Introduction to Complex Numbers

Welcome to one of the most exciting expansions of your mathematical toolkit! Up until now, you have likely been told that you cannot take the square root of a negative number. While that is true within the set of Real Numbers (\(\mathbb{R}\)), we are about to enter a larger world. Complex numbers allow us to solve equations that previously had "no solution" and provide a beautiful way to represent points and rotations in a two-dimensional plane. Don't worry if this seems a bit "imaginary" at first—by the end of this chapter, you’ll see that complex numbers are just as structured and logical as the numbers you’ve used since primary school.

1. The Imaginary Unit \(i\)

The foundation of this entire chapter is a single definition. We define the number \(i\) such that:

\(i^2 = -1\)

Or, more simply:

\(i = \sqrt{-1}\)

Why does this matter? It allows us to express the square root of any negative number. For example, \(\sqrt{-9}\) can be written as \(\sqrt{9 \cdot -1}\), which simplifies to \(3i\).
Quick Review: Powers of \(i\)
Because \(i^2 = -1\), the powers of \(i\) follow a repeating pattern:

\(i^1 = i\)
\(i^2 = -1\)
\(i^3 = i^2 \cdot i = -i\)
\(i^4 = i^2 \cdot i^2 = (-1)(-1) = 1\)

After \(i^4\), the pattern repeats every four steps!

2. The Cartesian Form \(z = a + bi\)

A complex number (usually denoted by the letter \(z\)) consists of two parts: a real part and an imaginary part. The standard Cartesian form is written as:

\(z = a + bi\)

Where:

\(a\) is the Real Part, written as \(\text{Re}(z) = a\)
\(b\) is the Imaginary Part, written as \(\text{Im}(z) = b\)

Note: Both \(a\) and \(b\) are themselves real numbers. The \(i\) just tells us which direction the \(b\) is "pointing" in the complex plane. Key Takeaway: If \(b = 0\), the number is purely real. If \(a = 0\), the number is called purely imaginary.

3. The Complex Plane (Argand Diagram)

Just as we use a 1D number line for real numbers, we use a 2D plane for complex numbers, called the Argand Diagram.

1. The horizontal axis is the Real Axis (Re).
2. The vertical axis is the Imaginary Axis (Im).

To plot the number \(z = 3 + 2i\), you simply move 3 units to the right and 2 units up. It looks exactly like plotting the coordinates \((3, 2)\) on a standard graph! Did you know? Visualizing numbers this way is why complex numbers are so useful in physics and engineering, especially when dealing with things that rotate, like wheels or alternating electric currents.

4. Modulus and Argument

When we look at a complex number on the Argand diagram, we can describe its position in two ways: its "address" (\(a + bi\)) or its distance and angle from the origin.

The Modulus \(|z|\)

The modulus is the magnitude or "length" of the complex number—the distance from the origin \((0,0)\) to the point \((a, b)\). We use Pythagoras' Theorem to find it:

\(|z| = \sqrt{a^2 + b^2}\)

Example: If \(z = 3 + 4i\), then \(|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\).

The Argument \(\text{arg}(z)\)

The argument is the angle \(\theta\) that the line connecting the origin to \(z\) makes with the positive real axis.

\(\tan(\theta) = \frac{b}{a}\)

Common Mistake to Avoid: When calculating \(\theta = \arctan(\frac{b}{a})\), always check which quadrant your number is in! If your number is in the 2nd or 3rd quadrant (where \(a\) is negative), your calculator might give you the wrong angle. Always sketch a small Argand diagram to be sure.

In the IB curriculum, we usually use the principal argument, which is restricted to the range: \(-\pi < \theta \leq \pi\).

5. The Complex Conjugate

For any complex number \(z = a + bi\), its complex conjugate is denoted as \(\bar{z}\) (pronounced "z-bar") and is defined as:

\(\bar{z} = a - bi\)

Geometrically, \(\bar{z}\) is a reflection of \(z\) in the real axis. Why is this useful? One of the coolest properties is that when you multiply a complex number by its conjugate, the result is always a real number:

\(z \bar{z} = (a + bi)(a - bi) = a^2 + b^2 = |z|^2\)

6. Arithmetic in Cartesian Form

Working with complex numbers is very similar to working with algebraic expressions, provided you remember that \(i^2 = -1\).

Addition and Subtraction

Simply group the real parts and the imaginary parts together.

\((a + bi) + (c + di) = (a + c) + (b + d)i\)

Multiplication

Use the FOIL method (First, Outside, Inside, Last) just like you would with \((x+2)(x+3)\).

\((a + bi)(c + di) = ac + adi + bci + bdi^2\)

Since \(i^2 = -1\), the last term becomes \(-bd\):

\(z_1 z_2 = (ac - bd) + (ad + bc)i\)

Division

To divide complex numbers, we want to get the \(i\) out of the denominator. We do this by multiplying the top and bottom by the conjugate of the denominator.

To find \(\frac{a + bi}{c + di}\), multiply by \(\frac{c - di}{c - di}\).

This turns the denominator into a real number (\(c^2 + d^2\)), making the fraction easy to split into \(a + bi\) form.

7. Summary Checklist

Before moving on to Polar and Euler form (covered in the next chapter), ensure you can:

• Identify the real part \(\text{Re}(z)\) and imaginary part \(\text{Im}(z)\).
• Plot a complex number on an Argand diagram.
• Calculate the modulus \(|z|\) and the principal argument \(\text{arg}(z)\).
• Find the conjugate \(\bar{z}\) and understand its geometric meaning.
• Add, subtract, multiply, and divide complex numbers in Cartesian form.

Key Takeaway: Treat \(i\) like a variable when doing algebra, but every time you see \(i^2\), immediately replace it with \(-1\). This simple rule will solve 90% of the common errors in this topic!