Pascal's triangle

Lore & Background

The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. In India, the Chandaḥśāstra by the ancient Indian poet and mathematician Piṅgala (3rd or 2nd century BC) describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Pingala's 10th-century commentator Halāyudha, his 'method of pyramidal expansion' (meru-prastāra) for counting metres is equivalent to Pascal's triangle. The 6th-century Indian mathematician and astronomer Varāhamihira later described a recursive method for computing binomial coefficients that is mathematically equivalent to Pascal's triangle, although arranged in a different orientation. Khayyám's distinctive contribution was to apply the triangle's binomial coefficients to a general method for extracting nth roots, an application not found in the earlier Indian or Karajian sources. Owing to this innovation and Khayyám's broad influence on later Islamic mathematics, the triangle is known in Iran as Khayyam's triangle or the Khayyam–Pascal triangle. In Europe, Pascal's triangle appeared for the first time in the Arithmetic of Jordanus de Nemore (13th century). The binomial coefficients were calculated by Gersonides during the early 14th century, using the multiplicative formula for them. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory.

Reader's Guide

Pascal's triangle is significant as an infinite triangular array of binomial coefficients, which are fundamental in probability theory, combinatorics, and algebra. The triangle is constructed by starting with a single 1 in row 0, and each subsequent row's entries are formed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. The entries are denoted as binomial coefficients (n choose k), which represent the number of ways of choosing k things from a collection of n things. This notation leads to Pascal's rule: (n choose k) = (n-1 choose k-1) + (n-1 choose k) for any positive integer n and any integer 0 ≤ k ≤ n. The triangle also determines the coefficients in binomial expansions, such as (x+y)^2 = x^2 + 2xy + y^2, where the coefficients 1, 2, 1 are the entries in the second row. Its legacy endures as a core tool in combinatorial mathematics and algebraic expansions.

Did You Know?

• The rows of Pascal's triangle are conventionally enumerated starting with row n=0 at the top, and entries within a row are numbered from k=0.
• In India, the Chandaḥśāstra by Piṅgala (3rd or 2nd century BC) describes a method of counting metres that is equivalent to Pascal's triangle.
• In Iran, the triangle is known as Khayyam's triangle or the Khayyam–Pascal triangle, due to Omar Khayyám's application of its coefficients to extracting nth roots.
• In China, the triangle is known as Yang Hui's triangle, after the 13th-century mathematician Yang Hui.