Q.E.D.

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For other uses of "QED", see QED.

Q.E.D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "which was to be demonstrated"). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven.

QED, may be written at the end of mathematical proofs to show that the result required for the proof to be complete has been obtained. It is not seen as frequently now as it once was, since formal geometry is less commonly taught as a separate subject.

The term is also used both formally and informally in a wide variety of disciplines, as well as in everyday conversation in many parts of the English-speaking world.

1 Historical usage
2 Modern-day usage and symbolism
3 Variations on the abbreviation
4 External links

[edit] Historical usage

Q.E.D. is a translation into Latin of the original Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai) which was used by many early mathematicians including Euclid and Archimedes. Benedict De Spinoza also makes extensive use of the abbreviation Q.E.D. in his various works.

[edit] Modern-day usage and symbolism

End-of-proof symbolism in the present day is often the symbol ■ (solid black square) called the tombstone, or the Halmos symbol (after Paul Halmos, who pioneered its use). The tombstone is sometimes open; □ (hollow black square). Another simple way of stating that the proof is complete is to simply write "proven" or "shown" in parentheses after the final step of the proof, or to draw two forward slashes (//).

Unicode provides the "End of Proof" character U+220E (∎), but also provides U+25A0 (■, black square) and U+2023 (‣, triangular bullet) as alternatives.

[edit] Variations on the abbreviation

In English speaking countries the letters have been humorously interpreted as "Quite Easily Done" or "Quickly and Easily Done" or, occasionally, "Quite Eloquently Done", or "Quite Enough Done". Other humorous expansions in the context of mathematical proofs are "Question Every Detail" or "Question Every Deduction", suggesting that the reader should check that the proof is indeed correct as claimed, or "Qualitatively Extracted Deduction" or "Quit and Eat Dinner." It has also been rendered as quod ego dico, "because I say so".

There is another Latin phrase, with a slightly different meaning, but a similar, if less common usage. Quod erat faciendum is translated as "which was to be done." This is usually shortened to Q.E.F.. As with Q.E.D., Q.E.F. is a translation of the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai). Euclid used this phrase to close propositions which were not precisely "proofs", but rather examplar constructions. The distinction between Q.E.D. and Q.E.F. is roughly equivalent to the distinction between a proof and an illustration of the proof.

Alternatively, some people prefer to use the more tongue-in-cheek WWWWW or W⁵ which stands for the English "Which Was What Was Wanted." or "Which Was What We Wanted."