This topic introduces the formal language of mathematics and the extension of the number system to include complex numbers.
| Term | Definition |
|---|---|
| Axiom | A statement accepted without proof |
| Definition | Precise meaning assigned to a term |
| Theorem | A statement proved from axioms and prior results |
| Lemma | A helper result used to prove a larger theorem |
| Corollary | A result following directly from a theorem |
Direct proof: Assume hypothesis, apply logic, reach conclusion.
Contrapositive: To prove $P \Rightarrow Q$, prove the equivalent $\neg Q \Rightarrow \neg P$.
Contradiction: Assume $\neg P$; derive a known falsehood.
Mathematical induction:
1. Base case: verify $P(1)$.
2. Inductive step: assume $P(k)$, prove $P(k+1)$.
3. Conclude $P(n)$ holds for all $n \geq 1$.
Induction example: Prove $\displaystyle\sum_{r=1}^{n} r = \dfrac{n(n+1)}{2}$.
Base case $n=1$: LHS $= 1$, RHS $= \frac{1 \cdot 2}{2} = 1$. True.
Inductive step: Assume $\displaystyle\sum_{r=1}^{k} r = \frac{k(k+1)}{2}$. Then:
$$\sum_{r=1}^{k+1} r = \frac{k(k+1)}{2} + (k+1) = \frac{(k+1)(k+2)}{2}.$$
This is the formula at $n = k+1$. $\square$
$\mathbb{C} = {a + bi : a, b \in \mathbb{R}}$ where $i^2 = -1$.
Division: $\dfrac{z_1}{z_2} = \dfrac{z_1 \bar{z}_2}{|z_2|^2}$.
Example: $z = 3+4i$: $|z| = 5$, $\arg(z) = \arctan(4/3)$, $\dfrac{1}{z} = \dfrac{3-4i}{25}$.
KEY TAKEAWAY: Proof and complex numbers are the twin foundations of Specialist Mathematics. Master the four proof methods and the arithmetic of $\mathbb{C}$ before attempting any other topic.
EXAM TIP: In induction proofs, always label the base case and inductive step. VCAA awards marks for structure.
COMMON MISTAKE: Confusing the contrapositive ($\neg Q \Rightarrow \neg P$, logically equivalent) with the converse ($Q \Rightarrow P$, not equivalent).
