Shortcuts

 Sitemap Contact Newsletter Store Books Features Gallery E-cards Games

Pascal's Triangle Builder

Construct your own Pascal's Triangle!

Pascal's Triangle or Binomial Coefficient

Select now ONE of the buttons below to display your Pascal's Triangle

The Historic Context

In 1653, a French mathematician-philosopher named Blaise Pascal described a triangular arrangement of numbers corresponding to the probabilities involved in flipping coins, or the number of ways to choose 'n' objects from a group of 'm' indistinguishable objects. Pascal's triangle has many uses in binomial expansions. Although Pascal never claimed recognition for his discovery, his name is inseparably linked with it. In fact, the triangle had been described centuries earlier... The first reference occurs in Indian mathematician Pingala's book on Sanskrit poetics that may be as early as 450 BC as Meru-prastaara, the "staircase of Mount Meru". The commentators of this book were also aware that the shallow diagonals of the triangle sum to the Fibonacci numbers. It was also known to Chinese mathematicians. It is said that the triangle was called "Yang Hui's triangle" (杨辉三角形) by the Chinese. Later, the Persian mathematician Karaji and the Persian astronomer-poet Omar Khayyám; thus the triangle is referred to as the "Khayyám triangle" (مثلث خیام) in Iran. Several theorems related to the triangle were known, including the binomial theorem. And to conclude, in Italy, it is referred to as "Triangolo di Tartaglia" (Tartaglia's triangle), named for the Italian algebraist Niccolò Fontana Tartaglia who lived a century before Pascal; Tartaglia is credited with the general formula for solving cubic polynomials.

The first seven rows of Pascal's Triangle look like:
 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 Note that every number in the interior of the triangle is the sum of the two numbers directly above it.

See: Astounding Pascal's Triangle

 Pascal's Arithmetical Triangle The Story of a Mathematical Idea Related link: Interesting fraction patterns

 Online translation

© 1992-2013 G. Sarcone, www.archimedes-lab.org
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.