Pascal’s triangle . Also called the Tartaglia Triangle for having proposed a rectangular scheme of the binomial coefficients, Nicola Tartaglia . Long before the previous ones it is known as the Yan Hui triangle (1303) in China, and in Iran as the Jayyam triangle (1100). And in Europe, as an arithmetic triangleit is a triangular arrangement of rows increasing in number of elements that are positive integers, from top to bottom. It is linked to the development of the successive positive integer powers of a binomial, where each number represents the coefficient of the terms that form the resulting polynomial: binomial coefficient. It can be used to calculate the probability of occurrence of a certain event in a random experiment.
Summary
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- 1 Features
- 2 Construction
- 3 Look in time
- 4 Contribution of Tartaglia
- 5 Sources
- 6 References
characteristics
- All rows start and end with 1, and are symmetric about the center element if the row contains an odd number of coefficients, but if there is an even number, the elements appear symmetrically about an imaginary point.
- Each number in the triangle corresponds to the sum of the two numbers above it. These coefficients represent the number of favorable cases of a certain event.
- The sum of all the elements in each row corresponds to the value 2 n, where n is the order of the row.
- Its construction can be followed, theoretically, infinitely and pragmatically up to a high number of rows.
- It is possible to develop an algorithm that allows designing the elements of the rows using a computer program.
Building
To build the triangle, start with a triangle formed by three numbers one, to obtain the third row one is placed at each end and the central value is obtained by adding the two numbers that are left on top.
The procedure is repeated for the other rows taking into account that each one increments one more number and that the ends always correspond to numbers one.
Look in time
- This arrangement was already known to the mathematicians of Ancient India in the 2nd century BC.
- Later, Chinese manuscripts record it precisely, in a text by Al-Kashi.
- The arithmetic triangleis also documented in the works of Stifel (1543), Stevin (1625) and Hérigone (1634) [1]
- It is presumable that Pascal knew the works of the mentioned European experts and “it can certainly be affirmed that he knew the Cursus mathematicusof Heritage” [2]
- Pascal wrote on this subject in his work Traité du triangle arithmétique(1664) and named the triangular arrangement the arithmetic triangle , which is also known as Pascal’s triangle or Tartaglia’s triangle.
Contribution of Tartaglia
Tartaglia in General trattato di numeri et misuri presents the following rectangular arrangement in 1556.
1 1 1 1 1 1 …
1 2 3 4 5 6 …
1 3 6 10 15 21 …
1 4 10 20 35 …
1 5 15 35 …
…………… [3]
