The word "proof" comes from the Latin probare (to test). Related modern words are English "probe", "probation", and "probability", Spanish probar (to smell or taste, or sometimes touch or test), Italian provare (to try), and

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As practiced, a proof is expressed in natural language and is a rigorous argumentintended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently

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Direct proof
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that the sum of two even integers is always even:

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A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor

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Visual proof
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The left-hand picture below is an example of a historic visual proof of the Pythagorean theoremin

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Sometimes, the abbreviation "Q.E.D." is written to indicate the end of a proof. This abbreviation stands for "quod erat demonstrandum", which is

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While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematicsdevelopments of the late

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