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Pronic Number
A pronic number is a number which is the product of two consecutive integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...s, that is, a number of the form .. The study of these numbers dates back to Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest .... They are also called oblong numbers, heteromecic numbers,. or rectangular numbers; however, the term "rectangular number" has also been applied to the composite number A composite number is a positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quant ...
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Pronic Number Cuisenaire Rods 12
A pronic number is a number which is the product of two consecutive integers, that is, a number of the form .. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers. The first few pronic numbers are: :0 (number), 0, 2 (number), 2, 6 (number), 6, 12 (number), 12, 20 (number), 20, 30 (number), 30, 42 (number), 42, 56 (number), 56, 72 (number), 72, 90 (number), 90, 110 (number), 110, 132 (number), 132, 156, 182, 210, 240, 272, 306, 342, 380, 420 (number), 420, 462 … . If is a pronic number, then the following is true: : \lfloor\rfloor \cdot \lceil\rceil = n As figurate numbers The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's ''Metaphysics (Aristotle), Metaphysics'', and their discovery has been attributed much earlier to the Pythagoreans.. ...
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Pronic Number Cuisenaire Rods 12 Square
A pronic number is a number which is the product of two consecutive integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...s, that is, a number of the form .. The study of these numbers dates back to Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio .... They are also called oblong numbers, heteromecic numbers,. or rectangular numbers; however, the term "rectangular number" has also been applied to the composite number A composite number is a positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quant ...
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Tetrahedral Number
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid (geometry), pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, : Te_n = \sum_^n T_k = \sum_^n \frac = \sum_^n \left(\sum_^k i\right) The tetrahedral numbers are: :1, 4, 10, 20 (number), 20, 35 (number), 35, 56 (number), 56, 84 (number), 84, 120 (number), 120, 165 (number), 165, 220 (number), 220, ... Formula The formula for the th tetrahedral number is represented by the 3rd rising factorial of divided by the factorial of 3: :Te_n= \sum_^n T_k = \sum_^n \frac = \sum_^n \left(\sum_^k i\right)=\frac = \frac The tetrahedral numbers can also be represented as binomial coefficients: :Te_n=\binom. Tetrahedral numbers can therefore be found in the fourth position either from left or right in Pascal's triangle. Proofs of formula This proof uses the fact that the th triangular num ...
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Partial Sum
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ... and its generalization, mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematic ...
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Telescoping Series
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a telescoping series is a series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ... whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums only consists of two terms of (a_n) after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. For example, the series :\sum_^\infty\frac (the series of reciprocal Reciprocal may refer to: In ...
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Centered Hexagonal Number
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered polygonal number, centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers: : Centered hexagonal numbers should not be confused with hexagonal number, cornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex. The sequence of hexagonal numbers starts out as follows : :1, 7, 19 (number), 19, 37 (number), 37, 61 (number), 61, 91 (number), 91, 127 (number), 127, 169 (number), 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919. Formula The th centered hexagonal number is given by the formula :H(n) = n^3 - (n-1)^3 = 3n(n-1)+1 = 3n^2 - 3n +1. \, Expressing the formula as :H(n) = 1+6\left(\frac\right) shows that the centered hexagonal number for is 1 more ...
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Even And Odd Numbers
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., parity is the property of an integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ... of whether it is even or odd. An integer's parity is even if it is divisible In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... by two with no remainders left and its parity is odd if it isn't; that is, its remainder is 1.. For exa ...
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Arithmetic Mean
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ... and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ..., the arithmetic mean (, stress on first and third syllables of "arithmetic"), or simply the mean There are several kinds of mean in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ... or the average (when the context is clear), is the sum of a collection of numbers divided by the count of nu ...
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Polygonal Number
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a polygonal number is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ... represented as dots or pebbles arranged in the shape of a regular polygon In , a regular polygon is a that is (all angles are equal in measure) and (all sides have the same length). Regular polygons may be either or . In the , a sequence of regular polygons with an increasing number of sides approximates a , if the .... The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numberThe term figurate nu ...
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Pythagoreans
Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ... and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone Crotone (, ; nap, label= Crotonese, Cutrone or ) is a city and ''comune The (; plural: ) is a of , roughly equivalent to a or . Importance and function The provides essential public services: of births and deaths, , and maintena ..., Italy Italy ( it, Italia ), officially the Italian Republic ( it, Repubblica Italiana, links=no ), is a country consisting of a peninsula delimited by the Alps The Alps ; german: Alpen ; it, Alpi ; rm, Alps; sl, Alpe ) are the highest .... Early Pythagorean co ...
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Metaphysics (Aristotle)
''Metaphysics'' (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...: τὰ μετὰ τὰ φυσικά, "things after the ones about the natural world"; Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...: ''Metaphysica'') is one of the principal works of Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ..., in which he develops the doctrine that he refers to somet ...
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