History and applications

History

The expansion \((a+b)^2 = a^2+2ab+b^2\) appears in Book 2 of Euclid. There it is stated in geometric terms:

If a line segment is cut at random, the square on the whole is equal to the square on each of the line segments which have been formed and twice the rectangle which has these line segments as length and width.

The triangle of numbers that we refer to as Pascal's triangle was known before Pascal. Pascal developed many uses of it and was the first one to organise all the information together in his 1653 treatise.

The triangle had been discovered centuries earlier in India and China. In the 13th century, Yang Hui (1238–1298) knew of this triangle of numbers. In China, Pascal's triangle is called Yang Hui's triangle. The triangle was known in China in the early 11th century by the mathematician Jia Xian (1010–1070). It was also discussed by the Persian poet-astronomer-mathematician Omar Khayyam (1048–1131). In Iran, the triangle is referred to as the Khayyam–Pascal triangle or simply the Khayyam triangle. Several theorems related to the triangle were known, including the binomial theorem for non-negative integer exponents. In Europe, it first appeared as the frontispiece of a book by Petrus Apianus (1495–1552) and this is the first printed record of the triangle in Europe. In Italy, it is still referred to as Tartaglia's triangle, named for the Italian algebraist Niccolo Tartaglia (1500–1577).

Although Pascal was not the first to study this triangle, his work on the topic was the most important. Pascal's work on the binomial coefficients led to Newton's discovery of the general binomial theorem for fractional and negative powers.

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