The progress of mankind is largely connected with the discoveries made by geniuses. One of them is Blaise Pascal. His creative biography once again confirms the truth of the expression of Lyon Feuchtwanger "Talented person, talented in everything." All the scientific achievements of this great scientist are difficult to count. These include one of the most elegant inventions in the world of mathematics - the Pascal triangle.
A few words about genius
Blaise Pascal, by modern standards, died early, at the age of 39. However, during his short life, he proved himself to be an outstanding physicist, mathematician, philosopher and writer. Grateful descendants named the pressure unit and the popular programming language Pascal in his honor. It has been used for almost 60 years for teaching writing different codes. For example, with its help, each student can write a program to calculate the area of a triangle on Pascal, as well as explore the properties of the circuit, which will be discussed below.
The activities of this scientist with extraordinary thinking cover a variety of fields of science. In particular, Blaise Pascal is one of the founders of hydrostatics of mathematical analysis, some areas of geometry and probability theory. In addition, he:
- created a mechanical calculator known as the Pascale wheel;
- presented experimental evidence that air is elastic and weight;
- found that the barometer can be used to predict the weather;
- invented a wheelbarrow;
- came up with an omnibus - horse-drawn carriages with fixed routes, which later became the first type of regular public transport, etc.
Pascal's arithmetic triangle
As already mentioned, this great French scientist made a huge contribution to mathematical science. One of his unconditional scientific masterpieces is The Treatise on the Arithmetic Triangle, which consists of binomial coefficients arranged in a specific order. The properties of this scheme are striking in its diversity, and it itself confirms the proverb "Everything ingenious is simple!"
A bit of history
In fairness, I must say that in fact the Pascal triangle was known in Europe at the beginning of the 16th century. In particular, his image can be seen on the cover of the arithmetic textbook of the famous astronomer Peter Apian of Ingolstadt University. A similar triangle is presented as an illustration in the book of Chinese mathematician Yang Hui, published in 1303. The remarkable Persian poet and philosopher Omar Khayyam was also aware of its properties at the beginning of the 12th century. Moreover, it is believed that he met him from treatises of Arab and Indian scholars written earlier.
Description
Before exploring the most interesting properties of the Pascal triangle, beautiful in its perfection and simplicity, it is worth knowing what it is.
In scientific terms, this numerical scheme is an endless table of a triangular shape formed from binomial coefficients arranged in a specific order. At its top and sides are the numbers 1. The remaining positions are occupied by numbers equal to the sum of the two numbers located above them next above. Moreover, all lines of the Pascal triangle are symmetric about its vertical axis.
Basic properties
Pascal's triangle is striking in its perfection. For any line numbered (n = 0, 1, 2 ...) it is true:
- the first and last numbers are 1;
- second and second to last - n;
- the third number is equal to the triangular number (the number of circles that can be arranged in the form of an equilateral triangle, i.e. 1, 3, 6, 10): T n -1 = n (n - 1) / 2.
- the fourth number is tetrahedral, that is, it is a pyramid with a triangle at the base.
In addition, relatively recently, in 1972, another property of the Pascal triangle was established. In order to detect it, you need to write the elements of this scheme in the form of a table with a line offset of 2 positions. Then mark the numbers divisible by line number. It turns out that the column number in which all the numbers are highlighted is a prime number.
The same trick can be done differently. To do this, in the Pascal triangle, the numbers are replaced by the residues from their division by the line number in the table. Then arrange the lines in the resulting triangle so that the next of them starts 2 columns to the right of the first element of the previous one. Then the columns having numbers that are prime numbers will consist only of zeros, and in those with composite numbers, there will be at least one zero.
Connection with Newton's binomial
As you know, this is the name of the formula for expansion into terms of a non-negative degree of the sum of two variables, which has the form:
The coefficients present in them are equal to C n m = n! / (m! (n - m)!), where m, is the serial number in line n of the Pascal triangle. In other words, having this table at hand, you can easily raise any numbers to a power by first decomposing them into two terms.
Thus, the Pascal triangle and Newton’s binom are closely interconnected.
Mathematical wonders
A careful study of the Pascal triangle reveals that:
- the sum of all numbers in the line with serial number n (counting from 0) is 2 n ;
- if the lines are left-aligned, then the sums of numbers that are located along the diagonals of the Pascal triangle, going from bottom to top and from left to right, are equal to Fibonacci numbers;
- the first “diagonal” consists of natural numbers in order;
- any element from the Pascal triangle, reduced by one, is equal to the sum of all numbers located inside the parallelogram, which is limited by the left and right diagonals intersecting on this number;
- in each line of the diagram, the sum of numbers in even places is equal to the sum of elements in odd places.
Sierpinski Triangle
Such an interesting mathematical scheme, quite promising from the point of view of solving complex problems, is obtained by coloring even numbers of the Pascal image in one color, and odd numbers in another.
Sierpinski’s triangle can be built in another way:
- in the shaded Pascal diagram, the middle triangle, which is formed by connecting the midpoints of the sides of the original, is repainted in a different color;
- do exactly the same with three unfilled, located in the corners;
- if the procedure is continued indefinitely, the result should be a two-color figure.
The most interesting property of the Sierpinski triangle is its self-similarity, since it consists of 3 of its copies, which are reduced by 2 times. It allows us to attribute this scheme to fractal curves, and, as recent studies show, they are best suited for mathematical modeling of clouds, plants, river deltas, and indeed the Universe itself.
Some interesting tasks
Where is the Pascal Triangle used? Examples of tasks that can be solved with its help are quite diverse and relate to various fields of science. Consider some of the most interesting of them.
Task 1. A large city surrounded by a fortified wall has only one entrance gate. At the first intersection, the main road diverges into two. The same thing happens on any other. 210 people enter the city. At each intersection, they are divided in half. How many people will find at each intersection, when sharing will be impossible. Her answer is the 10th line of the Pascal triangle (the formula of the coefficients is presented above), where the numbers 210 are located on both sides of the vertical axis.
Task 2. There are 7 kinds of colors. You need to make a bouquet of 3 flowers. Find out how many different ways this can be done. This problem is from the field of combinatorics. To solve it, we again use the Pascal triangle and get on the 7th line in the third position (numbering in both cases from 0) the number 35.
Now you know what the great French philosopher and scientist Blaise Pascal invented. When used correctly, its famous triangle can become a real lifesaver for solving many problems, especially from the field of combinatorics. In addition, it can be used to solve numerous puzzles related to fractals.