Q.E.D.

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 ὅπερ ἔδει δεῖξαι (hoper edei deixai) 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, since formal geometry is less commonly taught as a separate subject.

Modern-day usage

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". 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 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.

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."

Popular usage

• 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 affect a solution inherent to the movie's story line with the phrase "quod erat demonstrandum".

• In Voltaire's Candide, an overly optimistic tutor named Pangloss chooses to believe that all is for the best, and upon doing so, he will end his evidence by stating " quod erat demonstrandum, QED".

• 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:

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 could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don't. QED."

GOD: "Oh dear; I hadn't thought of that."

Whereupon he promptly vanishes in a puff of logic.

• One of the proofs in the Gospel of the Flying Spaghetti Monster ends with "QED, bitches."

• QED is referenced in Terry Pratchett's "Monstrous Regiment" and the popular FoxTrot comics by Bill Amend.

• Q.E.D. is used throughout the mathematics-ridden "Cryptonomicon" by Neal Stephenson.

• Q.E.D. appeares in Anne McCaffrey's novel "All the Weyrs of Pern".

• Q.E.D appears several times in Stephen King's novel "The Tommyknockers".

• Q.E.D. served as the title of two shows on American television:
  
  • A 1951 quiz show that featured a panel solving short mystery stories.
    
  • A 1982 CBS adventure/mystery drama starring Sam Waterston as Quentin E. Deverill, an American professor working in early 1900s England.

• In the song "Airhead" on the album "Aliens Ate My Buick" by Thomas Dolby, the phrase "Quod Erat Demonstrandum" is used:
  

You ask me do I love you

Does the Pope live in the woods?

"Quod Erat Demonstrandum, baby..."

"Ooh! You speak French!"

• In the T.V. series Monk on the episode "Mr. Monk Goes Back to School", Adrian Monk (played by actor Tony Shalhoub) uses the phrase "quod erat demonstrandum" against the science teacher who has murdered two people. Several characters translate the phrase as "thus it is proven," which would actually be a literal translation of "ita demonstratur."

• Internet celebrity George Ouzounian, better known as Maddox, uses the expression QED in one of his articles, as well as in his recent book, The Alphabet of Manliness.

• In the HBO series "Deadwood," the Francis Wolcott character answers his employer George Hearst's question of whether Wolcott has murdered three prostitutes, with the phrase "Quod erat demonstrandum." Hearst replied with, "I have no latin. What the hell are you saying?" with Wolcott's answer following as, "The thing has been proved." Season two, episode 12.

External links

• Earliest Known Uses of Some of the Words of Mathematics (Q)