Further pure mathematics

Proof by contradiction assumes the opposite of what you want to prove and shows it leads to something impossible: assuming the square root of 2 is rational leads to a fraction that cannot actually exist in lowest terms, so it must be irrational. Mathematical induction proves a base case, then shows that if a statement holds for some k it must hold for k+1, so it holds for every number from there on. Together these are core tools for proving general mathematical claims.

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