Welcome to the World of Rotation and Power!

In your previous studies, you learned about complex numbers in Cartesian form \( (z = a + bi) \). While that’s great for adding and subtracting, it can be a real headache when you need to multiply, divide, or find powers of complex numbers. In this chapter, we explore the Polar and Euler forms—mathematical "shortcuts" that make complex calculations much easier and reveal the beautiful geometric nature of complex numbers.

Think of it like this: If Cartesian form is like giving someone directions in a city ("Walk 3 blocks East and 4 blocks North"), Polar form is like giving them a compass and a distance ("Turn 53 degrees and walk 5 meters"). Both get you to the same spot, but one is often much more efficient!

1. Modulus-Argument (Polar) Form

Every complex number \( z = a + bi \) can be represented as a point on the Argand diagram. Instead of using coordinates \( a \) and \( b \), we can use:

  • Modulus \( r \): The distance from the origin to the point. \( r = |z| = \sqrt{a^2 + b^2} \).
  • Argument \( \theta \): The angle the line makes with the positive real axis.

The Formula

The Polar form is written as:
\( z = r(\cos \theta + i \sin \theta) \)

In the IB DP, we often use the shorthand cis notation:
\( z = r \text{ cis } \theta \)

Quick Tip: Always make sure your calculator is in Radians mode! The IB curriculum almost exclusively uses radians for these topics.

Key Takeaway: Polar form represents a complex number by its size (\( r \)) and its direction (\( \theta \)).

2. Euler Form

The Euler form is perhaps the most elegant way to write a complex number. It uses the natural base \( e \) to link trigonometry and exponents.

The Formula

\( z = r e^{i\theta} \)

Where \( r \) is the modulus and \( \theta \) is the argument in radians.

Why is this useful?
Because it follows all the standard laws of exponents! If you need to multiply numbers, you just add the exponents. If you need to divide, you subtract them. It turns complex trigonometry into simple algebra.

Did you know? If you set \( r = 1 \) and \( \theta = \pi \), you get the famous Euler's Identity: \( e^{i\pi} + 1 = 0 \). It connects five of the most important constants in mathematics!

3. Operations in Different Forms

The IB syllabus expects you to perform operations and understand their geometric meaning. Here is how to choose your "weapon":

Addition and Subtraction

Best form: Cartesian \( (a + bi) \).
If you are given numbers in Polar or Euler form and need to add them, it is usually easiest to convert them to Cartesian form first.

Multiplication and Division

Best form: Polar or Euler.
Let \( z_1 = r_1 e^{i\theta_1} \) and \( z_2 = r_2 e^{i\theta_2} \).

Multiplication: \( z_1 z_2 = (r_1 r_2) e^{i(\theta_1 + \theta_2)} \)
Geometry: Multiply the lengths (stretch) and add the angles (rotate).

Division: \( \frac{z_1}{z_2} = \left(\frac{r_1}{r_2}\right) e^{i(\theta_1 - \theta_2)} \)
Geometry: Divide the lengths and subtract the angles.

Common Mistake: Forgetting to adjust the final angle. If your resulting angle is \( \frac{5\pi}{2} \), remember that this is the same as \( \frac{\pi}{2} \) (subtracting \( 2\pi \)). We usually prefer the principal argument \( -\pi < \theta \leq \pi \).

4. De Moivre’s Theorem

Named after Abraham de Moivre, this theorem is a superpower for finding powers of complex numbers. If you tried to calculate \( (1+i)^{10} \) using Cartesian expansion, you’d be there all day. With De Moivre, it takes seconds.

The Theorem

For any integer \( n \):
\( [r(\cos \theta + i \sin \theta)]^n = r^n(\cos(n\theta) + i \sin(n\theta)) \)

In cis notation:
\( (r \text{ cis } \theta)^n = r^n \text{ cis }(n\theta) \)

In Euler form:
\( (r e^{i\theta})^n = r^n e^{in\theta} \)

Step-by-step Example:
To find \( z^3 \) where \( z = 2 \text{ cis } \frac{\pi}{6} \):
1. Cube the modulus: \( 2^3 = 8 \).
2. Multiply the argument by 3: \( 3 \times \frac{\pi}{6} = \frac{\pi}{2} \).
3. Result: \( 8 \text{ cis } \frac{\pi}{2} \) (which is simply \( 8i \)).

Key Takeaway: Powering a complex number involves powering the distance and "speeding up" the rotation.

5. Roots of Complex Numbers

Finding the \( n \)-th root of a complex number is like reversing De Moivre's theorem. However, there is a catch: every non-zero complex number has exactly \( n \) distinct \( n \)-th roots.

The Method

To find the \( n \)-th roots of \( z = r e^{i\theta} \):

  1. Write the argument in general form by adding \( 2k\pi \): \( \theta + 2k\pi \).
  2. Take the \( n \)-th root of the modulus: \( \sqrt[n]{r} \).
  3. Divide the general argument by \( n \): \( \frac{\theta + 2k\pi}{n} \).
  4. Plug in values for \( k \) (start with \( k = 0, 1, 2, \dots, n-1 \)) to get all distinct roots.

Geometric Interpretation

If you plot all the \( n \)-th roots of a complex number on an Argand diagram:
- They all lie on a circle with radius \( \sqrt[n]{r} \).
- They are equally spaced around the circle.
- They form the vertices of a regular \( n \)-sided polygon (like a square for 4th roots or a regular hexagon for 6th roots).

Quick Review:
- Cartesian: \( a + bi \) (Good for adding).
- Polar: \( r(\cos \theta + i \sin \theta) \) (Good for visualising rotation).
- Euler: \( r e^{i\theta} \) (Good for powers and roots).
- De Moivre: Multiply the angle, power the length.

Summary Checklist

  • Can you convert between Cartesian, Polar, and Euler forms?
  • Do you remember to use radians for all \( \theta \) values?
  • Can you multiply and divide in Euler form by adding/subtracting exponents?
  • Do you know that the \( n \)-th roots of a number are equally spaced on a circle?
  • When using a GDC (Paper 2 or 3), do you know how to switch your calculator between "Real" and "Complex" modes?