Q.E.D.

This article is about the Latin phrase. For the physical theory, see quantum electrodynamics. For other uses, see QED (disambiguation).

Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, meaning "which is what had to be proven". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation—and in the setting-out—has been exactly restated as the conclusion of the demonstration. The abbreviation thus signals the completion of the proof.

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

Philippe van Lansberge's 1604 Triangulorum Geometriæ used quod erat demonstrandum to conclude some proofs; others ended with phrases such as sigillatim deinceps demonstrabitur, magnitudo demonstranda est, and other variants.

In the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs.

Spinoza's original text of Ethics, Part 1. Q.E.D. is used at the end of DEMONSTRATIO of PROPOSITIO III. in the right page.

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 usually shortened to QEF.

Equivalents in other languages

Q.E.D. has acquired many translations in various languages, including:

Language	Abbreviations	Stands for...
Arabic	هـ.ط.ث	وهو المطلوب إثباته
Armenian	Ի.Պ.Ա. (rarely abbreviated)	ինչը և պահանջվում էր ապացուցել
Bengali	অ. সি.	অতঃ সিদ্ধ
Bosnian	Š.T.D.	što je trebalo dokazati
Bulgarian	КТДД	Което трябваше да докажем Което трябваше да се докаже
Chinese	证毕 證畢 证讫 證訖	证明完毕 證明完畢 证讫 證訖
Croatian	Š.T.D.	što je trebalo dokazati
Czech	C.B.D.	což bylo dokázati
Dutch	w.m.b.w. w.t.b.w	wat moest bewezen worden wat te bewijzen was
Estonian	M.O.T.T.	mida oligi tarvis tõestada
Finnish	M.O.T.	mikä oli todistettava
French	C.Q.F.D.	ce qu'il fallait démontrer
Catalan	C.V.D.	com volíem demostrar
Georgian	რ.დ.გ	რისი დამტკიცებაც გვსურდა
German	w.z.b.w.	was zu beweisen war
Greek	Ο.Ε.Δ.	όπερ έδει δείξαι
Hebrew	.מ.ש.ל	מה שהיה להוכיח
Hungarian	E.K.B. (rarely abbreviated)	Ezt kellett bizonyítani
Icelandic	Þ.S.S.Á.	Það sem sanna átti
Italian	C.V.D.	come volevasi dimostrare
Latvian	k.b.j.	kas bija jāpierāda
Polish	c.b.d.u.	co było do udowodnienia
Portuguese	C.Q.D.	como queríamos demonstrar
Romanian	c.c.t.d.	ceea ce trebuia demonstrat
Russian	ч.т.д.	что и требовалось доказать
Sanskrit	इ.सि.	इति सिद्धम्
Serbian	ш.т.д.	што је требало доказати
Slovak	č.b.t.d.	čo bolo treba dokázať
Slovenian	k.e.d.	konec enega dokaza
Spanish	C.Q.D. Q.E.D.	como queda demostrado queda entonces demostrado
Swedish	V.S.B. V.S.V	vilket skulle bevisas vilket skulle visas
Tamil	நி.வே.	நிரூபிக்கப்படவேண்டியது
Thai	ซ.ต.พ.	ซึ่งต้องพิสูจน์
Turkish	G.İ.B.	Gösterilmek istenen şey de buydu
Ukrainian	щ.с.б.д. щ.т.д.	що й слід було довести що і треба було довести
Vietnamese	đpcm.	Điều phải chứng minh

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.

Electronic forms

Main article: Tombstone (typography)

When typesetting was done by a compositor with letterpress printing, complex typography such as mathematics and foreign languages were called "penalty copy" (the author paid a "penalty" to have them typeset, as it was harder than plain text). With the advent of systems such as LaTeX, mathematicians found their options more open, so there are several symbolic alternatives in use, either in the input, the output, or both. When creating TeX, Knuth provided the symbol ■ (solid black square), also called by mathematicians tombstone or Halmos symbol (after Paul Halmos, who pioneered its use as an equivalent of Q.E.D.). The tombstone is sometimes open: □ (hollow black square). Unicode explicitly provides the "End of proof" character U+220E (∎), but also offers ▮ (U+25AE, black vertical rectangle) and ‣ (U+2023, triangular bullet) as alternatives. Some authors have adopted variants of this notation with other symbols, such as two forward slashes (//), or simply some vertical white space, implying no further statements need to be made in the proof.

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 mid eighties, 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.

See also

• Bob's your uncle
• A priori and a posteriori
• List of Latin abbreviations

External links

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