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Pascal's Triangle

Blaise Pascal (1623-1662), Omar Khayyam (1084-1131)

Lyndsey Pickett

on 11 May 2011

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Transcript of Pascal's Triangle

Pascal's Triangle Blaise Pascal was the first to write about this progression in 1654, although the Persian poet and mathmetician Omar Khayyam had previously known about the pattern. Even earlier mathmeticians of
India and China knew about the pattern before these mathmeticians. Pascal's triangle is a triangular array of the binomial
coefficients in a triangle.
For example: A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this way, the rows of the triangle go on infinitly. A number in the triangle can also be found by nCr (n Choose r) where n is the number of the row and r is the element in that row. For example, in row 3, 1 is the zeroth element, 3 is element number 1, the next three is the 2nd element, and the last 1 is the 3rd element. The formula for nCr is:


The set of numbers that form Pascal's triangle were well known before Pascal. However, Pascal developed many applications of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks' study of figurate numbers useful applications of pascals triangle Pascal's triangle determines the coefficients which arise in binomial expansions. For an example, consider the expansion

(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2.

Notice the coefficients are the numbers in row two of Pascal's triangle: 1, 2, 1. In general, when a binomial like x + y is raised to a positive integer power we have:

(x + y)n = a0xn + a1xn−1y + a2xn−2y2 + ... + an−1xyn−1 + anyn,
where the coefficients ai in this expansion are precisely the numbers on row n of Pascal's triangle. In other words,

Binomial Thereom
number combinations of n things taken k at a time (called n choose k) can be found by the equation

But this is also the formula for a cell of Pascal's triangle. Rather than performing the calculation, one can simply look up the appropriate entry in the triangle. For example, suppose a basketball team has 10 players and wants to know how many ways there are of selecting 8. Provided we have the first row and the first entry in a row numbered 0, the answer is entry 8 in row 10: 45. That is, the solution of 10 choose 8 is 45.
Combinations Blaise Pascal was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a Tax Collector. Mathmatician Donald Knuth once indicated that there are so many relations and patterns in Pascal's triangle that when someone finds a new indentity, there aren't many people who get excited about it anymore, except the discoverer. Fascinating studies have revealed countles wonders, including special geometric
patterns in the diagonals, the existence of perfect square patterns wit various
hexagonal properties, and an extension of the triangle and its patterns to
negative integers and to higher dimensions. When even numbers in the triangle are replaced by
dots and odd numbers by gaps, the resulting pattern
is a fractal. Fractals have intricate repeating patterns on different size scales. Fractal figures may have a practical importance in that they can provide models for materials scientists use to help produce new structures
for example: in 1986, researchers created wire gaskets on the micron size scale almost identical to Pascal's triangle, with holes for the odd numbers. In 1642, while still a teenager, he started some
pioneering work on calculating machines,
and after three years of effort and 50 prototypes
he invented the mechanical calculator. He built twenty of these machines (called the Pascaline) in the following ten years.[4] Pascal was a mathematician of the first order. He helped create two major new areas of research Along with finding the treatise known as
Pascal's triangle, he also wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids. Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit géométrique ("Of the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous "Petites-Ecoles de Port-Royal" ("Little Schools of Port-Royal"). The work was unpublished until over a century after his death. Pascal looked into the issue of discovering truths, arguing that the ideal of such a method would be to found all propositions on already established truths. At the same time, however, he claimed this was impossible because such established truths would require other truths to back them up—first principles, therefore, cannot be reached. Based on this, Pascal argued that the procedure used in geometry was as perfect as possible, with certain principles assumed and other propositions developed from them. Nevertheless, there was no way to know the assumed principles to be true. patterns in
pascal's triangle Each number is just the two numbers above it added together. Diagonals

The first diagonal is, of course, just "1"s, and the next diagonal has the Counting Numbers (1,2,3, etc).

The third diagonal has the triangular numbers

(The fourth diagonal, not highlighted, has the tetrahedral numbers.) Odds and Evens

If you color the Odd and Even numbers, you end up with a pattern the same as the Sierpinski Triangle Horizontal Sums
It doubles each time THE END
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