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Q.E.D. (also written QED) is an initialism of the Latin phrase quod erat demonstrandum, meaning "what was to be demonstrated", or, less formally, "thus it has been demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when the original proposition has been exactly restated as the conclusion of the demonstration. The abbreviation thus signals the completion of the proof.
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Etymology and early use
The phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). Translating from the Latin into English yields, "what was to be demonstrated"; however, translating the Greek phrase ὅπερ ἔδει δεῖξαι produces a slightly different meaning. Since the verb "δείκνυμι" also means to show or to prove, a better translation from the Greek would read, "The very thing it was required to have shown." The phrase was used by many early Greek mathematicians, including Euclid and Archimedes.
Modern philosophy
In the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs.
Perhaps the most famous use of Q.E.D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677. Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book is, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary.
Q.E.F.
There is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum, originating from the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai), meaning "which had to be done". Euclid used this phrase to close propositions which were not proofs of theorems, but constructions. For example, Euclid's first proposition showing how to construct an equilateral triangle given one side is concluded this way. The phrase is usually shortened to QEF.
Equivalents in other languages
Q.E.D. has acquired many translations in various languages, including:
There is no common formal English equivalent, though the end of a proof may be announced with a simple statement such as "this completes the proof", "as required", "hence proved", "ergo", or a similar locution. WWWWW or W5 - an abbreviation of "Which Was What Was Wanted" - has also been used. This is often considered to be more tongue-in-cheek than the usual Halmos symbol (see below) or Q.E.D.
Symbolic forms in typography
Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and/or end of proofs. In English language texts, the formal statements of theorems, lemmas, and propositions are typically set in italics. The beginning of a proof usually follows immediately thereafter and is indicated by the word "Proof" in boldface or italics. On the other hand, several symbolic conventions exist to indicate the end of a proof. While some authors still use the classical abbreviation Q.E.D., this practice is increasingly viewed as archaic or even pretentious. Paul Halmos pioneered the use of a black square at the end of a proof as a Q.E.D sign, a practice which has become standard though not universal. Halmos claimed to have adopted this symbol from magazine typography in which simple geometric shapes had been used to indicate the end of an article. This symbol was later called the tombstone or Halmos symbol or even a halmos by mathematicians. The Halmos symbol is also often drawn on chalkboard to signal the end of a proof during a lecture, although this practice is not as common as its use in printed text.
The tombstone symbol appears in TeX as the character
Modern humorous usage
In Joseph Heller's book Catch-22, the Chaplain, having been told to examine a forged letter allegedly signed by him (which he knew he didn't sign), verified that his name was in fact there. His investigator replied, "Then you wrote it. Q.E.D." The chaplain said he didn't write it and that it wasn't his handwriting, and the investigator replied, "Then you signed your name in somebody else's handwriting again."
In the 1980s, BBC ran a series called Q.E.D. which showed how certain things were made or put together.
In the 1978 sci-fi radio comedy, and later in the TV and novel adaptations of The Hitchhiker's Guide to the Galaxy, "Q.E.D." is referred to in the Guide's entry for the babel fish, when it is claimed that the babel fish is used as evidence for existence and non-existence of God.
In Neal Stephenson's 1999 novel Cryptonomicon, Q.E.D. is used as a punchline to several humorous anecdotes in which characters go to great lengths to prove something non-mathematical.
Singer-songwriter Thomas Dolby's 1988 song "Airhead" includes the lyric, "Quod erat demonstrandum, baby," referring to the self-evident vacuousness of the eponymous subject; and in response, a female voice squeals, delightedly, "Oooh... you speak French!"