- For other meanings of the abbreviation "QED", see QED (disambiguation).
Q.E.D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "which was to be demonstrated"). This is a translation of the Greek Template:Polytonic (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. Q.E.D. 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 in earlier centuries.
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.
Variations on the abbreviation
In English speaking countries the letters have been humorously interpreted as "Quite Easily Done" or, occasionally, "Quite Eloquently Done", or "Quite Enough Done", "Quite Elegantly 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."
There exists another Latin phrase, with a slight difference in meaning, but a similar, if less common usage. Quod erat faciendum translates into English 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 oper edei poihsai. Euclid used this phrase to close propositions which were not precisely "proofs", but rather constructions for example.
Incidentally, 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."
In the 1987 American movie "No Way Out," the character Scott Pritchard (played by actor Will Patton) concludes an explanation of his character's power and ability to effect a solution inherent to the movie's story line with the phrase "quod erat demonstrandum".
In The Hitchhiker's Guide to the Galaxy, Douglas Adams hypothesises a dialogue between Man and God concerning whether a creature, the babel fish, which allows anyone who places it in their ear to understand any language, is too useful to have evolved purely by chance and therefore must have been divinely created (see intelligent design).
GOD: "I refuse to prove that I exist, for proof denies faith and without faith I am nothing."
MAN: "But the babel fish is a dead giveaway, isn't it? It proves you exist and so therefore You don't. Q.E.D."
GOD: "Oh dear; I hadn't thought of that."
Whereupon he promptly vanishes in a puff of logic.