Pascal’s Triangle Essay Samples
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Pascal's triangle is an endless table of binomial coefficients, which has a triangular shape. In the upper line of the triangle is a single unit. In other lines, each number is the sum of its two neighbors on the floor above - left and right. If some of the neighbors is not available, it is considered to be zero. The lines of triangle are symmetrical about a vertical axis. It is named in honor of Blaise Pascal. The triangle has application in the theory of probability.
The first mention of the triangular sequence of binomial coefficients called meru-prastaara found in the commentary of X century Indian mathematician Halayudhi to the works of other mathematicians Pingala. Triangle is also studied by Omar Khayyam around 1100, so in Iran, this scheme is called a triangle Khayyam. In 1303 he was released book "Jasper mirror the four elements" of the Chinese mathematician Zhu Shijie, which was depicted Pascal's triangle on one of the illustrations; it is believed that it was invented by another Chinese mathematician Yang Hui (Chinese call it so triangle Hui Yang). On the title page of the textbook of arithmetic, written in 1529 by Peter Apianom, an astronomer from the University of Ingolstadt also depicted Pascal's triangle. And in 1653 (other sources in 1655), a book of Blaise Pascal "Treatise on the arithmetical triangle".
First 15 rows of the Pascal’s Triangle:
We denote by n the line number of the triangle, and the letter k - number of numbers in a row (starting in both cases from scratch). Most often, the number in the n-th row and k-th place in this line is denoted Cnk, rarely – nk. We name only some of the facts relating to the Pascal triangle.
The numbers in the n-th row of the triangle are the binomial coefficients, i.e., coefficients in the expansion of the n-th degree binomial:
The sum of all the numbers in the n-th row is equal to the n-th power of two:
This formula is obtained from the binomial formula, if we set a = b = 1. It can be proved an explicit formula for the calculation of the binomial coefficient:
If we deduct from the center of a row with an even number of adjacent numbers in the same row, you get a number of Catalan.
All numbers in the n-th row except the units divided by the number n, if and only if n is a prime number (a consequence of Theorem Luc).
If in a line with an odd number add up all the numbers with numbers of the form 3n, 3n + 1, 3n + 2, the first two sums will be equal, and the third will be less by 1.
Each number in the triangle is equal to the number of ways to get to it from the top, or moving to the right and down or left and down.
Pascal's triangle is so simple that it can be written by even preteen. At the same time, it is fraught with the unsearchable riches and binds together the different aspects of mathematics that are not at first glance anything in common. These unusual properties suggest Pascal's triangle one of the most elegant schemes in all of mathematics.
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