H3 Mathematics revision reference

This lookup covers the parts of 2027 H3 Mathematics (9820) that have lessons here so far. It is not a complete syllabus summary.

Statements and their related forms

For the conditional P ⇒ Q:

• P ⇒ Q is false only when P is true and Q is false.
• If P ⇒ Q: P is sufficient for Q, and Q is necessary for P.
• P ⇔ Q means both directions hold.

Read more: Conditionals, quantifiers and negation.

Negations

Negate nested quantifiers one at a time from the outside. “There exists a unique x” claims existence and that any two such x are equal.

Proof methods

• Direct: unfold the definitions in the hypothesis and work forwards to the defined form of the conclusion.
• Contrapositive: prove not Q ⇒ not P instead of P ⇒ Q.
• Contradiction: assume the statement is false, deduce something impossible, and name the contradiction.
• Cases: split into cases that together cover every possibility, and prove each one.
• Counterexample: one example that satisfies every hypothesis and fails the conclusion disproves a universal claim. Confirming examples never prove it.
• Existence: construct an object and check it, or show one must exist (for example by cases or the intermediate value theorem). Uniqueness: assume two and prove them equal.
• Pigeonhole: more than kn objects in n boxes put at least k + 1 in some box. Choose the boxes so that two objects in one box have the property you want.
• Combinatorial proof: show both sides count the same set; for a split into cases, say why the cases are disjoint and cover everything.
• Induction: prove P(n₀), and prove P(k) ⇒ P(k + 1) for every k ≥ n₀. Strong induction may assume all of P(n₀),…,P(k); a step reaching back j places needs j base cases.

Read more: Contrapositive, contradiction, cases and counterexamples, Proof by mathematical induction, Existence, uniqueness and construction and The pigeonhole principle and combinatorial proofs.

Inequalities

• AM–GM: (a + b)/2 ≥ square root of ab for a,b ≥ 0, with equality exactly when a = b.
• Cauchy–Schwarz: (∑ aᵢbᵢ)² ≤ ∑ aᵢ²∑ bᵢ², with equality when the lists are proportional.
• Triangle: |x + y| ≤ |x| + |y|, with equality exactly when xy ≥ 0.

Read more: AM–GM, Cauchy–Schwarz and the triangle inequality.

Congruences

a≡ bpmod n when n ∣ a-b. Congruences can be added, subtracted, multiplied and raised to powers; cancel only factors coprime to n. Every integer is congruent to exactly one of 0,1,…,n-1.

Read more: Congruences and modular arithmetic.

Problem solving and investigation

Work backwards, solve a simpler or similar problem, look for pattern and structure, consider cases and restate the problem. Test a conjecture on further and extreme cases before proving it, and when critiquing a solution check every step that is not reversible.

Read more: Heuristics, conjectures and critique.

Symmetry, series, integrals and counting

• Symmetry principle: quantities swapped by a symmetry of the problem are equal; for example ∫₀^(π/2)f(sin x, cos x) dx with x↦π/2-x.
• Method of differences: write uᵣ = f(r)-f(r + k); k terms survive at each end.
• Reduction formulae: integrate by parts to link Iₙ to Iₙ₋₁ (or Iₙ₋₂).
• Improper integrals are limits: ∫₁^∞ x^(-p) dx converges exactly when p > 1.
• Growth rates: x^k/(e^x) → 0 and (ln x)/(x^k) → 0 as x → ∞, for k > 0.
• Bijection principle: r identical objects into n boxes: ((r + n-1) choose (n-1)) ways.
• Inclusion–exclusion: |A∪ B∪ C| = ∑|A|-∑|A∩ B| + |A∩ B∩ C|.

Read more: The symmetry principle, Differences, reduction formulae and limits and Bijections and inclusion–exclusion.