Ultimate Guide to Complex Arithmetic (Part 2): Operations and Conjugates

Introduction

Now that we have established the existence of the imaginary unit $i$ , it's time to learn how to manipulate these new numbers algebraically. In Complex Arithmetic (Part 2), you will learn how the set of complex numbers fits into the broader hierarchy of mathematics, how to perform basic operations like addition and multiplication, and how to utilize the "complex conjugate"—a vital tool for simplifying complex expressions.

Executive Summary

This guide breaks down the core algebraic mechanics of complex numbers:

Number Sets: Classifying numbers into Natural ( $\mathbb{N}$ ), Integers ( $\mathbb{Z}$ ), Rational ( $\mathbb{Q}$ ), Real ( $\mathbb{R}$ ), and Complex ( $\mathbb{C}$ ).
Real and Imaginary Parts: Defining $\text{Re}(z)$ and $\text{Im}(z)$ , and establishing the condition for equality.
Basic Arithmetic: Adding, subtracting, and multiplying complex numbers algebraically.
The Complex Conjugate ( $\bar{z}$ ): Defining the conjugate and using it to turn complex products into purely real numbers.

What is this about?

When you first learned algebra, you learned how to collect "like terms"—grouping all the $x$ 's together and all the constant numbers together. Complex arithmetic works on the exact same principle. A complex number $z = x + iy$ has a "real" part and an "imaginary" part. You cannot simply mush them together into a single number. Instead, you must keep them separate, adding real parts to real parts, and imaginary parts to imaginary parts. This section formalizes the rules for operating on these two-part numbers.

Main Content

1. Classifying Numbers and Number Sets

Mathematics is built on an expanding hierarchy of number sets. Each set is an extension of the previous one:

Natural Numbers ( $\mathbb{N}$ ): The counting numbers ( $1, 2, 3, \dots$ ).
Integers ( $\mathbb{Z}$ ): Adding zero and negative whole numbers ( $\dots, -2, -1, 0, 1, 2, \dots$ ).
Rational Numbers ( $\mathbb{Q}$ ): Fractions of integers (e.g., $\frac{1}{2}, -0.75$ ).
Real Numbers ( $\mathbb{R}$ ): Adding irrational numbers that cannot be written as fractions (e.g., $\pi, \sqrt{2}$ ).
Complex Numbers ( $\mathbb{C}$ ): Adding the imaginary unit $i$ to form numbers like $3 + 4i$ .

Because every real number $x$ can be written as $x + 0i$ , the real numbers are actually a subset of the complex numbers! ( $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$ ).

2. Real and Imaginary Parts

For a complex number $z = x + iy$ (where $x, y \in \mathbb{R}$ ):

We call $x$ the real part of $z$ , denoted by $\text{Re}(z) = x$ .
We call $y$ the imaginary part of $z$ , denoted by $\text{Im}(z) = y$ .

Equality of Complex Numbers: Two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ are equal if and only if their real parts are equal and their imaginary parts are equal. $z_1 = z_2 \iff a = c \text{ and } b = d$ This is a powerful tool. It means a single complex equation actually represents two simultaneous real equations.

3. Addition, Subtraction, and Multiplication

Addition and Subtraction: Treat $i$ just like an algebraic variable (like $x$ ). Collect the real terms and collect the imaginary terms. If $z_1 = a + ib$ and $z_2 = c + id$ :

$z_1 + z_2 = (a + c) + i(b + d)$
$z_1 - z_2 = (a - c) + i(b - d)$

Multiplication: Expand the brackets using FOIL (First, Outer, Inner, Last), and then apply the crucial rule $i^2 = -1$ . $(a + ib)(c + id) = ac + iad + ibc + i^2bd$ Since $i^2 = -1$ , the last term becomes $-bd$ . Grouping the real and imaginary parts: $= (ac - bd) + i(ad + bc)$

4. The Complex Conjugate ( $\bar{z}$ )

The complex conjugate of a complex number $z = x + iy$ is defined by flipping the sign of the imaginary part. It is denoted by a bar over the variable: $\bar{z} = x - iy$

Why is this useful? Watch what happens when you multiply a complex number by its conjugate: $z\bar{z} = (x + iy)(x - iy)$ This is a "difference of two squares" expansion: $= x^2 - (iy)^2 = x^2 - i^2 y^2$ Since $i^2 = -1$ : $= x^2 + y^2$

The result is a purely real number! Multiplying by the conjugate is the primary method used to eliminate imaginary numbers from denominators (which we will cover in Part 3).

Simple Worked Example

Question: Let $z = 3 + 2i$ and $w = 1 - 4i$ . Evaluate: (a) $2z - w$ (b) $z \cdot \bar{w}$

Solution: (a) Evaluate $2z - w$ Substitute the complex numbers: $2(3 + 2i) - (1 - 4i)$ Expand the brackets: $= 6 + 4i - 1 + 4i$ Collect real and imaginary parts: $= (6 - 1) + i(4 + 4)$ $= 5 + 8i$

(b) Evaluate $z \cdot \bar{w}$ First, find the conjugate of $w$ . If $w = 1 - 4i$ , then $\bar{w} = 1 + 4i$ . Now multiply $z$ and $\bar{w}$ : $(3 + 2i)(1 + 4i)$ Expand using FOIL: $= (3)(1) + (3)(4i) + (2i)(1) + (2i)(4i)$ $= 3 + 12i + 2i + 8i^2$ Since $i^2 = -1$ , the last term is $-8$ : $= 3 + 14i - 8$ $= -5 + 14i$

mini-FAQ page

Q: Can I use my calculator for complex arithmetic? A: Yes! Most modern scientific calculators (like the Casio fx-100AU PLUS or fx-82/100 series with complex modes) can add, subtract, and multiply complex numbers in Cartesian form. However, you must still show your working in HSC exams, particularly for algebraic proofs.

Q: What is the conjugate of a purely real number, like $z = 7$ ? A: The imaginary part is $0$ , so $z = 7 + 0i$ . The conjugate is $\bar{z} = 7 - 0i = 7$ . The conjugate of a real number is just itself!

Common mistakes to avoid

Mishandling the $i^2$ term: When expanding $(a + ib)(a - ib)$ , many students write $a^2 - b^2$ . Remember that $(ib)^2 = -b^2$ , so the expansion evaluates to $a^2 + b^2$ .
Equating variables incorrectly: If $x + iy = 5$ , do not assume $x=5$ and $y=5$ . This equation means the real part is $5$ and the imaginary part is $0$ . Therefore, $x=5$ and $y=0$ .
Conjugating both parts: The conjugate of $3 + 4i$ is $3 - 4i$ . Do not change the sign of the real part ( $-3 - 4i$ is incorrect).

Practice on Vu's Maths Hub

Complex arithmetic must become second nature before moving on to geometry and polynomial roots. Build your fluency with our resources:

Practice addition, multiplication, and conjugate expansions in the HSC Complex Numbers Booklet.
Use complex equality to solve algebraic puzzles in the HSC Last Resorts Booklet.

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