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In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, [1] India, [2] China, Germany, and Italy.
Jia Xian triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD. Yang Hui referred to Jia Xian's Shi Suo Suan Shu in the Yongle Encyclopedia Jia Xian ( simplified Chinese : 贾宪 ; traditional Chinese : 賈憲 ; pinyin : Jiǎ Xiàn ; Wade–Giles : Chia Hsien ; ca. 1010–1070) was a Chinese ...
The earliest extant Chinese illustration of 'Pascal's triangle' is from Yang's book Xiángjiě Jiǔzhāng Suànfǎ (詳解九章算法) [1] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian [2] who expounded it around 1100 AD, about 500 years before Pascal.
He is also known for naming Pascal's triangle after Blaise Pascal, calling it "Table de M. Pascal pour les combinaisons." Another of de Montmort's interests was the subject of finite differences. He determined in 1713 the sum of n terms of a finite series of the form
Al-Karaji gave an early formulation of the binomial coefficients and the first description of Pascal's triangle. [13] [14] [15] He is also presumed to have discovered the binomial theorem. [16] In a now lost work known only from subsequent quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by mathematical induction. [17] As ...
One of Khayyam's predecessors, al-Karaji, had already discovered the triangular arrangement of the coefficients of binomial expansions that Europeans later came to know as Pascal's triangle; [49]: 60 Khayyam popularized this triangular array in Iran, so that it is now known as Omar Khayyam's triangle. [39]
He neglected his teaching duties. Apianus's work included in mathematics – in 1527 he published a variation of Pascal's triangle, and in 1534 a table of sines – as well as astronomy. In 1531, he observed Halley's Comet and noted that a comet's tail always point away from the sun. [17]
Blaise Pascal [a] (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer.. Pascal was a child prodigy who was educated by his father, a tax collector in Rouen.