Q.E.D.

From Freepedia

For other meanings of the abbreviation "QED", see QED.

Q. E. D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "that which was to be demonstrated"). This is a translation of the Greek ὅπερ ἔδει δεῖξαι (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. Q.E.D. may be written at the end of mathematical proofs to show that the result required for the proof to be complete has been obtained. It is not seen as frequently now as in earlier centuries.

Modern-day usage

End-of-proof symbolism in the present day is often the symbol ■ (solid black square) called the tombstone or halmos (after Paul Halmos who pioneered its use). The tombstone is sometimes open; □ (hollow black square). Another simple way of stating that the proof is complete is to simply write "proven" or "shown" in parentheses after the final step of the proof, or to draw two forward slashes (//).

Unicode provides the "End of Proof" character U+220E (∎), but also provides U+25A0 (■, black square) and U+2023 (‣, triangular bullet) as alternatives.

Variations on the abbreviation

In English speaking countries the letters can also mean "Quite Easily Done" or, occasionally, "Quite Eloquently Done", or humourously "Quite Enough Done", "Quite Elegantly Done". Other humorous expansions in the context of mathematical proofs are "Question Every Detail" or "Question Every Deduction", suggesting that the reader should check that the proof is indeed correct as claimed.

In Asian countries, the letters are sometimes taken to mean (mistakenly and ungrammatically) "Question Easy Done."

There exists another Latin phrase, with a slight difference in meaning, but a similar, if less common usage. Quod erat faciendum translates into English 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 oper edei poihsai. Euclid used this phrase to close propositions which were not precisely "proofs", but rather constructions for example.

Incidentally, some people prefer to use the more tongue-in-cheek WWWWW or <math>W^5</math> which stands for the English "Which Was What Was Wanted."