Complex numbers expressed in cartesian form and Argand diagrams

Welcome to the World of Complex Numbers!

Ever felt stuck when solving a quadratic equation like \(x^2 + 1 = 0\)? In secondary school, we simply said there were "no real roots." But in H2 Mathematics, we unlock a whole new dimension of numbers that allows us to solve these "impossible" problems. Complex numbers aren't just a mathematical trick; they are essential in electrical engineering, quantum physics, and even computer graphics!

In this guide, we will explore the Cartesian form of complex numbers and how to visualize them using Argand diagrams. Don't worry if it seems a bit "imaginary" at first—we'll break it down step-by-step.

1. Extending the Number System: What is \(i\)?

The foundation of complex numbers is the imaginary unit, denoted by \(i\). We define it as:

\(i = \sqrt{-1}\) or \(i^2 = -1\)

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

\(z = x + iy\)

Where:
• \(x\) is the real part, written as \(Re(z)\).
• \(y\) is the imaginary part, written as \(Im(z)\). (Note: \(y\) is a real number!)

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

Quick Review: Powers of \(i\)

Because \(i^2 = -1\), the powers of \(i\) follow a repeating pattern:
• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = i^2 \times i = -i\)
• \(i^4 = (i^2)^2 = 1\)

2. Complex Roots of Quadratic Equations

When the discriminant (\(b^2 - 4ac\)) of a quadratic equation is negative, the roots are complex. These roots always appear in conjugate pairs if the coefficients are real.

Step-by-Step Example: Solve \(x^2 - 4x + 13 = 0\).
1. Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
2. Substitute values: \(x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(13)}}{2(1)}\)
3. Simplify: \(x = \frac{4 \pm \sqrt{16 - 52}}{2} = \frac{4 \pm \sqrt{-36}}{2}\)
4. Since \(\sqrt{-36} = \sqrt{36} \times \sqrt{-1} = 6i\):
5. Result: \(x = \frac{4 \pm 6i}{2} = 2 \pm 3i\)

Key Takeaway: If a quadratic has real coefficients and one root is \(2 + 3i\), the other root must be \(2 - 3i\).

3. The Argand Diagram

Think of an Argand diagram as a coordinate plane for complex numbers. Instead of \(x\) and \(y\) axes, we use:
• The Horizontal Axis: Real axis (\(Re\))
• The Vertical Axis: Imaginary axis (\(Im\))

The complex number \(z = x + iy\) is represented by the point \((x, y)\) or by a vector from the origin to that point.

Analogy: Just like you use GPS coordinates (Latitude, Longitude) to find a place on a map, we use (Real, Imaginary) to find a complex number on the Argand diagram.

4. Modulus, Argument, and Conjugate

The Conjugate (\(\bar{z}\))

If \(z = x + iy\), its conjugate is \(\bar{z} = x - iy\). Geometrically, this is a reflection of the point in the Real axis.

The Modulus (\(|z|\))

The modulus is the distance of the point from the origin. We use Pythagoras' Theorem:
\(|z| = \sqrt{x^2 + y^2}\)

The Argument (\(\arg z\))

The argument is the angle \(\theta\) the vector makes with the positive Real axis. By convention, we use the Principal Argument, where \(-\pi < \arg z \le \pi\) (or \(-180^\circ < \theta \le 180^\circ\)).

• For a point in the 1st quadrant: \(\theta = \tan^{-1}(\frac{y}{x})\)
• Common Mistake: Always check which quadrant your point is in before calculating the angle! Don't just rely on the calculator's \(\tan^{-1}\) value.

5. Operations with Complex Numbers

Addition and Subtraction

Simply treat \(i\) like a variable (like \(x\)) and group "like terms."
\((a + bi) + (c + di) = (a + c) + (b + d)i\)

Multiplication

Use the "FOIL" method (expanding brackets) and remember that \(i^2 = -1\).
Example: \((2 + 3i)(1 - 2i) = 2 - 4i + 3i - 6i^2\)
= \(2 - i - 6(-1)\)
= \(2 - i + 6 = 8 - i\)

Division

To divide, we "rationalize the denominator" by multiplying the top and bottom by the conjugate of the denominator.
\(\frac{z_1}{z_2} = \frac{x_1 + iy_1}{x_2 + iy_2} \times \frac{x_2 - iy_2}{x_2 - iy_2}\)

Quick Review Box: The product of a complex number and its conjugate is always a real number: \((x + iy)(x - iy) = x^2 + y^2\).

Equality of Complex Numbers

If \(a + bi = c + di\), then \(a = c\) and \(b = d\). You can solve equations by "comparing real and imaginary parts."

6. Geometrical Effects on the Argand Diagram

Operations on complex numbers have beautiful geometric meanings:

• Negation (\(-z\)): A reflection through the origin.
• Conjugation (\(\bar{z}\)): A reflection across the Real axis.
• Addition (\(z_1 + z_2\)): Follows the Parallelogram Law (just like adding vectors).
• Multiplication by \(i\): A 90° anticlockwise rotation about the origin.
• Multiplication by \(-i\): A 90° clockwise rotation about the origin.

Did you know? Multiplying by \(i\) twice gives \(i^2 = -1\). Geometrically, two 90° rotations equal a 180° rotation, which is exactly what happens when you multiply a number by -1!

7. Conjugate Roots Theorem

For any polynomial equation \(P(z) = 0\) with real coefficients, if a complex number \(w\) is a root, then its conjugate \(\bar{w}\) is also a root.

Important Point: This theorem only works if all coefficients of the polynomial are real. If the equation is \(z^2 + iz + 2 = 0\), the roots will not necessarily be conjugates.

Summary of Key Takeaways

• Cartesian form: \(z = x + iy\).
• \(i^2 = -1\): The most important rule in multiplication.
• Argand Diagram: Real part on the x-axis, Imaginary part on the y-axis.
• Conjugate: Change the sign of the imaginary part to reflect across the real axis.
• Comparing Parts: Solve equations by setting Re = Re and Im = Im.
• Rotation: Multiplying by \(i\) rotates the vector 90° anticlockwise.

Don't worry if this seems tricky at first! Complex numbers are a new way of thinking. Practice plotting them on Argand diagrams, and the algebra will start to feel more natural.