The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top. The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. 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. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.
This construction is related to the binomial coefficients by Pascal's rule, which states that if
then
for any nonnegative integer n and any integer k between 0 and n.[2]
Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices.
The set of numbers that form Pascal's triangle were well known before Pascal. But, 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.[3]
The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala between the 5th and 2nd century BC. While Pingala's work only survives in fragments, the commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle sum to the Fibonacci numbers.
At around the same time, it was discussed in Persia (Iran) by the Persian mathematician, Al-Karaji (953–1029).[4] It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám (1048–1131); thus the triangle is referred to as the Khayyam triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
In 13th century, Yang Hui (1238–98) presented the arithmetic triangle that is the same as Pascal's triangle. Pascal's triangle is called Yang Hui's triangle in China. The "Yang Hui's triangle" was known in China in the upper half of the 11th century by the Chinese mathemtician Jia Xian (1010-1070).
Petrus Apianus (1495–1552) published the triangle on the frontispiece of his book on business calculations in the 16th century. This is the first record of the triangle in Europe.
In Italy, it is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolò Fontana Tartaglia (1500–77). Tartaglia is credited with the general formula for solving cubic polynomials, (which may be really from Scipione del Ferro but was published by Gerolamo Cardano 1545).
Traité du triangle arithmétique (Treatise on Arithmetical Triangle) has been published posthumously in 1665. In the Treatise Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name
No comments:
Post a Comment