Pascal's triangle.

His Traité du triangle arithmétique (" Treatise on the Arithmetical Triangle ") of 1653 described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle.

The binomial coefficients can be arranged to form Pascal's triangle.

Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle.

The notation was introduced by Andreas von Ettingshausen in 1826, although the numbers were already known centuries before that ( see Pascal's triangle ).

The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.

These coefficients for varying n and b can be arranged to form Pascal's triangle.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

Pascal's triangle

The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the third row of Pascal's triangle.

The arithmetical triangle — a graphical diagram showing relationships among the binomial coefficients — was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.

* Pascal's triangle, a geometric arrangement of the binomial coefficients in a triangle

Consequently, the number of m-faces of an n-simplex may be found in column ( m + 1 ) of row ( n + 1 ) of Pascal's triangle.

The first six rows of Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.

The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top.

Pascal's triangle has higher dimensional generalizations.

Yang Hui ( Pascal's ) triangle, as depicted by the Chinese using Counting rods | rod numerals.

Pascal's earliest work was in the natural and applied sciences where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

Pascal's work was so precocious that Descartes was convinced that Pascal's father had written it.

In 2008, the system was brought up to a new level and the resulting language termed " Pascaline " ( after Pascal's calculator ).

He also sent to France his famous " formulary ", that was to be signed by all the clergy as a means of detecting and extirpating Jansenism and which inflamed public opinion, leading to Blaise Pascal's defense of Jansenism.

This concept was first formulated in a slightly extended form by French mathematician and philosopher Blaise Pascal in 1647 and became known as Pascal's Law.

SNOBOL4's programmer-defined data type facility was advanced at the time — it is similar to the earlier Cobol's and the later Pascal's records.

Gottfried Leibniz ( 1646 – 1716 ), building on Pascal's work, became one of the most prolific inventors in the field of mechanical calculators ; he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator.

Some people believe that although Torricelli's experiment was crucial, it was Blaise Pascal's experiments that proved the top space really contained vacuum.

Pascal's Traité du triangle arithmétique ( Treatise on Arithmetical Triangle ) was published posthumously in 1665.

The triangle was later named after Pascal by Pierre Raymond de Montmort ( 1708 ) who called it " Table de M. Pascal pour les combinaisons " ( French: Table of Mr. Pascal for combinations ) and Abraham de Moivre ( 1730 ) who called it " Triangulum Arithmeticum PASCALIANUM " ( Latin: Pascal's Arithmetic Triangle ), which became the modern Western name.

Blaise Pascal's Écrits sur la grâce, based on what Michel Serres has called his " anamorphotic method ," attempted to conciliate the contradictory positions of Molinists and Calvinists by stating that both were partially right: Molinists, who claimed God's choice concerning a person's sin and salvation was a posteriori and contingent, while Calvinists claimed that it was a priori and necessary.

Pascal's religious conversion led him into a life of asceticism, and the Pensées was in many ways his life's work.

It was not until the beginning of the 20th century that scholars began to understand Pascal's intention.

( See, also, the monumental edition of his Oeuvres complètes ( 1964 – 1991 ), which is known as the Tercentenary Edition and was realized by Jean Mesnard ; this edition reviews the dating, history, and critical bibliography of each of Pascal's texts.

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri ( 1000 ) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.

Pascal's Wager ( also known as Pascal's Gambit ) is an argument in apologetic philosophy which was devised by the seventeenth-century French philosopher, mathematician, and physicist, Blaise Pascal.

He translated Blaise Pascal's Provincial Letters in 1657, under the title of Les Provinciales, or the Mystery of Jesuitisme, discovered in certain letters written upon occasion of the present differences at Sorbonne between the jansenists and the molinists, London, Royston, 1657 ).

Wilhelm Schickard ( 22 April 1592 – 24 October 1635 ) was a German professor of Hebrew and Astronomy who became famous in the second part of the 20th century after Dr. Franz Hammer, a biographer of Johannes Kepler, claimed that the drawings of a calculating clock, predating the public release of Pascal's calculator by twenty years, had been discovered in two unknown letters written by Schickard to Johannes Kepler in 1623 and 1624.

This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian ( 贾宪 ).

* The earliest extant Chinese illustration of ' Pascal's Triangle ' is from Yang Hui's book Xiangjie Jiuzhang Suanfa, published in this year, although knowledge of Pascal's Triangle existed in China by at least 1100.

Pascal's triangle is called Yang Hui's triangle in China.

Yang Hui was also the first person in history to discover and prove " Pascal's Triangle ", along with its binomial proof ( although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD ).

Pascal's triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa ( 详解九章算法 ), although it was described earlier around 1100 by Jia Xian.