# Complex numbers: definition, arithmetic and polar form

> IB Mathematics AI HL · Unit 1: Number and algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u1-complex-numbers-definition-arithmetic-and/

This subtopic introduces complex numbers, their basic arithmetic operations, and conversion between Cartesian and polar forms. It is a foundational tool for solving higher-degree polynomials and advanced trigonometric problems, regularly tested in IB AI HL exams.

**Prerequisites:** Basic algebraic manipulation of binomials; Unit circle trigonometry and inverse tangent

## Learning objectives

- Define complex numbers and their key components in Cartesian form
- Perform all standard arithmetic operations on complex numbers in Cartesian form
- Convert between Cartesian and polar forms of complex numbers correctly
- Apply modulus and argument properties for products and quotients in polar form

## Definition and Cartesian Form

**Complex Number (Cartesian Form)** — A number of the form $a + bi$, where $a$ and $b$ are real numbers, and $i^2 = -1$. $a = \text{Re}(z)$ (real part) and $b = \text{Im}(z)$ (imaginary part).

*Notation:* z = a + bi

*Example:* For $z = 3 - 2i$, $\text{Re}(z) = 3$, $\text{Im}(z) = -2$

All complex numbers can be plotted as points $(a, b)$ on the complex plane, with the horizontal axis as the real axis and the vertical axis as the imaginary axis.

**Worked example:** Find $\text{Re}(z)$ and $\text{Im}(z)$ for $z = i(4 - 2i)$, after simplifying to Cartesian form.

1. Expand the product, using $i^2 = -1$:
2. $$i(4 - 2i) = 4i - 2i^2 = 4i - 2(-1) = 2 + 4i$$
3. Match to the form $a + bi$ to get the components:
4. $\text{Re}(z) = 2$, $\text{Im}(z) = 4$

> **Exam tip:** Always simplify your expression fully before identifying real and imaginary parts, and do not forget the sign of the imaginary part.

## Arithmetic of Complex Numbers (Cartesian Form)

Arithmetic operations on complex numbers follow the same rules as real numbers, with the additional simplification $i^2 = -1$. Addition, subtraction, and multiplication are straightforward, while division requires the use of the complex conjugate.

**Complex Conjugate** — For $z = a + bi$, the conjugate is $\bar{z} = a - bi$. The product $z\bar{z} = a^2 + b^2$, which is always a positive real number. This property lets us rationalize denominators when dividing complex numbers.

*Notation:* \bar{z}

*Example:* Conjugate of $1 - i$ is $1 + i$, and $(1 - i)(1 + i) = 2$

**Worked example:** Simplify $\frac{3 + 2i}{1 - 2i}$ to Cartesian form $a + bi$.

1. 1. Get the conjugate of the denominator: $\overline{1 - 2i} = 1 + 2i$
2. 2. Multiply numerator and denominator by the conjugate to keep the expression equivalent:
3. $$\frac{(3 + 2i)(1 + 2i)}{(1 - 2i)(1 + 2i)}$$
4. 3. Expand and simplify denominator:
5. $$(1)^2 - (2i)^2 = 1 - 4(-1) = 5$$
6. 4. Expand and simplify numerator:
7. $$3(1) + 3(2i) + 2i(1) + 2i(2i) = 3 + 6i + 2i + 4i^2 = 3 + 8i - 4 = -1 + 8i$$
8. 5. Divide by the denominator to get final result:
9. $$\frac{-1 + 8i}{5} = -\frac{1}{5} + \frac{8}{5}i$$

## Polar Form of Complex Numbers

Using the polar coordinate system on the complex plane, we can write any non-zero complex number in terms of its modulus (distance from origin) and argument (angle from positive real axis). This form simplifies multiplication and division of multiple complex numbers.

**Polar (Modulus-Argument) Form** — Where $r = |z| \geq 0$ (modulus), and $\theta = \text{arg}(z)$ (argument), usually given in the principal range $-\pi < \theta \leq \pi$. For any Cartesian complex number $a + bi$, $r = \sqrt{a^2 + b^2}$ and $\tan\theta = \frac{b}{a}$.

*Notation:* z = r(\cos\theta + i\sin\theta)

**Worked example:** Convert $z = -1 + i$ from Cartesian to polar form.

1. 1. Calculate the modulus $r$:
2. $$r = \sqrt{(-1)^2 + (1)^2} = \sqrt{2}$$
3. 2. Check the quadrant of $z$: $(-1, 1)$ is in the second quadrant.
4. 3. Calculate $\tan\theta = \frac{b}{a} = \frac{1}{-1} = -1$. The reference angle is $\frac{\pi}{4}$, adjust to the second quadrant:
5. $$\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$
6. 4. Write the final polar form:
7. $$z = \sqrt{2}\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)$$

For products and quotients of two complex numbers in polar form, we use the properties: if $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then:

- $z_1z_2 = r_1r_2\left(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\right)$
- $\frac{z_1}{z_2} = \frac{r_1}{r_2}\left(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right)$

> **Exam tip:** Always verify the quadrant of your complex number when calculating the argument. The arctangent function on your calculator only returns values for the first and fourth quadrants, so you will need to add $\pi$ for points in the second and third quadrants.

## Common pitfalls

- **Wrong:** Stating $\text{Im}(3 - 2i) = 2$, ignoring the negative sign
  - Why it fails: The imaginary part is defined as the coefficient of $i$, which includes its sign
  - Correct: $\text{Im}(3 - 2i) = -2$
- **Wrong:** When dividing complex numbers, only multiplying the numerator by the conjugate of the denominator
  - Why it fails: This changes the value of the expression, as it is equivalent to multiplying the entire fraction by the conjugate
  - Correct: Always multiply both numerator and denominator by the same conjugate to preserve equality
- **Wrong:** Using the direct calculator output of $\tan^{-1}\left(\frac{1}{-1}\right) = -\frac{\pi}{4}$ as the argument for $z = -1 + i$
  - Why it fails: The point $(-1, 1)$ lies in the second quadrant, while $\tan^{-1}$ only outputs angles between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$
  - Correct: Adjust the output by adding $\pi$ to get the correct argument $\frac{3\pi}{4}$
- **Wrong:** Writing polar form with a negative modulus $r = -2$
  - Why it fails: Modulus is a distance from the origin, so it is always non-negative. Negative signs are absorbed into the argument
  - Correct: Use $r \geq 0$ and add $\pi$ to the argument to account for negative coordinates

## Cheatsheet

| Concept | Cartesian Form $z=a+bi$ | Polar Form $z=r(\cos\theta+i\sin\theta)$ |
| --- | --- | --- |
| Modulus | $\sqrt{a^2+b^2}$ | $r$ |
| Conjugate | $a - bi$ | $r(\cos(-\theta)+i\sin(-\theta))$ |
| Addition | Add real/imaginary parts separately | Convert to Cartesian first |
| Multiplication | Expand binomials, replace $i^2=-1$ | $r_1r_2(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))$ |
| Division | Multiply by conjugate of denominator | $\frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2))$ |
| Argument | $\tan\theta = \frac{b}{a}$ (adjust for quadrant) | $\theta$ (given directly) |

## What's next

Mastering the definition, arithmetic, and polar form of complex numbers is a critical foundation for further work with complex numbers in IB AI HL. This topic regularly appears in both Paper 1 (non-calculator) and Paper 2 (calculator) IB exams, and the skills you learn here are also applied to advanced trigonometric identity problems later in the course.

- [Functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-overview/)

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