Ultimate Guide to Complex Arithmetic (Part 1): Introducing the Imaginary Unit

Introduction

For years, whenever you encountered a quadratic equation where the discriminant ( $\Delta = b^2 - 4ac$ ) was less than zero, you simply wrote "no real solutions" and moved on. In HSC Mathematics Extension 2, we refuse to accept that limitation. Complex Arithmetic introduces a revolutionary new mathematical concept: the imaginary unit, denoted as $i$ . This singular addition to our number system unlocks the ability to solve any polynomial equation.

Executive Summary

This guide is the first part of a deep dive into the Arithmetic of Complex Numbers:

Defining the Imaginary Unit: Understanding the foundational definition $i^2 = -1$ .
Solving Simple Quadratics: Using $i$ to solve equations like $x^2 + a^2 = 0$ .
The Complex Number Set ( $\mathbb{C}$ ): Defining numbers in the form $z = x + iy$ .
The Quadratic Formula: Expressing roots for equations where the discriminant is negative ( $\Delta < 0$ ).

What is this about?

When you take the square root of a positive number, like $\sqrt{9}$ , you get $3$ (and $-3$ ). But what happens if you try to evaluate $\sqrt{-9}$ ? No real number multiplied by itself can yield a negative result.

To bridge this gap, mathematicians defined a new number, $i$ , specifically designed so that its square is $-1$ . By combining this "imaginary" number with the "real" numbers we already know, we create a 2D number system called the Complex Numbers. This system is not just a mathematical trick; it has profound applications in quantum mechanics, electrical engineering, and fluid dynamics.

Main Content

1. Defining the Imaginary Unit, $i$

The cornerstone of the entire complex number syllabus is a single definition: $i^2 = -1$

From this, it naturally follows that $i = \sqrt{-1}$ . Using index laws, we can find higher powers of $i$ :

$i^1 = i$
$i^2 = -1$
$i^3 = i^2 \cdot i = -1 \cdot i = -i$
$i^4 = i^2 \cdot i^2 = (-1)(-1) = 1$
$i^5 = i^4 \cdot i = 1 \cdot i = i$

Notice the repeating cycle: $i, -1, -i, 1$ . This pattern of four allows you to evaluate any integer power of $i$ rapidly. For example, $i^{400} = (i^4)^{100} = 1^{100} = 1$ .

2. Solving Simple Quadratic Equations

Consider the equation $x^2 + 9 = 0$ . In the real number system, $x^2 = -9$ has no solutions. With the introduction of $i$ , we can rewrite $-9$ as $9i^2$ . $x^2 = 9i^2$ Taking the square root of both sides: $x = \pm \sqrt{9i^2} = \pm 3i$

In general, to solve equations of the form $x^2 + a^2 = 0$ (where $a$ is a positive real number): $x^2 = -a^2$ $x = \pm \sqrt{-a^2} = \pm \sqrt{a^2 \cdot (-1)} = \pm a\sqrt{-1} = \pm ai$

3. The Set of Complex Numbers ( $\mathbb{C}$ )

A complex number is formed by joining a real number and an imaginary number together. We define the set of complex numbers ( $\mathbb{C}$ ) as the set of all numbers of the form: $z = x + iy$ where $x$ and $y$ are real numbers.

For example, $z = 3 + 4i$ . Here, $3$ is the real part, and $4$ is the imaginary part. (Note: The imaginary part is the real number $4$ , not $4i$ . We will cover this distinction further in Part 2).

4. Quadratic Equations with Negative Discriminants

We can now use complex numbers to express the roots of any quadratic equation $ax^2 + bx + c = 0$ , where $a, b, c$ are real numbers, even when the discriminant $\Delta = b^2 - 4ac$ is negative ( $\Delta < 0$ ).

The standard quadratic formula is: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$

When $\Delta < 0$ , we can factor out $-1$ inside the square root: $\sqrt{\Delta} = \sqrt{-1 \cdot |\Delta|} = \sqrt{-1}\sqrt{|\Delta|} = i\sqrt{|\Delta|}$

Thus, the roots become: $x = \frac{-b \pm i\sqrt{|\Delta|}}{2a}$

Simple Worked Example

Question: Solve the quadratic equation $x^2 - 4x + 13 = 0$ .

Solution: Step 1: Identify $a, b,$ and $c$ . $a = 1, b = -4, c = 13$ .

Step 2: Calculate the discriminant ( $\Delta$ ). $\Delta = b^2 - 4ac$ $\Delta = (-4)^2 - 4(1)(13)$ $\Delta = 16 - 52$ $\Delta = -36$

Step 3: Apply the quadratic formula. Since $\Delta < 0$ , we will have complex roots. $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ $x = \frac{-(-4) \pm \sqrt{-36}}{2(1)}$ $x = \frac{4 \pm \sqrt{36 \cdot -1}}{2}$ $x = \frac{4 \pm 6i}{2}$

Step 4: Simplify. Divide both terms by $2$ : $x = 2 \pm 3i$

The two roots are $x_1 = 2 + 3i$ and $x_2 = 2 - 3i$ . Notice that they form a "conjugate pair" (the imaginary parts are opposites). We will explore this vital concept in the upcoming sections.

mini-FAQ page

Q: Are complex numbers actually "imaginary"? Do they exist? A: "Imaginary" is just an unfortunate historical name given by mathematician René Descartes, who was skeptical of them. Complex numbers are just as "real" as negative numbers or fractions—none of them exist as physical objects you can hold, but they are all perfectly logically consistent tools used to model the real world.

Q: Can $x$ and $y$ in $z = x + iy$ be fractions or decimals? A: Yes! As long as $x$ and $y$ belong to the set of real numbers ( $\mathbb{R}$ ), they can be any fraction, decimal, or irrational number (like $\sqrt{2}$ ). For example, $z = \pi + \frac{1}{2}i$ is a valid complex number.

Common mistakes to avoid

Writing the imaginary part with $i$ : If $z = 3 + 4i$ , the imaginary part is $4$ , not $4i$ .
Messing up the $\pm$ sign: When solving $x^2 = -16$ , students often write $x = 4i$ and forget the $-4i$ solution. Every non-zero number has exactly two square roots!
Squaring a complex number incorrectly: Remember that $(3i)^2 = 3^2 \cdot i^2 = 9(-1) = -9$ . Many students accidentally write $+9$ .

Practice on Vu's Maths Hub

Ready to master complex arithmetic? Put your skills to the test with our comprehensive resources:

Practice evaluating powers of $i$ and solving complex quadratics in the HSC Complex Numbers Booklet.
See how complex roots factor into larger polynomial equations in the HSC Polynomials Booklet.
Tackle challenging exam-style algebra questions in the HSC Last Resorts Booklet.

Connect with me

Want to master Complex Numbers and lock in top marks for HSC Mathematics Extension 2? Visit Vu's Maths Hub for in-depth booklets, detailed video explanations, and expert advice to help you ace your exams!