Pascal's triangle

The Pascalian triangle is a form of graphical representation of the binomial coefficients, which also allows for a simple calculation of these. They are arranged in a triangle such that each entry is the sum of the two entries above it. This situation is represented by the equation

Described. The variable can be interpreted as a row index and a column index, the counting starts with zero ( ie first row, first column). If you start at the edges with entries with the value, so this could offer exactly the binomial coefficients.

The name goes back to Blaise Pascal. The Pascalian triangle, however, was known earlier and is therefore also today named yet by other mathematicians. In China, this is known as Yang Hui triangle (after Yang Hui ), in Italy by Tartaglia Triangle ( by Nicolo Tartaglia ) and Iran from the Khayyam triangle (after Omar Khayyam ).

History

The earliest detailed account of a triangle of binomial coefficients appeared in the 10th century in commentaries on Chandas Shastra, an Indian book on Sanskrit prosody, which was written by Pingala between the fifth and second centuries BC. While Pingalas work only preserved only in fragments remained, the commentator Halayudha used to 975 the triangle to make dubious relations with Meru prastaara the " stages of Mount Meru ". It was already known that the sum of the shallow diagonals of the triangle make the Fibonacci numbers. From the Indian mathematician Bhattotpala (about 1068 ) are delivered the first 17 lines of the triangle.

Almost at the same time the Pascalian triangle in Persia by Al- Karaji (953-1029) and Omar Khayyam was treated and is therefore in present-day Iran as Khayyam triangle known. There were various mathematical theorems for triangular known, including the binomial theorem. Indeed, it is quite sure that Khayyam has used a method of calculation of the - th root, based on the binomial expansion, and thus the binomial coefficients.

The earliest Chinese illustration of Pascal's triangle is identical to the arithmetic triangle can be found in Yang Hui book Xiangjie Jiuzhang Suanfa of 1261, which has remained fragmentary preserved in the Yongle Encyclopedia. Yang writes in the triangle of Jia Xian ( 1050 ) and its li cheng shi shuo ( "Determination of coefficients via chart out") called method of calculating square and cube roots to have.

Peter Apian published the triangle 1531/32 on the cover of his book on commercial calculations, the earlier version of 1527 is the first written record of the pascal 's triangle in Europe.

1655 Blaise Pascal wrote the book " Traité du triangle arithmétique " ( treatise on the arithmetical triangle), in which he collected several results concerning the triangle and used these to solve problems in probability theory. The triangle was later named by Pierre Raymond de Mont Mort ( 1708) and Abraham de Moivre ( 1730) by Pascal.

Application

Pascal's triangle is a handle, fast auszumultiplizieren any powers of binomials. For instance, in the third row (), the coefficients 1, 2, 1 of the first two binomial formulas:

In the next line you will find the coefficients 1, 3, 3, 1 for:

This list can be continued as desired, keeping in mind that for the binomial the minus sign from " " to take is always and that while the exponent of always decreases by 1 in each formula, the exponent of increases by 1. A generalization provides the binomial theorem.

Furthermore, the signs alternate in the application of Pascal's triangle on the binomial with any exponent - and regularly (there is always a minus if the exponent of odd). This means, for example,

A two-dimensional generalization is the trinomial Triangle, in which each number is the sum of three (instead of Pascal's triangle: two ) is entries. An extension in the third dimension is the Pascal's pyramid.

Consequences in Pascal's triangle

In Pascal 's Triangle, many known sequences of numbers find.

The diagonals

The first diagonal contains only ones and the second diagonal, the sequence of natural numbers. In the third diagonal there are the triangular numbers and in the fourth the tetrahedral numbers. General can be found in the -th diagonal regular figurate numbers of the order. Is the sequence of partial sums for the series which is on the diagonal above in each diagonal. Conversely, each diagonal sequence is the difference sequence to the standing in diagonal sequence below.

Thus, in general applies to the triangular numbers

For the tetrahedral numbers

And for the regular figurate numbers of the order

The Fibonacci numbers

In general therefore

The lines

The sum of the entries in a row is called a row total. From top to bottom to double the row sums from row to row. This stems from the Education Act of Pascal's triangle. Each entry of a line is used in the following line for the calculation of two entries. Here one has to generalize the left and right of each line, so that the outer ones each row are generated by the addition of the overlying entries the Education Act by adding imaginary zeros. Since the row sum of the first row is equal to one, the row sum of the -th row is the same. This corresponds to the following law for binomial coefficients:

Then string together the digits in each of the first five rows of Pascal's triangle, obtained with 1, 11, 121, 1331 and 14641, the first powers of 11

Formal follows both from x = 1 and x = 10

The alternating sum of each row is zero:

Mean binomial coefficients

The sequence of the middle binomial coefficients begins with 1, 2, 6, 20, 70, 252, ... ( in sequence A000984 OEIS ).

Related to the Sierpinski triangle

See How pascalschem and Sierpinski triangle

Powers with arbitrary base

For powers with any basis exists a number triangle of another kind:

This triangular matrix is accessed by inverting the matrix of the coefficients of those terms, which represent the combinations without repetition of the form data etc..

The Education Act of the coefficients for the coefficient in row and column is:

Therefore, it is also true with the Stirling number.

With the help of this triangle to win immediate insight into the divisibility of powers. So is any prime power for equivalent modulo. This is essentially the content of Fermat's little theorem; In addition, however, shown that the expression of all is not only, but also for a multiple of 6. The greatest common divisor of the matrix coefficients from the second coefficient of the prime exponent is always equal to the denominator of the respective Bernoulli's numbers (example :: denominator = 6;: Denominator = 30, etc. )

With this number triangle can be easily proved, for example, that 24 is divisible by:

Is always divisible by 24 because because are.