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Q.E.D. or QED is an
initialism An acronym is a type of abbreviation consisting of a phrase whose only pronounced elements are the initial letters or initial sounds of words inside that phrase. Acronyms are often spelled with the initial letter of each word in all caps wi ...
of the
Latin phrase 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). Lists of pages * List of Latin phrases (A) * List of Latin phrases ( ...
, meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of
mathematical proofs A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof c ...
and
philosophical Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
arguments An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persua ...
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 languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
from the
Greek Greek may refer to: 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 of all kno ...
(; abbreviated as ''ΟΕΔ''). The meaning of the Latin phrase is "that hingwhich was to be demonstrated" (with ''demonstrandum'' in the
gerundive In Latin grammar, a gerundive () is a verb form that functions as a verbal adjective. In Classical Latin, the gerundive has the same form as the gerund, but is distinct from the present active participle. In Late Latin, the differences were lar ...
). The Greek phrase was used by many early Greek mathematicians, including
Euclid Euclid (; ; 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 geometry that largely domina ...
and
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
. The Latin phrase is attested in a 1501 Euclid translation of
Giorgio Valla Giorgio Valla (Latin: ''Georgius Valla''; Piacenza 1447–Venice January 23, 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 ...
. 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 Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem ...
in the form ''Q.E.D.'', and subsequently by many post-
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
mathematicians and philosophers.


Modern philosophy

During 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 Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...
'' of
Baruch Spinoza Baruch (de) Spinoza (24 November 163221 February 1677), also known under his Latinized pen name Benedictus de Spinoza, was a philosopher of Portuguese-Jewish origin, who was born in the Dutch Republic. A forerunner of the Age of Enlightenmen ...
, published posthumously in 1677. Written in Latin, it is considered by many to be Spinoza's ''
magnum opus A masterpiece, , or ; ; ) 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, skill, profundity, or workmanship. Historically, ...
''. The style and system of the book are, as Spinoza says, "demonstrated in geometrical order", with
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s and definitions followed by
propositions A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
. For Spinoza, this is a considerable improvement over
René Descartes René Descartes ( , ; ; 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 Modern science, science. Mathematics was paramou ...
's writing style in the ''
Meditations ''Meditations'' () is a series of personal writings by Marcus Aurelius, Roman Emperor from 161–180 AD, recording his private notes to himself and ideas on Stoic philosophy. Composition Marcus Aurelius wrote the 12 books of the ''Meditations'' i ...
'', which follows the form of a
diary A diary is a written or audiovisual memorable 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 digita ...
.


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 (; ; 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 geometry that largely domina ...
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 An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, given one side, is concluded this way.


Equivalent forms

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


Typographical forms used symbolically

Due to the paramount importance of proofs in mathematics, mathematicians since the time of
Euclid Euclid (; ; 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 geometry that largely domina ...
have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements of
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s, lemmas, 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 (; 3 March 1916 – 2 October 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and probabilist who made fundamental advances in the areas of mathematical logic, probability theory, operat ...
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 Typesetting, arranging type to make written language legibility, legible, readability, readable and beauty, appealing when displayed. The arrangement of type involves selecting typefaces, Point (typogra ...
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 A gravestone or tombstone is a marker, usually stone, that is placed over a grave. A marker set at the head of the grave may be called a headstone. An especially old or elaborate stone slab may be called a funeral stele, stela, or slab. The us ...
'', 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, TeX, TEX, may refer to: People and fictional characters * Tex (nickname), a list of people and fictional characters with the nickname * Tex Earnhardt (1930–2020), U.S. businessman * Joe Tex (1933–1982), stage name of American soul singer ...
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. Latices are found in nature, but synthetic latices are common as well. In nature, latex is found as a wikt:milky, milky fluid, which is present in 10% of all floweri ...
, the hollow square is the default end-of-proof symbol.
Unicode Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...
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's 1961 novel '' 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 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."


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 experience. knowledge is independent from any experience. Examples include ...
* 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. (Contras ...
'' * 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