Complex numbers

Welcome to the World of Complex Numbers!

In H2 Mathematics, you’ve already met complex numbers in their Cartesian form (\(x + iy\)). In Further Mathematics (9649), we take this a step further. We are going to look at complex numbers as "vectors" that can rotate and stretch. This chapter is vital because it connects algebra, geometry, and trigonometry in a way that is incredibly useful in physics and engineering.

Don't worry if this seems tricky at first—once you master the "geometry" of these numbers, the algebra becomes much easier!

1. Beyond \(x + iy\): Polar and Exponential Forms

In H2 Math, we used coordinates (left/right and up/down). In Further Math, we use distance and direction. This is called the Polar Form.

The Polar Form

A complex number \(z\) can be written as:
\(z = r(\cos \theta + i\sin \theta)\)

• \(r\) (Modulus): The distance from the origin. Always \(r > 0\).
• \(\theta\) (Argument): The angle measured from the positive real axis.
• The Range: We use the "Principal Argument," which means \(-\pi < \theta \le \pi\).

The Exponential Form (Euler's Form)

This is a super-convenient shorthand for the polar form:
\(z = re^{i\theta}\)

Did you know? This comes from Euler’s Formula, often called the most beautiful formula in mathematics! It allows us to treat complex numbers like powers, making multiplication and division a breeze.

Quick Review:
If \(z = 1 + i\):
1. Find \(r\): \(\sqrt{1^2 + 1^2} = \sqrt{2}\)
2. Find \(\theta\): \(\tan^{-1}(1/1) = \pi/4\)
3. Polar Form: \(\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\)
4. Exponential Form: \(\sqrt{2}e^{i\pi/4}\)

Key Takeaway: Polar and Exponential forms focus on how far (\(r\)) and what angle (\(\theta\)) the number is from the center.

2. Multiplication and Division: The Geometrical Magic

In Cartesian form (\(x+iy\)), multiplying is messy. In Polar form, it’s a joy!

Multiplication

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

• The Rule: Multiply the lengths (\(r\)), add the angles (\(\theta\)).
• Interpretation: Multiplying by a complex number stretches the vector by \(r\) and rotates it by \(\theta\).

Division

If you divide \(z_1\) by \(z_2\):
\(\frac{z_1}{z_2} = (\frac{r_1}{r_2})e^{i(\theta_1 - \theta_2)}\)

• The Rule: Divide the lengths (\(r\)), subtract the angles (\(\theta\)).
• Interpretation: Dividing shrinks the vector and rotates it backwards.

Common Mistake: When adding or subtracting angles, always check if your final answer is still between \(-\pi\) and \(\pi\). If it's \(1.2\pi\), subtract \(2\pi\) to get \(-0.8\pi\).

Key Takeaway: Multiplication = Rotate + Stretch. Division = Rotate back + Shrink.

3. De Moivre’s Theorem (DMT)

De Moivre’s Theorem is like a "power-up" for complex numbers. It states that for any integer \(n\):
\((r(\cos \theta + i\sin \theta))^n = r^n(\cos n\theta + i\sin n\theta)\)

Or in exponential form: \((re^{i\theta})^n = r^n e^{in\theta}\)

Why is this useful?

1. Finding Powers: Imagine trying to calculate \((1+i)^{10}\) by expanding brackets. It would take forever! With DMT, you just turn it into polar form, raise the \(r\) to the power of 10, and multiply the angle by 10.

2. Deriving Trig Identities: You can use DMT and the Binomial Expansion to find formulas for \(\cos n\theta\) and \(\sin n\theta\) in terms of powers of \(\cos \theta\) and \(\sin \theta\).

Step-by-Step for Trig Identities:
Suppose you want \(\cos 3\theta\):
1. Write \((\cos \theta + i\sin \theta)^3 = \cos 3\theta + i\sin 3\theta\).
2. Expand the left side using \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).
3. Group the Real parts (the bits without \(i\)).
4. Set the Real parts equal to \(\cos 3\theta\).

Key Takeaway: DMT turns difficult powers into simple multiplication of angles.

4. Finding \(n\)-th Roots

How do you solve \(z^n = w\)? For example, what are the cube roots of \(8i\)?

The Analogy: Imagine a pizza. The roots are always perfectly evenly spaced slices around a circle.

The Process:

1. Convert: Write the number \(w\) in polar form: \(Re^{i(\phi + 2k\pi)}\). Note: We add \(2k\pi\) because rotating a full circle brings you back to the same spot!
2. Apply Power: Take the \(1/n\) power: \(z = R^{1/n} e^{i(\frac{\phi + 2k\pi}{n})}\).
3. Substitute: Let \(k = 0, 1, 2, \dots, n-1\). This gives you exactly \(n\) distinct roots.

Example (Square roots of \(i\)):
\(i = e^{i(\pi/2 + 2k\pi)}\)
Roots are \(e^{i(\pi/4 + k\pi)}\)
For \(k=0\), \(z_1 = e^{i\pi/4}\)
For \(k=1\), \(z_2 = e^{i5\pi/4}\) (which is \(e^{-i3\pi/4}\))

Memory Aid: All roots of \(z^n = w\) lie on a circle of radius \(\sqrt[n]{|w|}\) and are separated by an angle of \(\frac{2\pi}{n}\).

Key Takeaway: Roots are just points on a circle, spread out equally like the numbers on a clock face.

5. Loci: Drawing in the Complex Plane

A "locus" (plural: loci) is just a set of points that follow a rule. Think of it like a "path" a point \(z\) can move on.

The Circle: \(|z - c| \le r\)

• Meaning: The distance between \(z\) and point \(c\) is less than or equal to \(r\).
• Visual: A solid circle with center \(c\) and radius \(r\). If it was just \(|z-c| = r\), it would be just the boundary line.

The Perpendicular Bisector: \(|z - a| = |z - b|\)

• Meaning: \(z\) is always the same distance from \(a\) as it is from \(b\).
• Visual: A straight line that cuts the segment \(ab\) in half at a 90-degree angle.

The Ray: \(\arg(z - a) = \alpha\)

• Meaning: The angle of the line starting from \(a\) to \(z\) is fixed at \(\alpha\).
• Visual: A half-line (ray) starting from point \(a\).
• Crucial Tip: The point \(a\) itself is not included. We usually draw a small open circle at \(a\) to show this.

Common Mistake: For \(\arg(z-a) = \alpha\), students often draw a full line. Remember, it only goes in one direction from \(a\)!

Key Takeaway: Modulus (\(|\dots|\)) equations usually describe circles or lines; Argument (\(\arg\)) equations describe rays.

Summary Checklist

• Can I convert between Cartesian, Polar, and Exponential forms?
• Do I remember that multiplying complex numbers means adding their arguments?
• Can I apply De Moivre's Theorem to find \(z^n\)?
• Do I know the steps to find all \(n\) roots of a complex number?
• Can I sketch the three basic loci (circle, bisector, ray)?

Great job! Complex numbers can feel abstract, but if you keep thinking about them as points and arrows on a grid, you'll find them much more intuitive.