# Pascal's Triangle

Pronunciation: /pɑˈskɑlz ˈtraɪˌæŋ.gəl/ Explain

Pascal's triangle is a simplified version of the binomial theorem. Pascal's triangle starts at row 1 with the number 1. The second row contains two numbers: 1 and 1. In each iteration, adjacent values are added together to make the number for the next iteration. Figure 2: Yang Hui triangle published in 1303 by Zhu Shijie. Click on the image to see a larger version.

Pascal's triangle in the western world is named for the French mathematician Blaise Pascal. However, it was studied earlier in India, Persia, China, and Italy. See figure 2.

#### Properties of Pascal's Triangle

• The sum of the numbers of each row is twice that of the previous row.
• The first number after the 1 in each row evenly divides all the other numbers if and only if the first number after the 1 is a prime number.
• The 'shallow diagonals' of Pascal's triangle add up to the Fibonacci numbers. See figure 3.
• The equation for Pascal's triangle is where n is the row number and r is the column in the row. Figure 3: Shallow diagonals of Pascal's Triangle

1. McAdams, David E.. All Math Words Dictionary, Pascal's triangle. 2nd Classroom edition 20150108-4799968. pg 135. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. pascal's triangle. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 12/3/2018. http://www.merriam-webster.com/dictionary/pascal's triangle. Buy the book

McAdams, David E. Pascal's Triangle. 4/27/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/pascalstriangle.html.

### Revision History

4/27/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)