Contradiction assumes the exact negation of the target and derives an impossibility — not a surprise, a genuine clash with a premise or known fact. The template is √2: assume it equals a/b in lowest terms, force both a and b even, and contradict lowest terms. √3 and ∛2 run the same shape with a divisibility lemma pushed through numerator and denominator; log₂5 collapses differently, with 2^a even against 5^b odd.
Negating correctly is half the battle: 'not (If P then Q)' is 'P and not Q', so the proof that even squares have even roots opens with n² even and n odd — squaring 2k + 1 gives an odd square, and there is the clash. Existence denials ('no smallest positive rational', 'no largest integer', 'no largest rational below 2') negate to an extremal object — smallest r, largest N — which you then beat: r/2 is smaller, N + 1 larger, (r + 2)/2 sits strictly inside. Euclid's infinite primes assumes a complete finite list, forms N = p₁…pₖ + 1, and takes a prime divisor N must have outside the list.
Small cases motivate but never substitute: n² + n + 41 looks prime for n = 0, 1, 2, … until n = 41 gives 41 × 43 — testing builds the conjecture, contradiction-style argument seals it. Traps: assuming the conclusion itself instead of its negation; deriving something merely unexpected; a 'contradiction' that contradicts nothing cited. Note lowest terms costs nothing — every rational has one — and that N in Euclid's argument need not itself be prime; only its prime divisor matters. When reviewing misses, rewrite the correct opening assumption from memory: if you cannot state the exact negation you assumed, you never really started the proof.
Study method: attempt five timed questions, log every miss with its topic code, redo misses from scratch within 48 hours, then re-attempt the topic a week later. Pair each hub with its drill link below and a fortnightly full mock.
Place this topic in the wider test with the syllabus checklist and Paper 1 vs Paper 2, learn the timing system in how to prepare, and check what scores mean in scores explained.