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Euclid was a mathematician associated with Alexandria around 300 BCE. His Euclid's Elements organized geometry and number theory into definitions, assumptions and chains of proofs, drawing together earlier results as well as material attributed to him. Very little reliable biographical information survives, so the familiar portraits are later imagined likenesses.^[ref] The lasting attraction of the Euclid's Elements is its deductive structure: a construction or mathematical theorem becomes reusable in subsequent proofs, rather than remaining an isolated calculation. Its Euclidean algorithm and proof that there are infinitely many prime numbers remain basic tools of number theory.

A mathematical theorem is a statement established by a mathematical proof from specified assumptions and previously justified results. Its hypotheses belong to the claim: proving a conclusion for one class of objects does not automatically prove it for a larger class.

For every nonnegative integer $n$, the sum of the first $n$ odd positive integers is $n^2$. A proof by mathematical induction starts with the empty sum at $n=0$; adding the next term gives $n^2+(2n+1)=(n+1{)}^2$. This argument establishes all cases from one implication. Checking many numerical cases can suggest the result, but does not replace the inductive step unless the full stated domain is finite and exhaustively covered.