Complex numbers (HL)

Q: What is the definition of the imaginary unit \(i\)?

The imaginary unit \(i\) is defined by the rule \(i^2 = -1\), which implies \(i = \sqrt{-1}\).

Q: What are the components of a complex number written in Cartesian form?

A complex number \(z = a + bi\) consists of a real part \(a = Re(z)\) and an imaginary part \(b = Im(z)\).

Q: How do you find the complex conjugate of \(z = a + bi\)?

The complex conjugate \(z^*\) is found by flipping the sign of the imaginary part: \(z^* = a - bi\).

Q: How are complex numbers added or subtracted?

Combine like terms by adding or subtracting the real parts together and the imaginary parts together.

Q: How do you multiply two complex numbers in Cartesian form?

Use the FOIL method to expand the brackets and then substitute \(i^2 = -1\) to simplify the result.

Q: What is an Argand Diagram?

An Argand Diagram is a geometric representation of complex numbers where the x-axis represents the Real axis and the y-axis represents the Imaginary axis .

Q: How are the modulus and argument of a complex number defined?

The modulus \(|z| = r = \sqrt{a^2 + b^2}\) is the distance from the origin, and the argument \(\theta\) is the angle from the positive real axis where \(\tan \theta = \frac{b}{a}\).

Q: What is the Conjugate Root Theorem for quadratic equations?

If a quadratic equation with real coefficients has complex roots, those roots will always occur in conjugate pairs (\(a \pm bi\)).

Q: What are Polar and Exponential forms of a complex number?

Polar: \(z = r(\cos \theta + i \sin \theta)\) or \(r \text{ cis } \theta\). Exponential: \(z = r e^{i\theta}\).

Q: How do you multiply and divide complex numbers in exponential form?

Multiplication: Multiply the moduli and add the arguments. Division: Divide the moduli and subtract the arguments.

Mathematics - Applications and Interpretation · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: Sep 7, 2026

Welcome to the World of Complex Numbers!

Ever been told in math class that you can't take the square root of a negative number? Well, in this chapter, we are going to break that rule! Complex numbers allow us to solve equations that were previously "unsolvable" and provide essential tools for engineering, physics, and even describing the behavior of electricity. Don't worry if it sounds "complex" — we will break it down step-by-step.

1. The Basics: What is \(i\)?

The foundation of this chapter is the imaginary unit, denoted by the letter \(i\). It is defined by one simple rule:

\(i^2 = -1\)

This means that \(i = \sqrt{-1}\). While we call it "imaginary," it is just as mathematically valid as the "real" numbers you’ve used since primary school.

Cartesian Form

A complex number \(z\) is usually written in Cartesian form:

\(z = a + bi\)

Where:
- \(a\) is the real part, written as \(Re(z)\).
- \(b\) is the imaginary part, written as \(Im(z)\). (Note: the imaginary part is just the number \(b\), not \(bi\)!)

Example: If \(z = 3 - 4i\), then \(Re(z) = 3\) and \(Im(z) = -4\).

The Complex Conjugate

The conjugate of a complex number \(z = a + bi\) is denoted as \(z^*\) (or sometimes \(\bar{z}\)). You simply flip the sign of the imaginary part:

\(z^* = a - bi\)

Why do we need this? When you multiply a complex number by its conjugate, the result is always a real number!
\((a + bi)(a - bi) = a^2 + b^2\).

Quick Takeaway

Complex numbers combine real and imaginary parts. Remember: \(i^2\) is always \(-1\)!

2. Arithmetic with Complex Numbers

You can add, subtract, multiply, and divide complex numbers just like algebraic expressions, keeping in mind that \(i\) behaves like a variable — until you see \(i^2\).

Addition and Subtraction

Just "combine like terms." Add the real parts together and the imaginary parts together.

\((3 + 2i) + (5 - 4i) = (3+5) + (2-4)i = 8 - 2i\)

Multiplication

Use the FOIL method (First, Outside, Inside, Last) and remember to replace \(i^2\) with \(-1\).

Example: \((2 + 3i)(1 - 2i)\)
\(= 2(1) + 2(-2i) + 3i(1) + 3i(-2i)\)
\(= 2 - 4i + 3i - 6i^2\)
\(= 2 - i - 6(-1)\)
\(= 2 - i + 6 = 8 - i\)

Division (by hand)

To divide, multiply the top and bottom by the conjugate of the denominator. This clears the imaginary unit from the bottom.

Quick Review: While you should know how to do these by hand, the IB AI HL syllabus expects you to be comfortable using your GDC (Graphic Display Calculator) for these calculations! Most modern calculators have a dedicated \(i\) button.

3. The Complex Plane (Argand Diagrams)

We visualize complex numbers on a 2D plane called an Argand Diagram.
- The x-axis is the Real axis.
- The y-axis is the Imaginary axis.

A complex number \(z = a + bi\) is plotted as the point \((a, b)\).

Modulus and Argument

Every complex number can also be described by its distance from the origin and its angle.

1. Modulus (\(|z|\)): The distance from the origin to the point. Using Pythagoras:
\(|z| = r = \sqrt{a^2 + b^2}\)

2. Argument (\(arg(z)\)): The angle \(\theta\) measured from the positive real axis.
\(\tan \theta = \frac{b}{a}\)

Common Mistake: Be careful with the quadrant! If your complex number is in the 2nd or 3rd quadrant (where the real part \(a\) is negative), your calculator’s \(\arctan\) function might give you the wrong angle. Always sketch the point to check!

4. Solving Quadratic Equations

In your earlier math studies, if the discriminant (\(b^2 - 4ac\)) was negative, you said "no real solutions." Now, we can find complex solutions.

If \(b^2 - 4ac < 0\), the roots will always appear in conjugate pairs (e.g., \(2 + 3i\) and \(2 - 3i\)).

Example: Solve \(x^2 + 9 = 0\)
\(x^2 = -9\)
\(x = \pm \sqrt{-9} = \pm \sqrt{9 \times -1} = \pm 3i\)

5. Polar and Exponential Forms

Working in Cartesian form (\(a+bi\)) is great for adding, but Polar form is much better for multiplying and finding powers.

Polar Form

\(z = r(\cos \theta + i \sin \theta)\)
This is often abbreviated as: \(z = r \text{ cis } \theta\)

Exponential Form (Euler’s Form)

Using Euler’s formula, we can write a complex number as:
\(z = r e^{i\theta}\)

Note: In both forms, \(\theta\) must be in radians for the math to work correctly!

Converting Between Forms

To Cartesian: Use \(a = r \cos \theta\) and \(b = r \sin \theta\).
To Polar/Exponential: Find \(r\) using Pythagoras and \(\theta\) using trigonometry.

Operations in Polar/Exponential Form

If you have \(z_1 = r_1 e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\):

Multiplication: Multiply the moduli, add the angles.
\(z_1 z_2 = (r_1 r_2) e^{i(\theta_1 + \theta_2)}\)

Division: Divide the moduli, subtract the angles.
\(\frac{z_1}{z_2} = (\frac{r_1}{r_2}) e^{i(\theta_1 - \theta_2)}\)

Powers: Raise the modulus to the power, multiply the angle by the power.
\(z^n = r^n e^{in\theta}\)

6. Application: Adding Sinusoidal Functions

One of the unique "Applications" parts of this syllabus is adding two waves (sinusoidal functions) together. If you have two waves with the same frequency but different phase shifts:

\(y_1 = A_1 \cos(\omega t + \alpha)\)
\(y_2 = A_2 \cos(\omega t + \beta)\)

You can represent these as complex numbers (phasors):
\(z_1 = A_1 e^{i\alpha}\) and \(z_2 = A_2 e^{i\beta}\)

To find the resulting wave (\(y_{total} = y_1 + y_2\)):
1. Convert \(z_1\) and \(z_2\) to Cartesian form (\(a+bi\)).
2. Add them together to get \(z_{new} = a_{total} + b_{total}i\).
3. Convert \(z_{new}\) back to Polar form to find the new amplitude (\(R\)) and the new phase shift (\(\phi\)).
4. The result is \(R \cos(\omega t + \phi)\).

Did you know? This is exactly how noise-canceling headphones work! They create a sound wave with a phase shift that "cancels out" the incoming noise through addition.

Final Tips for Success

Mode Check: Ensure your GDC is in Complex Mode (usually \(a+bi\) or \(re^{i\theta}\)) and Radian Mode before starting exam questions.
Visualise: Always draw a small Argand diagram. It helps you see if your argument (\(\theta\)) makes sense.
Don't Panic: If a question looks scary, try converting the numbers into a different form. Often, what is hard in Cartesian form is easy in Polar form!

Key Takeaway Summary

Cartesian: Best for addition/subtraction. \(z = a + bi\).
Polar/Exponential: Best for multiplication/division/powers. \(z = r e^{i\theta}\).
Conjugate: Flip the imaginary sign to help with division and roots of equations.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the definition of the imaginary unit \(i\)?

The imaginary unit \(i\) is defined by the rule \(i^2 = -1\), which implies \(i = \sqrt{-1}\).

What are the components of a complex number written in Cartesian form?

A complex number \(z = a + bi\) consists of a real part \(a = Re(z)\) and an imaginary part \(b = Im(z)\).

How do you find the complex conjugate of \(z = a + bi\)?

The complex conjugate \(z^*\) is found by flipping the sign of the imaginary part: \(z^* = a - bi\).

How are complex numbers added or subtracted?

Combine like terms by adding or subtracting the real parts together and the imaginary parts together.

How do you multiply two complex numbers in Cartesian form?

Use the FOIL method to expand the brackets and then substitute \(i^2 = -1\) to simplify the result.

What is an Argand Diagram?

An Argand Diagram is a geometric representation of complex numbers where the x-axis represents the Real axis and the y-axis represents the Imaginary axis.

How are the modulus and argument of a complex number defined?

The modulus \(|z| = r = \sqrt{a^2 + b^2}\) is the distance from the origin, and the argument \(\theta\) is the angle from the positive real axis where \(\tan \theta = \frac{b}{a}\).

What is the Conjugate Root Theorem for quadratic equations?

If a quadratic equation with real coefficients has complex roots, those roots will always occur in conjugate pairs (\(a \pm bi\)).

What are Polar and Exponential forms of a complex number?

Polar: \(z = r(\cos \theta + i \sin \theta)\) or \(r \text{ cis } \theta\). Exponential: \(z = r e^{i\theta}\).

How do you multiply and divide complex numbers in exponential form?

Multiplication: Multiply the moduli and add the arguments. Division: Divide the moduli and subtract the arguments.

How can complex numbers be used to add sinusoidal waves?

Represent waves as phasors (\(A e^{i\theta}\)), add them in Cartesian form, and convert the result back to find the new amplitude and phase shift.

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