where $i^{2} = - 1$ , and $a, b$ are real numbers.

Example:

For $z = 3 - 2 i$ , $Re (z) = 3$ , $Im (z) = - 2$ (not $- 2 i$ ).

Basic operations on complex numbers in rectangular form follow standard algebraic rules, with one key adjustment: replace every occurrence of $i^{2}$ with $- 1$ when simplifying.

Addition: $(a + bi) + (c + d i) = (a + c) + (b + d) i$ , add real and imaginary parts separately
Subtraction: $(a + bi) - (c + d i) = (a - c) + (b - d) i$ , same structure as addition
Multiplication: Expand like a binomial product, substitute $i^{2} = - 1$ , then group terms

📐 Worked Example

Given $z_{1} = 4 + 2 i$ and $z_{2} = 1 - 3 i$ , calculate (a) $z_{1} + z_{2}$ , (b) $z_{1} z_{2}$ .

1
For addition, group real and imaginary components:
2
$z_1 + z_2 = (4 + 1) + (2 + (-3))i = 5 - i$
3
For multiplication, expand the binomial product:
4
$(4 + 2i)(1 - 3i) = 4(1) + 4(-3i) + 2i(1) + 2i(-3i) = 4 - 12i + 2i - 6i^2$
5
Substitute $i^{2} = - 1$ and simplify:
6
$= 4 - 10i - 6(-1) = 10 - 10i$

2. Geometric Representation: The Argand Diagram★★☆☆☆⏱ 4 min

Every complex number can be represented as a point or position vector on a special coordinate plane called the Argand diagram (or complex plane).

📘 Definition

Argand Diagram

A coordinate plane where the horizontal axis is the real axis (for $Re (z)$ ) and the vertical axis is the imaginary axis (for $Im (z)$ ). A complex number $z = a + bi$ maps to the point $(a, b)$ .

📐 Worked Example

Plot $z_{1} = 3 + i$ , $z_{2} = - 2 - 2 i$ , and $z_{1} + z_{2}$ on an Argand diagram, and confirm the addition matches vector addition.

1
First, identify coordinates for each point: $z_{1} = (3, 1)$ , $z_{2} = (- 2, - 2)$
2
Calculate the sum to get its coordinates:
3
$z_1 + z_2 = (3-2) + (1-2)i = 1 - i \rightarrow (1, -1)$
4
When plotted, the point $(1, - 1)$ is the resultant of the two position vectors, which follows the parallelogram rule for vector addition.

3. Modulus and Principal Argument★★★☆☆⏱ 6 min

Polar form of a complex number is defined by two key properties: the modulus (distance from the origin) and the argument (angle from the positive real axis).

📘 Definition

Modulus and Principal Argument

$∣ z ∣$ , $ar g (z)$

For $z = a + bi$ , modulus $∣ z ∣ = a^{2} + b^{2}$ is the distance from the origin. The principal argument $ar g (z)$ is the anticlockwise angle from the positive real axis, defined for IB in the range $- π < ar g (z) \leq π$ .

📐 Worked Example

Find the modulus and principal argument of $z = - 1 + 3 i$ .

1
Calculate the modulus using Pythagoras' theorem:
2
$|z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$
3
Identify the quadrant: $Re (z) < 0$ , $Im (z) > 0$ , so $z$ lies in the second quadrant. Calculate the reference angle:
4
$\tan \alpha = \left| \frac{\text{Im}(z)}{\text{Re}(z)} \right| = \left| \frac{\sqrt{3}}{-1} \right| = \sqrt{3} \implies \alpha = \frac{\pi}{3}$
5
Adjust the angle to get the principal argument for the second quadrant:
6
$\arg(z) = \pi - \alpha = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$

4. Polar (Modulus-Argument) Form★★★☆☆⏱ 5 min

Using right triangle trigonometry on the Argand diagram, we can write any non-zero complex number in polar form, which is more useful for multiplication, division, and raising to powers than rectangular form.

📘 Definition

Polar Form of a Complex Number

$z = r (cos θ + i sin θ)$

For any non-zero complex number with modulus $r = ∣ z ∣$ and principal argument $θ = ar g (z)$ , $z$ can be written as $r (cos θ + i sin θ)$ , where $a = r cos θ$ and $b = r sin θ$ for $z = a + bi$ .

📐 Worked Example

Convert $z = 1 - i$ from rectangular form to polar form.

1
First calculate the modulus $r$ :
2
$r = |z| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$
3
Find the principal argument $θ$ : $z$ is in the fourth quadrant, so:
4
$\tan \theta = \frac{-1}{1} = -1 \implies \theta = -\frac{\pi}{4}$
5
Write the final polar form:
6
$z = \sqrt{2} \left( \cos \left(-\frac{\pi}{4}\right) + i \sin \left(-\frac{\pi}{4}\right) \right)$

5. Common Pitfalls

Wrong move:

Stating $Im (z) = bi$ for $z = a + bi$ , instead of just $b$ .

Why:

The imaginary part is defined as the real scalar multiple of $i$ , it does not include $i$ itself.

Correct move:

For $z = 5 - 3 i$ , $Im (z) = - 3$ , not $- 3 i$ .

Wrong move:

Writing $ar g (z) = arctan (\frac{b}{a})$ without adjusting for quadrant.

Why:

Arctangent only returns values between $- \frac{π}{2}$ and $\frac{π}{2}$ , so it is incorrect for second and third quadrants.

Correct move:

Find the reference angle first, then adjust it to get the correct principal argument based on the quadrant of $z$ .

Wrong move:

Simplifying $- a \times - b = ab$ for positive $a, b$ .

Why:

The square root identity $x y = x y$ only holds for non-negative real $x, y$ .

Correct move:

Rewrite $- a = i a$ and $- b = i b$ first, then multiply to get $- ab$ .

Wrong move:

Using the range $0 \leq ar g (z) < 2 π$ for principal argument.

Why:

The IB AA HL syllabus requires principal argument to be in the range $- π < ar g (z) \leq π$ unless stated otherwise.

Correct move:

Subtract $2 π$ from any angle greater than $π$ to get it into the principal range.

Wrong move:

Calculating modulus as $a^{2} - b^{2}$ instead of the sum.

Why:

Confusion between Pythagoras' theorem and difference of squares.

Correct move:

Modulus is distance from the origin, so always use $∣ z ∣ = a^{2} + b^{2}$ .

6. Quick Reference Cheatsheet

Concept	Formula/Rule
Rectangular Form	$z = a + bi$ , $Re (z) = a$ , $Im (z) = b$ , $i^{2} = - 1$
Modulus	$∣ z ∣ = a^{2} + b^{2}$
Principal Argument Range	$- π < ar g (z) \leq π$
Addition	$(a + bi) + (c + d i) = (a + c) + (b + d) i$
Multiplication	$(a + bi) (c + d i) = (a c - b d) + (a d + b c) i$
Polar Form	$z = r (cos θ + i sin θ)$ , $r = ∣ z ∣$ , $θ = ar g (z)$
Argand Coordinates	$z = a + bi \leftrightarrow (a, b)$

7. Frequently Asked

Is the principal argument range always $- π < ar g (z) \leq π$ for IB?

Yes, this is the standard range required by the IB AA HL syllabus, unless the question explicitly asks for an angle in $0 \leq θ < 2 π$ .

Can I write $i = - 1$ ?

This is acceptable for introductory contexts, but remember that the square root function is only defined for non-negative reals, so avoid using $- a = i a$ in proofs unless you explicitly define it first.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find argument of a complex number
2022 · 2
Convert polar to rectangular form
2023 · 1
Calculate modulus of product

Going deeper

What's Next

A strong grasp of complex number fundamentals is required for all subsequent complex number topics in IB AA HL, from multiplication of complex numbers in polar form to De Moivre's theorem and finding roots of complex numbers. It also underpins the complex conjugate root theorem for polynomials, which is a common exam question topic. Geometric intuition for complex numbers built here will help you recognize rotation and scaling properties of complex multiplication, which often appear in extended response questions.