Pascals Triangle Essay Example

Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle.It is also known as the figurate triangle, the combinatorial triangle, and the binomial triangle. Even though Pascal’s Triangle is named after seventeenth century mathematician, Blaise Pascal, several other mathematicians knew about and applied their knowledge of the triangle hundreds of years before the birth of Pascal in 1623. The Arabs, Persians and the Chinese discovered Pascal’s triangle in earlier centuries. The earliest depictions of a triangle of binomial coefficients occur in the tenth century in commentaries on the Chandas Shastra. The Chandas Shastra was an ancient Indian book on Sanskrit prosody written by Pingala between the fifth and second centuries BC. Though Pingala's work was found in fragments, 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 were sums to the Fibonacci numbers. The Indian mathematician Bhattotpala later gave rows 0-16 of the triangle.
Around the same time, the traingle was discussed by mathematicians in Persia, Al-Karaji,and mathematician Omar Khayyám, reffered to the traingle as the "Khayyam triangle" in Iran. Many theorems related to the triangle were known, including the binomial theorem. In fact Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
Chinese mathematician Chia Hsien showed that he was using the triangle to extract square and cube roots of numbers. In China, after Chia Hsien’s discovery of the relationship between extracting roots and the binomial coefficients of the triangle, work continued on this topic by several Chinese algebraists to solve higher than cubic equations. In 13th century, Yang Hui presented the arithmetic triangle, which was the same as Pascal's Triangle. Today Pascal's triangle in China is reffered