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Quod Erat Demonstrandum
Q.E.D. or QED is an initialism of the Latin phrase , 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 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 (; 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 and Archimedes. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initialism
An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in ''NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, as in ''Benelux'' (short for ''Belgium, the Netherlands, and Luxembourg''). They can also be a mixture, as in ''radar'' (''Radio Detection And Ranging''). Acronyms can be pronounced as words, like ''NASA'' and ''UNESCO''; as individual letters, like ''FBI'', ''TNT'', and ''ATM''; or as both letters and words, like '' JPEG'' (pronounced ') and ''IUPAC''. Some are not universally pronounced one way or the other and it depends on the speaker's preference or the context in which it is being used, such as '' SQL'' (either "sequel" or "ess-cue-el"). The broader sense of ''acronym''—the meaning of which includes terms pronounced as letters—is sometimes criticized, but it is the term's original meaning and is in common use. Dictionary and st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baruch Spinoza
Baruch (de) Spinoza (born Bento de Espinosa; later as an author and a correspondent ''Benedictus de Spinoza'', anglicized to ''Benedict de Spinoza''; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Portuguese-Jewish origin, born in Amsterdam. One of the foremost exponents of 17th-century Rationalism and one of the early and seminal thinkers of the Enlightenment and modern biblical criticism including modern conceptions of the self and the universe, he came to be considered "one of the most important philosophers—and certainly the most radical—of the early modern period." Inspired by Stoicism, Jewish Rationalism, Machiavelli, Hobbes, Descartes, and a variety of heterodox religious thinkers of his day, Spinoza became a leading philosophical figure during the Dutch Golden Age. Spinoza's given name, which means "Blessed", varies among different languages. In Hebrew, his full name is written . In most of the documents and records contemporary with Spinoza's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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), and letter-spacing (tracking), as well as adjusting the space between pairs of letters (kerning). The term ''typography'' is also applied to the style, arrangement, and appearance of the letters, numbers, and symbols created by the process. Type design is a closely related craft, sometimes considered part of typography; most typographers do not design typefaces, and some type designers do not consider themselves typographers. Typography also may be used as an ornamental and decorative device, unrelated to the communication of information. Typography is the work of typesetters (also known as compositors), typographers, graphic designers, art directors, manga artists, comic book artists, and, now, anyone who arranges words, letters, numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians. Early life and education Born in Hungary into a Jewish family, Halmos arrived in the U.S. at 13 years of age. He obtained his B.A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, and was only 19 when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus; but, after failing his masters' oral exams, he shifted to mathematics, graduating in 1938. Joseph L. Doob supervised his dissertation, titled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek ("anything which is received", such as a gift, profit, or a bribe). Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 diary may include a person's experiences, thoughts, and/or feelings, excluding comments on current events outside the writer's direct experience. Someone who keeps a diary is known as a diarist. Diaries undertaken for institutional purposes play a role in many aspects of human civilization, including government records (e.g. ''Hansard''), business ledgers, and military records. In British English, the word may also denote a preprinted journal format. Today the term is generally employed for personal diaries, normally intended to remain private or to have a limited circulation amongst friends or relatives. The word "journal" may be sometimes used for "diary," but generally a diary has (or intends to have) daily entries (from the Latin wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meditations On First Philosophy
''Meditations on First Philosophy, in which the existence of God and the immortality of the soul are demonstrated'' ( la, Meditationes de Prima Philosophia, in qua Dei existentia et animæ immortalitas demonstratur) is a philosophical treatise by René Descartes first published in Latin in 1641. The French translation (by the Duke of Luynes with Descartes' supervision) was published in 1647 as ''Méditations Métaphysiques''. The title may contain a misreading by the printer, mistaking ''animae immortalitas'' for ''animae immaterialitas'', as suspected by A. Baillet. The book is made up of six meditations, in which Descartes first discards all belief in things that are not absolutely certain, and then tries to establish what can be known for sure. He wrote the meditations as if he had meditated for six days: each meditation refers to the last one as "yesterday". (In fact, Descartes began work on the ''Meditations'' in 1639.) One of the most influential philosophical texts ever w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Mathematics was central to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. Descartes spent much of his working life in the Dutch Republic, initially serving the Dutch States Army, later becoming a central intellectual of the Dutch Golden Age. Although he served a Protestant state and was later counted as a deist by critics, Descartes considered himself a devout Catholic. Many elements of Descartes' philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differed from the schools on two major points: first, he rejected the splitting of corporeal substance into mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proposition (mathematics)
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 the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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