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 mention Gaussian elimination, both the concept and history of it, which is a more well-known algorithm for solving systems of linear equation discovered in 18th century. Finally, the importance of elimination in mathematical history will be shown. A comparison of eastern and western approach to Gaussian elimination would also be given.

The Pascal’s Triangle

Long before Pascal’s Triangle has been named after 17th century French mathematician Blaise Pascal (1623-1662), several other mathematicians had studied the concept as well as the application about it hundreds years ahead independently. 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…