Pascal's Triangle Essay

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Introduction I’m going to start this paper with introducing the history of Pascal’s Triangle, time ranging from 2nd century BC to 18th century AD. Among all the academic works by ancient mathematicians, I’m going to focus on Jiu Zhang Suan Shu [Nine Chapters on the Mathematical Art] composed by Chinese scholars around 2nd century BC-10th century AD. I will use the binomial expansions derived from Pascal’s triangle as an example and then illustrate the coefficients expansion in general. After that, I will introduce Jade Mirror of the Four Unknowns written by Zhu Shijie around 14th century AD. My focus will be his idea of using the “four unknowns” to convert a problem into a mathematical system of polynomial equations. Later on, I will also …show more content…
A literature of Pascal’s Triangle emerged in 1068 discovered by Hindu mathematician Bhattotpala (c.1068) who recorded the first 16 rows of the triangle (Wilson, 2013, p170). Meanwhile, in Persia, Al-Karaji (953-1029) also found the binomial theorem according to Pascal’s Triangle as well as several theorems related to it (Coolidge, 1949, p151). Although the original work from Al-Karaji had lost, the later Persian mathematician Khayyam (1048-1122) referred Al-Karaji’s work about Pascal’s Triangle and uses it to find the nth roots according to the binomial expansion. Also around 11th century, Chinese mathematician Jia Xian (1010-1070) used Pascal’s Triangle to extract square roots and cube roots – more details will be mentioned later in this paper. He wrote down his discovery in the book Shi Shuo Suan Shu [The key to Mathematics]. Although the original book has lost, the later mathematician Yang Hui (1238-1298) mentioned in his book that he was inspired by Jia’s work and found out more theorems and applications related to Pascal’s Triangle (Qian, 1981, p56). Moreover, the earliest display of Pascal’s Triangle was also from Yang Hui’s work. The following is how Yang Hui display the …show more content…
The book was composed around 2nd century BC to 10th century AD with no exact date could be tracked. But what we can know is, The Nine Chapters on the Mathematical Art is the basis of ancient Chinese mathematics, which is designated as the math textbook in Tang and Song dynasty (7th – 13th century BC) and widely spread in Asian countries. That period of time is also called the golden age of Chinese mathematics (Straffin, 1993,

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