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Devised by Niklaus Wirth in the late 1960s and early 1970s, Pascal is a programming language.Originally produced by Borland Software Corporation, Embarcadero Delphi is composed of an IDE, set of standard libraries, and a Pascal-based language commonly called either Object Pascal, Delphi Pascal, or simply 'Delphi' (Embarcadero's current documentation refers to it as 'the Delphi language (Object ...
This algorithm draws all eight octants simultaneously, starting from each cardinal direction (0°, 90°, 180°, 270°) and extends both ways to reach the nearest multiple of 45° (45°, 135°, 225°, 315°).
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix , an upper-triangular matrix , or a symmetric matrix .
Each Pascal's m-simplex is a semi-infinite object, which consists of an infinite series of its components. Let ∧ {\displaystyle \wedge } m n denote its n th component, itself a finite ( m − 1) - simplex with the edge length n , with a notational equivalent n m − 1 {\displaystyle \vartriangle _{n}^{m-1}} .
Singmaster's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times).
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, [1] India, [2] China, Germany, and Italy.
The starting point is on the line (,) =only because the line is defined to start and end on integer coordinates (though it is entirely reasonable to want to draw a line with non-integer end points).
Pascal was born in Clermont-Ferrand, which is in France's Auvergne region, by the Massif Central. He lost his mother, Antoinette Begon, at the age of three. [14] His father, Étienne Pascal, also an amateur mathematician, was a local judge and member of the "Noblesse de Robe". Pascal had two sisters, the younger Jacqueline and the elder Gilberte.