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Q.E.D. or QED is an initialism of the
Latin phrase __NOTOC__ This is a list of Wikipedia articles of Latin phrases and their translation into English. ''To view all phrases on a single, lengthy document, see: List of Latin phrases (full)'' The list also is divided alphabetically into twenty page ...
, meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and
philosophical Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
arguments An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
in print publications, to indicate that the proof or the argument is complete.


Etymology and early use

The phrase ''quod erat demonstrandum'' is a translation into
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
(; abbreviated as ''ΟΕΔ''). Translating from the Latin phrase into English yields "what was to be demonstrated". However, translating the Greek phrase can produce a slightly different meaning. In particular, since the verb also means ''to show'' or ''to prove'', a different translation from the Greek phrase would read "The very thing it was required to have shown."Euclid's Elements translated from Greek by Thomas L. Heath. 2003 Green Lion Press pg. xxiv The Greek phrase was used by many early Greek mathematicians, including
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
and Archimedes. The Latin phrase is attested in a 1501 Euclid translation of
Giorgio Valla Giorgio Valla (Latin: ''Georgius Valla''; Piacenza 1447–Venice 1500) was an Italian academic, mathematician, philologist and translator. Life He was born in Piacenza in 1447. He was the son of Andrea Valla and Cornelia Corvini. At the age of ...
. Its abbreviation ''q.e.d.'' is used once in 1598 by Johannes Praetorius, more in 1643 by Anton Deusing, extensively in 1655 by Isaac Barrow in the form ''Q.E.D.'', and subsequently by many post-
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history The history of Europe is traditionally divided into four time periods: prehistoric Europe (prior to about 800 BC), classical antiquity (800 BC to AD ...
mathematicians and philosophers.


Modern philosophy

During the European
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history The history of Europe is traditionally divided into four time periods: prehistoric Europe (prior to about 800 BC), classical antiquity (800 BC to AD ...
, 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 Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concer ...
'' of Baruch Spinoza, published posthumously in 1677. Written in Latin, it is considered by many to be Spinoza's ''
magnum opus A masterpiece, ''magnum opus'' (), or ''chef-d’œuvre'' (; ; ) in modern use is a creation that has been given much critical praise, especially one that is considered the greatest work of a person's career or a work of outstanding creativity, ...
''. The style and system of the book are, as Spinoza says, "demonstrated in
geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
order", with axioms and definitions followed by
propositions In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the n ...
. For Spinoza, this is a considerable improvement over
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
's writing style in the ''
Meditations ''Meditations'' () is a series of personal writings by Marcus Aurelius, Roman Emperor from AD 161 to 180, recording his private notes to himself and ideas on Stoic philosophy. Marcus Aurelius wrote the 12 books of the ''Meditations'' in Koine ...
'', which follows the form of a
diary A diary is a written or audiovisual record with discrete entries arranged by date reporting on what has happened over the course of a day or other period. Diaries have traditionally been handwritten but are now also often digital. A personal ...
.


Difference from Q.E.F.

There is another Latin phrase with a slightly different meaning, usually shortened similarly, but being less common in use. , originating from the Greek geometers' closing (), meaning "which had to be done". Because of the difference in meaning, the two phrases should not be confused.
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
used the Greek original of Quod Erat Faciendum (Q.E.F.) to close propositions that were not proofs of theorems, but constructions of geometric objects. For example, Euclid's first proposition showing how to construct an
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
, given one side, is concluded this way.


English equivalent

There is no common formal English equivalent, although the end of a proof may be announced with a simple statement such as "this completes the proof", "as required", "as desired", "as expected", "hence proved", "ergo", "so correct", or other similar locutions.


Typographical forms used symbolically

Due to the paramount importance of proofs in mathematics, mathematicians since the time of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements of
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
s,
lemmas Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), ...
, and propositions are set in italics by tradition. 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., it is relatively uncommon in modern mathematical texts.
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
claims to have pioneered the use of a solid black square (or rectangle) at the end of a proof as a Q.E.D. symbol, a practice which has become standard, although not universal. Halmos noted that he adopted this use of a symbol from magazine
typography Typography is the art and technique of arranging type to make written language legible, readable and appealing when displayed. The arrangement of type involves selecting typefaces, point sizes, line lengths, line-spacing ( leading), ...
customs in which simple geometric shapes had been used to indicate the end of an article, so-called '' end marks''. This symbol was later called the '' tombstone'', the ''Halmos symbol'', or even a ''halmos'' by mathematicians. Often the Halmos symbol is drawn on chalkboard to signal the end of a proof during a lecture, although this practice is not so common as its use in printed text. The tombstone symbol appears in
TeX Tex may refer to: People and fictional characters * Tex (nickname), a list of people and fictional characters with the nickname * Joe Tex (1933–1982), stage name of American soul singer Joseph Arrington Jr. Entertainment * ''Tex'', the Italian ...
as the character \blacksquare (filled square, \blacksquare) and sometimes, as a \square (hollow square, \square or \Box). In the AMS Theorem Environment for
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
, the hollow square is the default end-of-proof symbol.
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, wh ...
explicitly provides the "end of proof" character, U+220E (∎). Some authors use other Unicode symbols to note the end of a proof, including, ▮ (U+25AE, a black vertical rectangle), and ‣ (U+2023, a triangular bullet). Other authors have adopted two forward slashes (//) or four forward slashes (////). In other cases, authors have elected to segregate proofs typographically—by displaying them as indented blocks.


Modern humorous use

In
Joseph Heller Joseph Heller (May 1, 1923 – December 12, 1999) was an American author of novels, short stories, plays, and screenplays. His best-known work is the 1961 novel ''Catch-22'', a satire on war and bureaucracy, whose title has become a synonym for ...
's 1961 book ''
Catch-22 ''Catch-22'' is a satirical war novel by American author Joseph Heller. He began writing it in 1953; the novel was first published in 1961. Often cited as one of the most significant novels of the twentieth century, it uses a distinctive non-ch ...
'', 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 did not write it and that it was not his handwriting, to which the investigator replied, "Then you signed your name in somebody else's handwriting again." In the 1978 science-fiction radio comedy, and later in the television, novel, and film adaptations of ''
The Hitchhiker's Guide to the Galaxy ''The Hitchhiker's Guide to the Galaxy'' (sometimes referred to as ''HG2G'', ''HHGTTG'', ''H2G2'', or ''tHGttG'') is a comedy science fiction franchise created by Douglas Adams. Originally a 1978 radio comedy broadcast on BBC Radio 4, it ...
'', "Q.E.D." is referred to in the Guide's entry for the babel fish, when it is claimed that the babel fish – which serves the "mind-bogglingly" useful purpose of being able to translate any spoken language when inserted into a person's ear – is used as evidence for existence and non-existence of God. The exchange from the novel is as follows: "'I refuse to prove I exist,' says God, 'for proof denies faith, and without faith I am nothing.' 'But,' says Man, '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.' 'Oh dear,' says God, 'I hadn't thought of that,' and promptly vanishes in a puff of logic." In Neal Stephenson's 1999 novel ''
Cryptonomicon ''Cryptonomicon'' is a 1999 novel by American author Neal Stephenson, set in two different time periods. One group of characters are World War II-era Allied codebreakers and tactical-deception operatives affiliated with the Government Code an ...
'', 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 Thomas Morgan Robertson (born 14 October 1958), known by the stage name Thomas Dolby, is an English musician, producer, composer, entrepreneur and teacher. Dolby came to prominence in the 1980s, releasing hit singles including "She Blinded Me ...
'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 delightedly squeals, "Oooh... you speak French!"


See also

*
List of Latin abbreviations This is a list of common Latin abbreviations. Nearly all the abbreviations below have been adopted by Modern English. However, with some exceptions (for example, ''versus'' or '' modus operandi''), most of the Latin referent words and phrases a ...
*
A priori and a posteriori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
* Bob's your uncle * ''
Ipso facto is a Latin phrase, directly translated as "by the fact itself", which means that a specific phenomenon is a ''direct'' consequence, a resultant ''effect'', of the action in question, instead of being brought about by a previous action. It is a ...
'' * Q.E.A. *


References


External links

{{Wiktionary, quod erat demonstrandum, QED
Earliest Known Uses of Some of the Words of Mathematics (Q)
Latin logical phrases Latin philosophical phrases Mathematical proofs Mathematical terminology