Pronunciation: /pɑˈskɑlz ˈtraɪˌæŋ.gəl/ Explain
Pascal's triangle is a simplified version of the
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.
Click on figure 1 to see the algorithm for building a Pascal's triangle.
||Figure 1: Pascal's Triangle
Figure 1 contains the first eight iterations
of Pascal's triangle. Click on the image to see how the triangle is constructed.
|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
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|
- 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
- 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
Cite this article as:
McAdams, David E. Pascal's Triangle. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/pascalstriangle.html.
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/1/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
10/25/2008: Initial version. (McAdams, David E.)