Proof and Complex Numbers

In VCE Specialist Mathematics, the study of complex numbers extends the real number system $\mathbb{R}$ to the complex field $\mathbb{C}$. This involves understanding the algebraic properties of the imaginary unit $i$, geometric representations in the Argand plane, and the application of mathematical proof to verify properties and solve equations.

1. Fundamentals of Complex Numbers

A complex number $z$ is expressed in the form $z = a + bi$, where $a, b \in \mathbb{R}$ and $i$ is the imaginary unit defined by $i^2 = -1$.

Key Definitions

• Real Part: $\text{Re}(z) = a$
• Imaginary Part: $\text{Im}(z) = b$
• Complex Conjugate: If $z = a + bi$, then $\bar{z} = a - bi$.
• Equality: $a + bi = c + di$ if and only if $a = c$ and $b = d$.

Basic Arithmetic

1. Addition/Subtraction: Combine real and imaginary parts separately.
   $$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$$
2. Multiplication: Expand using the distributive law and substitute $i^2 = -1$.
   $$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$
3. Division: Multiply the numerator and denominator by the conjugate of the denominator to “realise” the divisor.
   $$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$

EXAM TIP: When simplifying powers of $i$, remember the cyclic nature: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. For any $i^n$, divide $n$ by 4 and use the remainder.

2. Geometric Representations and Polar Form

Complex numbers can be represented as points or vectors on an Argand Diagram, where the horizontal axis is the Real axis ($\text{Re}$) and the vertical axis is the Imaginary axis ($\text{Im}$).

Modulus and Argument

• Modulus: The distance from the origin, $|z| = \sqrt{a^2 + b^2}$.
• Argument: The angle $\theta$ the vector makes with the positive $\text{Re}$ axis.
• Principal Argument: $\text{Arg}(z) = \theta$, where $-\pi < \theta \le \pi$.

Polar (Mod-Arg) Form

A complex number can be written as:
$$z = r(\cos\theta + i\sin\theta) = r\text{cis}(\theta)$$
where $r = |z|$ and $\theta = \text{Arg}(z)$.

Geometric Transformations

Operations on complex numbers correspond to geometric transformations in the Argand plane:

KEY TAKEAWAY: Multiplying two complex numbers in polar form results in multiplying their moduli and adding their arguments: $|z_1 z_2| = |z_1||z_2|$ and $\text{Arg}(z_1 z_2) = \text{Arg}(z_1) + \text{Arg}(z_2)$.

3. Solving Equations over the Complex Field

The Fundamental Theorem of Algebra states that a polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicity).

Quadratic Equations

For $az^2 + bz + c = 0$ where the discriminant $\Delta = b^2 - 4ac < 0$, the roots are complex conjugates:
$$z = \frac{-b \pm i\sqrt{|\Delta|}}{2a}$$

Square Roots of a Complex Number

To find the square root of $w = x + iy$, set $(a + bi)^2 = x + iy$:
1. Expand: $(a^2 - b^2) + 2abi = x + iy$
2. Equate parts: $a^2 - b^2 = x$ and $2ab = y$
3. Solve the simultaneous equations for $a$ and $b$.

Factorising Polynomials

• Conjugate Root Theorem: If $P(z)$ is a polynomial with real coefficients, and $z = u + vi$ is a root, then $\bar{z} = u - vi$ must also be a root.
• Factor Theorem: If $P(\alpha) = 0$, then $(z - \alpha)$ is a linear factor of $P(z)$.

COMMON MISTAKE: Students often apply the Conjugate Root Theorem to polynomials with complex coefficients. This is incorrect; the theorem only applies if all coefficients of the polynomial are real.

4. Mathematical Proof

VCE Specialist Mathematics requires the use of formal logic to establish the truth of mathematical statements.

Methods of Proof

1. Direct Proof: A clear sequence of logical steps starting from known facts to reach a conclusion.
2. Proof by Contrapositive: To prove $P \implies Q$, we prove $\text{not } Q \implies \text{not } P$.
3. Proof by Contradiction: Assume the statement is false and show this leads to a logical impossibility.
4. Proof by Induction: (Primarily used for sequences and series or divisibility).

Proofs in Complex Numbers

Common proofs involve verifying properties of conjugates and moduli:
* Property: $\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$
* Property: $|z|^2 = z\bar{z}$
* Triangle Inequality: $|z_1 + z_2| \le |z_1| + |z_2|$

Divisibility Proofs

Often involve showing that an expression $f(n)$ is divisible by an integer $k$ for all $n \in \mathbb{Z}^+$. These are frequently tested via Mathematical Induction in Unit 3.

VCAA FOCUS: Expect questions that require you to “Show that…” a particular complex number is a root of a cubic equation and then “Hence” find the remaining roots. The “Hence” usually implies using the Conjugate Root Theorem or polynomial long division.

5. Summary Table: Properties of Conjugates and Modulus

STUDY HINT: Practice converting between Cartesian form ($a+bi$) and Polar form ($r\text{cis}\theta$) quickly. This is a foundational skill that makes multiplication, division, and finding powers of complex numbers significantly easier.