Complex Numbers Key Terms: A Comprehensive Glossary

Introduction

Complex Numbers expand the mathematical universe by allowing the square roots of negative numbers. In HSC Mathematics Extension 2, this topic introduces an entirely new set of algebra and 2D geometric visualization. This glossary provides the definitive mathematical definitions and illustrative examples for the core terminology used throughout the Complex Numbers syllabus.

Executive Summary

This guide serves as a quick-reference dictionary for:

Formats: Cartesian form, Modulus-Argument (Polar) form.
Operations: Conjugation, Dilation, Rotation.
The Complex Plane: Vectors, modulus, principal argument.
Polynomials: Roots of unity, discriminant, Complex Conjugate Root Theorem.

What is this about?

A single complex number can be viewed algebraically (as $x + iy$ ), trigonometrically (as $r(\cos\theta + i\sin\theta)$ ), or geometrically (as a vector on an Argand diagram). Understanding vocabulary like "principal argument", "conjugate", and "roots of unity" is crucial because exam questions constantly require you to switch between these different perspectives to find the most efficient solution.

Main Content: Key Terms

A–C

Argument: The angle $\theta$ $θ$ a complex number makes with the positive real axis on the complex plane, measured counter-clockwise.
- Example: The argument of $z = i$ is $\frac{\pi}{2}$ .
Cartesian Form of a Complex Number: A complex number expressed in rectangular coordinates as $z = x + iy$ $z = x + i y$ , where $x$ $x$ and $y$ $y$ are real numbers.
- Example: $z = 3 - 4i$ .
Complex Conjugate Root Theorem: A theorem stating that if a polynomial with purely real coefficients has a complex root $z = a + ib$ $z = a + ib$ , then its complex conjugate $\bar{z} = a - ib$ $\overset{z}{ˉ} = a - ib$ must also be a root.
- Example: If $2 + 3i$ is a root of $x^2 - 4x + 13 = 0$ , then $2 - 3i$ is definitely the other root.
Complex Number: A number belonging to the set $\mathbb{C}$ $C$ , which can be written in the form $x + iy$ $x + i y$ , where $x, y \in \mathbb{R}$ $x, y \in R$ and $i^2 = -1$ $i^{2} = - 1$ .
- Example: $5 + 2i$ , or even $7$ (since it can be written as $7 + 0i$ ).
Complex Plane: A 2D geometric plane (also called an Argand diagram) where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number.
- Example: The number $3 + 4i$ is plotted at the coordinate $(3, 4)$ on the complex plane.
Conjugate: The complex conjugate of $z = x + iy$ $z = x + i y$ is $\bar{z} = x - iy$ $\overset{z}{ˉ} = x - i y$ . Geometrically, it is the reflection of $z$ $z$ across the real axis.
- Example: The conjugate of $-2 + 5i$ is $-2 - 5i$ .

D–M

De Moivre’s Theorem: A powerful formula linking complex numbers to trigonometry: $(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$ $(r (cos θ + i sin θ))^{n} = r^{n} (cos (n θ) + i sin (n θ))$ .
- Example: $(\cos\pi + i\sin\pi)^3 = \cos(3\pi) + i\sin(3\pi)$ .
Dilation: A geometric transformation that changes the scale or magnitude (length) of a complex vector without changing its angle, achieved by multiplying by a real scalar.
- Example: Multiplying $z = 1 + i$ by the scalar $3$ dilates it to $3 + 3i$ .
Discriminant: The expression under the square root in the quadratic formula, $\Delta = b^2 - 4ac$ $Δ = b^{2} - 4 a c$ . If $\Delta < 0$ $Δ < 0$ , the roots are complex conjugates.
- Example: For $x^2 + 2x + 5 = 0$ , $\Delta = (2)^2 - 4(1)(5) = -16$ .
Integer: A whole number ( $\dots, -2, -1, 0, 1, 2, \dots$ $\dots, - 2, - 1, 0, 1, 2, \dots$ ).
- Example: $5$ and $-10$ are integers.
Modulus: The absolute magnitude or distance of a complex number from the origin on the complex plane, denoted by $|z| = \sqrt{x^2 + y^2}$ $∣ z ∣ = x^{2} + y^{2}$ .
- Example: The modulus of $3 + 4i$ is $\sqrt{3^2 + 4^2} = 5$ .
Modulus–Argument Form: (Also known as Polar Form). Expressing a complex number by its length $r$ $r$ and angle $\theta$ $θ$ , as $z = r(\cos\theta + i\sin\theta)$ $z = r (cos θ + i sin θ)$ .
- Example: $z = 2(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})$ .

P–R

Polar Form: Another name for Modulus-Argument form.
- Example: $z = 5 \text{cis}(\pi)$ .
Polynomial: A mathematical expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.
- Example: $P(z) = z^3 - 2z^2 + z - 5$ .
Principal Argument: The unique argument of a complex number restricted to the domain $-\pi < \theta \le \pi$ $- π < θ \leq π$ . Denoted by $\text{Arg}(z)$ $Arg (z)$ with a capital A.
- Example: While $\frac{3\pi}{2}$ is a valid argument for $z = -i$ , its Principal Argument is $-\frac{\pi}{2}$ .
Proof: A rigorous logical argument demonstrating that a mathematical statement is true.
- Example: Proving geometrically that $|z_1 + z_2| \le |z_1| + |z_2|$ .
Quadratic Equation: A polynomial equation of the second degree, $az^2 + bz + c = 0$ $a z^{2} + b z + c = 0$ .
- Example: $z^2 - 6z + 10 = 0$ .
Rational Number: Any number that can be expressed as a fraction of two integers $\frac{p}{q}$ $\frac{p}{q}$ .
- Example: $\frac{2}{3}$ .
Real Number: Any number on the continuous 1D number line, including rationals and irrationals, but excluding numbers with imaginary parts.
- Example: $\pi, \sqrt{2}, 5$ .
Roots of Unity: The complex solutions to the equation $z^n = 1$ $z^{n} = 1$ . They form a regular $n$ $n$ -sided polygon centered at the origin on the complex plane.
- Example: The cube roots of unity ( $z^3=1$ ) are $1$ , $\text{cis}(\frac{2\pi}{3})$ , and $\text{cis}(\frac{-2\pi}{3})$ .
Rotation: A geometric transformation that rotates a complex vector around the origin, achieved by multiplying it by a complex number with a modulus of 1 (like $\text{cis}(\theta)$ $cis (θ)$ or $i$ $i$ ).
- Example: Multiplying $z = 2 + 0i$ by $i$ rotates it $90^\circ$ to become $0 + 2i$ .

T–V

Triangle Inequality: For complex numbers, it states that the length of the sum of two vectors is less than or equal to the sum of their individual lengths: $|z_1 + z_2| \le |z_1| + |z_2|$ $∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$ .
- Example: $|(3+4i) + (1+i)| \le |3+4i| + |1+i|$ .
Vector: A quantity with both magnitude and direction. Every complex number can be represented as a position vector originating from $0+0i$ $0 + 0 i$ on the complex plane.
- Example: $z = 4 - 3i$ is a vector pointing right 4 units and down 3 units.

mini-FAQ page

Q: Do I need to write 'cis' or fully write out $\cos\theta + i\sin\theta$ ? A: Both are perfectly acceptable in the HSC, but writing 'cis' saves a lot of time. Just remember it is shorthand; it is not a real trigonometric function on your calculator.

Q: What happens if I use an argument outside the principal argument range? A: If a question asks specifically for the Modulus-Argument form, you MUST use the principal argument ( $-\pi < \theta \le \pi$ ). If you write $\text{cis}(\frac{3\pi}{2})$ instead of $\text{cis}(-\frac{\pi}{2})$ , you will likely lose a mark.

Common mistakes to avoid

Messing up the Principal Argument quadrants: Calculating $\theta = \tan^{-1}(\frac{y}{x})$ on your calculator will only ever give you answers in the 1st and 4th quadrants. If your complex number is in the 2nd or 3rd quadrant (e.g., $z = -1 + i$ ), you must manually adjust the angle by adding or subtracting $\pi$ .
Forgetting that $|z|^2 = z \bar{z}$ : This simple algebraic identity is the secret to unlocking almost all complex number algebraic proofs. Do not try to expand $(x+iy)^2$ when doing modulus proofs!

Practice on Vu's Maths Hub

Fluency in switching between Cartesian and Polar vocabulary is vital.

Practice finding the principal argument and plotting vectors in the HSC Complex Numbers Booklet.
Review modulus properties and roots of unity at Vu's Maths Hub.

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