Q.E.D.: Difference between revisions
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=== Q.E.F. === |
=== Q.E.F. === |
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There is another Latin phrase with a slightly different meaning, |
There is another Latin phrase with a slightly different meaning, and less common in usage. {{lang|la|''Quod erat faciendum''}} is translated as "which was to be done." This is usually shortened to Q.E.F.. As with Q.E.D., Q.E.F. is a translation of the Greek geometers' closing {{Polytonic|ὅπερ ἔδει ποιῆσαι}} ({{lang|grc-Latn|''hoper edei poiēsai''}}). [[Euclid]] used this phrase to close propositions which were not precisely "proofs", but rather exemplar constructions. The distinction between Q.E.D. and Q.E.F. is roughly equivalent to the distinction between a proof and an illustration of the proof. |
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=== Foreign Languages === |
=== Foreign Languages === |
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Revision as of 18:21, 22 August 2007
Q.E.D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "which was to be demonstrated", and figuratively, "I rest my case"). The phrase is usually written in its abbreviated form at the end of mathematical proofs, to signify the completion of a previously asserted claim.
Etymology and Early Use
The phrase is a translation into Latin of the original Greek ὅπερ ἔδει δει̂ξαι. ([hoper edei deixai] Error: {{Lang}}: text has italic markup (help)) which was used by many early mathematicians including Euclid[1] and Archimedes. These mathematicians, in particular Euclid, are credited with founding axiomatic mathematics with its emphasis on establishing truths by logical deduction (rather than experimentation or assertion); their use of this phrase symbolizes this emphasis, as well as marking this important step in the development of mathematical philosophy.
Modern Use
In modern times, it has become so symbolic of irrefutable logic that "Q.E.D." is occasionally used in non-mathematical contexts as well to intensify assertions; in this context it has little connection with rigorous deduction, however, and is more tongue-in-cheek.
Alternatives
Q.E.D. is the classical way to finish a proof and while it is still used, there are other ways of expressing the same thing, a number of which are very recent and have arisen through the use of computers.
Q.E.F.
There is another Latin phrase with a slightly different meaning, and less common in usage. [Quod erat faciendum] Error: {{Lang}}: text has italic markup (help) is translated as "which was to be done." This is usually shortened to Q.E.F.. As with Q.E.D., Q.E.F. is a translation of the Greek geometers' closing ὅπερ ἔδει ποιῆσαι ([hoper edei poiēsai] Error: {{Lang}}: text has italic markup (help)). Euclid used this phrase to close propositions which were not precisely "proofs", but rather exemplar constructions. The distinction between Q.E.D. and Q.E.F. is roughly equivalent to the distinction between a proof and an illustration of the proof.
Foreign Languages
As mathematics has come to be written in languages other than Latin, Q.E.D. has acquired many translations; in French and German (two languages a mathematician might see often) it is respectively C.Q.F.D, for "ce qu'il fallait démontrer", and W.Z.B.W, for "was zu beweisen war". There does not appear to be a common English equivalent, though the end of a proof may be announced with a simple statement "this completes the proof" or similar locution.
Electronic Forms
With computers frequently being used to "write" proofs (see LaTeX), there are several symbolic alternatives in use. The most popular symbol is ■ (solid black square), also called tombstone or Halmos symbol (after Paul Halmos, who pioneered its use). The tombstone is sometimes open: □ (hollow black square). Unicode explicitly provides the "End of Proof" character U+220E (∎), but also offers ■ (U+25A0, black square) 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.
Notes
- ^ Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005