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Pascal's triangle has many properties and contains many patterns of numbers. Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent 1 and dark pixels 0. The numbers of compositions of n +1 into k +1 ordered partitions form Pascal's triangle.
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.
1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients, right-aligned. The left boundary of the image corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence .
Pressure in water and air. Pascal's law applies for fluids. Pascal's principle is defined as: A change in pressure at any point in an enclosed incompressible fluid at rest is transmitted equally and undiminished to all points in all directions throughout the fluid, and the force due to the pressure acts at right angles to the enclosing walls.
In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. [1] Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle , which contains the binomial numbers and relates to the binomial ...
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 .
In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae , [ 1 ] a textbook on number theory published in 1801, in Articles 234 - 244.
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}} .