fbpx
Wikipedia

Q.E.D.

May 18, 2023

For other uses, see QED.

Q.E.D. or QED is an initialism of the Latin phrase quod erat demonstrandum, meaning "which was to be demonstrated". Literally it states "what was to be shown".[1] Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments 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 from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). Translating from the Latin phrase into English yields "what was to be demonstrated". However, translating the Greek phrase ὅπερ ἔδει δεῖξαι can produce a slightly different meaning. In particular, since the verb "δείκνυμι" also means to show or to prove,[2] a different translation from the Greek phrase would read "The very thing it was required to have shown."[3]

The Greek phrase was used by many early Greek mathematicians, including Euclid[4] and Archimedes.

The Latin phrase is attested in a 1501 Euclid translation of Giorgio Valla.[5] Its abbreviation q.e.d. is used once in 1598 by Johannes Praetorius,[6] more in 1643 by Anton Deusing,[7] extensively in 1655 by Isaac Barrow in the form Q.E.D.,[8] and subsequently by many post-Renaissance mathematicians and philosophers.[9]

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.[10]

During 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 on the right hand 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.[11] Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book are, 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.[12]

Difference from Q.E.F.

There is another Latin phrase with a slightly different meaning, usually shortened similarly, but being less common in use. Quod erat faciendum, originating from the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai), meaning "which had to be done".[13] Because of the difference in meaning, the two phrases should not be confused.

Euclid 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.[14] For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way.[15]

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

Main article: Tombstone (typography)

Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements of theorems, 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 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,[16] a practice which has become standard, although not universal. Halmos noted that he adopted this use of a symbol from magazine typography customs in which simple geometric shapes had been used to indicate the end of an article, so-called end marks.[17][18] 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 as the character   (filled square, \blacksquare) and sometimes, as a   (hollow square, \square or \Box).[19] In the AMS Theorem Environment for LaTeX, the hollow square is the default end-of-proof symbol. Unicode 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 (////,  ).[20] In other cases, authors have elected to segregate proofs typographically—by displaying them as indented blocks.[21]

Modern humorous use

In Joseph Heller's 1961 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 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."[22]

In the 1978 science-fiction radio comedy, and later in the television, novel, and film 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 – 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."[23]

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.[24]

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 delightedly squeals, "Oooh... you speak French!" [25]

See also

References

  1. ^ "Definition of QUOD ERAT DEMONSTRANDUM". www.merriam-webster.com. Retrieved 2017-09-03.
  2. ^ Entry δείκνυμι at LSJ.
  3. ^ Euclid's Elements translated from Greek by Thomas L. Heath. 2003 Green Lion Press pg. xxiv
  4. ^ Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005
  5. ^ Valla, Giorgio. "Georgii Vallae Placentini viri clariss. De expetendis, et fugiendis rebus opus. 1".
  6. ^ Praetorius, Johannes. "Ioannis Praetorii Ioachimici Problema, quod iubet ex Quatuor rectis lineis datis quadrilaterum fieri, quod sit in Circulo".
  7. ^ Deusing, Anton. "Antonii Deusingii Med. ac Philos. De Vero Systemate Mundi Dissertatio Mathematica : Quâ Copernici Systema Mundi reformatur: Sublatis interim infinitis penè orbibus, quibus in Systemate Ptolemaico humana mens distrahitur".
  8. ^ Barrow, Isaac. "Elementa geometrie : libri XV".
  9. ^ "Earliest Known Uses of some of the Words of Mathematics (Q)". jeff560.tripod.com. Retrieved 2019-11-04.
  10. ^ Philippe van Lansberge (1604). Triangulorum Geometriæ. Apud Zachariam Roman. pp. 1–5. quod-erat-demonstrandum 0-1700.
  11. ^ "Baruch Spinoza (1632–1677) – Modern Philosophy". opentextbc.ca. Retrieved 2019-11-04.
  12. ^ The Chief Works of Benedict De Spinoza, translated by R. H. M. Elwes, 1951. ISBN 0-486-20250-X.
  13. ^ Gauss, Carl Friedrich; Waterhouse, William C. (7 February 2018). Disquisitiones Arithmeticae. ISBN 9781493975600.
  14. ^ Weisstein, Eric W. "Q.E.F." mathworld.wolfram.com. Retrieved 2019-11-04.
  15. ^ "Euclid's Elements, Book I, Proposition 1". mathcs.clarku.edu. Retrieved 2019-11-04.
  16. ^ This (generally accepted) claim was made in Halmos's autobiography, I Want to Be a Mathematician. The first usage of the solid black rectangle as an end-of-proof symbol appears to be in Halmos's Measure Theory (1950). The intended meaning of the symbol is explicitly given in the preface.
  17. ^ Halmos, Paul R. (1985). I Want to Be a Mathematician: An Automathography. p. 403. ISBN 9781461210849.
  18. ^ Felici, James (2003). "The complete manual of typography : a guide to setting perfect type". Berkeley, CA : Peachpit Press.
  19. ^ See, for example, list of mathematical symbols for more.
  20. ^ Rudin, Walter (1987). Real and Complex Analysis. McGraw-Hill. ISBN 0-07-100276-6.
  21. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X.
  22. ^ Heller, Joseph (1971). Catch-22. ISBN 978-0-573-60685-4. Retrieved 15 July 2011.
  23. ^ Adams, Douglas (2005). The Hitchhiker's Guide to the Galaxy. The Hitchhiker's Guide to the Galaxy (Film tie-in ed.). Basingstoke and Oxford: Pan Macmillan. pp. 62–64. ISBN 0-330-43798-4.
  24. ^ Stephenson, Neal (1999). Cryptonomicon. New York, NY: Avon Books. ISBN 978-0-06-051280-4.
  25. ^ "Airhead – Thomas Dolby". play.google.com. Retrieved 2016-09-15.

External links

 
Look up quod erat demonstrandum or QED in Wiktionary, the free dictionary.
  • Earliest Known Uses of Some of the Words of Mathematics (Q)

May 18, 2023
other, uses, initialism, latin, phrase, quod, erat, demonstrandum, meaning, which, demonstrated, literally, states, what, shown, traditionally, abbreviation, placed, mathematical, proofs, philosophical, arguments, print, publications, indicate, that, proof, ar. For other uses see QED Q E D or QED is an initialism of the Latin phrase quod erat demonstrandum meaning which was to be demonstrated Literally it states what was to be shown 1 Traditionally the abbreviation is placed at the end of mathematical proofs and philosophical arguments in print publications to indicate that the proof or the argument is complete Contents 1 Etymology and early use 2 Modern philosophy 3 Difference from Q E F 4 English equivalent 5 Typographical forms used symbolically 6 Modern humorous use 7 See also 8 References 9 External linksEtymology and early use EditThe phrase quod erat demonstrandum is a translation into Latin from the Greek ὅper ἔdei deῖ3ai hoper edei deixai abbreviated as OED Translating from the Latin phrase into English yields what was to be demonstrated However translating the Greek phrase ὅper ἔdei deῖ3ai can produce a slightly different meaning In particular since the verb deiknymi also means to show or to prove 2 a different translation from the Greek phrase would read The very thing it was required to have shown 3 The Greek phrase was used by many early Greek mathematicians including Euclid 4 and Archimedes The Latin phrase is attested in a 1501 Euclid translation of Giorgio Valla 5 Its abbreviation q e d is used once in 1598 by Johannes Praetorius 6 more in 1643 by Anton Deusing 7 extensively in 1655 by Isaac Barrow in the form Q E D 8 and subsequently by many post Renaissance mathematicians and philosophers 9 Modern philosophy Edit Philippe van Lansberge s 1604 Triangulorum Geometriae used quod erat demonstrandum to conclude some proofs others ended with phrases such as sigillatim deinceps demonstrabitur magnitudo demonstranda est and other variants 10 During 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 on the right hand 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 11 Written in Latin it is considered by many to be Spinoza s magnum opus The style and system of the book are as Spinoza says demonstrated in geometrical order with axioms and definitions followed by propositions For Spinoza this is a considerable improvement over Rene Descartes s writing style in the Meditations which follows the form of a diary 12 Difference from Q E F EditThere is another Latin phrase with a slightly different meaning usually shortened similarly but being less common in use Quod erat faciendum originating from the Greek geometers closing ὅper ἔdei poiῆsai hoper edei poiesai meaning which had to be done 13 Because of the difference in meaning the two phrases should not be confused Euclid 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 14 For example Euclid s first proposition showing how to construct an equilateral triangle given one side is concluded this way 15 English equivalent EditThere 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 EditMain article Tombstone typography Due to the paramount importance of proofs in mathematics mathematicians since the time of Euclid have developed conventions to demarcate the beginning and end of proofs In printed English language texts the formal statements of theorems 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 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 16 a practice which has become standard although not universal Halmos noted that he adopted this use of a symbol from magazine typography customs in which simple geometric shapes had been used to indicate the end of an article so called end marks 17 18 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 as the character displaystyle blacksquare filled square blacksquare and sometimes as a displaystyle square hollow square square or Box 19 In the AMS Theorem Environment for LaTeX the hollow square is the default end of proof symbol Unicode 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 displaystyle or four forward slashes displaystyle 20 In other cases authors have elected to segregate proofs typographically by displaying them as indented blocks 21 Modern humorous use EditIn Joseph Heller s 1961 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 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 22 In the 1978 science fiction radio comedy and later in the television novel and film 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 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 23 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 24 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 delightedly squeals Oooh you speak French 25 See also EditList of Latin abbreviations A priori and a posteriori Bob s your uncle Ipso facto Q E A List of Latin phrases E ergoReferences Edit Definition of QUOD ERAT DEMONSTRANDUM www merriam webster com Retrieved 2017 09 03 Entry deiknymi at LSJ Euclid s Elements translated from Greek by Thomas L Heath 2003 Green Lion Press pg xxiv Elements 2 5 by Euclid ed J L Heiberg retrieved 16 July 2005 Valla Giorgio Georgii Vallae Placentini viri clariss De expetendis et fugiendis rebus opus 1 Praetorius Johannes Ioannis Praetorii Ioachimici Problema quod iubet ex Quatuor rectis lineis datis quadrilaterum fieri quod sit in Circulo Deusing Anton Antonii Deusingii Med ac Philos De Vero Systemate Mundi Dissertatio Mathematica Qua Copernici Systema Mundi reformatur Sublatis interim infinitis pene orbibus quibus in Systemate Ptolemaico humana mens distrahitur Barrow Isaac Elementa geometrie libri XV Earliest Known Uses of some of the Words of Mathematics Q jeff560 tripod com Retrieved 2019 11 04 Philippe van Lansberge 1604 Triangulorum Geometriae Apud Zachariam Roman pp 1 5 quod erat demonstrandum 0 1700 Baruch Spinoza 1632 1677 Modern Philosophy opentextbc ca Retrieved 2019 11 04 The Chief Works of Benedict De Spinoza translated by R H M Elwes 1951 ISBN 0 486 20250 X Gauss Carl Friedrich Waterhouse William C 7 February 2018 Disquisitiones Arithmeticae ISBN 9781493975600 Weisstein Eric W Q E F mathworld wolfram com Retrieved 2019 11 04 Euclid s Elements Book I Proposition 1 mathcs clarku edu Retrieved 2019 11 04 This generally accepted claim was made in Halmos s autobiography I Want to Be a Mathematician The first usage of the solid black rectangle as an end of proof symbol appears to be in Halmos s Measure Theory 1950 The intended meaning of the symbol is explicitly given in the preface Halmos Paul R 1985 I Want to Be a Mathematician An Automathography p 403 ISBN 9781461210849 Felici James 2003 The complete manual of typography a guide to setting perfect type Berkeley CA Peachpit Press See for example list of mathematical symbols for more Rudin Walter 1987 Real and Complex Analysis McGraw Hill ISBN 0 07 100276 6 Rudin Walter 1976 Principles of Mathematical Analysis New York McGraw Hill ISBN 0 07 054235 X Heller Joseph 1971 Catch 22 ISBN 978 0 573 60685 4 Retrieved 15 July 2011 Adams Douglas 2005 The Hitchhiker s Guide to the Galaxy The Hitchhiker s Guide to the Galaxy Film tie in ed Basingstoke and Oxford Pan Macmillan pp 62 64 ISBN 0 330 43798 4 Stephenson Neal 1999 Cryptonomicon New York NY Avon Books ISBN 978 0 06 051280 4 Airhead Thomas Dolby play google com Retrieved 2016 09 15 External links Edit Look up quod erat demonstrandum or QED in Wiktionary the free dictionary Earliest Known Uses of Some of the Words of Mathematics Q Retrieved from https en wikipedia org w index php title Q E D amp oldid 1150519075, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.