The progress of mankind is largely due to the discoveries made by geniuses. One of them is Blaise Pascal. His creative biography once again confirms the truth of Lion Feuchtwanger's expression "A talented person, talented in everything." All the scientific achievements of this great scientist are hard to count. Among them is one of the most elegant inventions in the world of mathematics - Pascal's triangle.
A few words about genius
Blaise Pascal died early by modern standards, at the age of 39. However, in his short life he distinguished himself as an outstanding physicist, mathematician, philosopher and writer. Grateful descendants named the unit of pressure and the popular programming language Pascal in his honor. It has been used for almost 60 years to teach how to write various codes. For example, with its help, each student can write a program to calculate the area of a triangle in Pascal, as well as explore the properties of the circuit, aboutwhich will be discussed below.
The activity of this scientist with extraordinary thinking spans a wide variety of fields of science. In particular, Blaise Pascal is one of the founders of hydrostatics, mathematical analysis, some areas of geometry and probability theory. Also, he:
- created a mechanical calculator known as the Pascal wheel;
- provided experimental evidence that air has elasticity and weight;
- established that a barometer can be used to predict the weather;
- invented the wheelbarrow;
- invented the 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 absolute scientific masterpieces is the "Treatise on the Arithmetic Triangle", which consists of binomial coefficients arranged in a certain order. The properties of this scheme are striking in their diversity, and it itself confirms the proverb "Everything ingenious is simple!".
A bit of history
To be fair, it must be said that in fact Pascal's triangle was known in Europe as early as the beginning of the 16th century. In particular, his image can be seen on the cover of an arithmetic textbook by the famous astronomer Peter Apian from the University of Ingolstadt. A similar triangle is also shown as an illustration.in a book by the 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 the treatises of Arab and Indian scientists written earlier.
Description
Before exploring the most interesting properties of Pascal's triangle, beautiful in its perfection and simplicity, it is worth knowing what it is.
Scientifically speaking, this numerical scheme is an endless triangular table formed from binomial coefficients arranged in a certain order. At its top and on the sides are the numbers 1. The remaining positions are occupied by numbers equal to the sum of the two numbers located above them next to each other. Moreover, all lines of Pascal's triangle are symmetrical about its vertical axis.
Basic Features
Pascal's triangle strikes with its perfection. For any line numbered n (n=0, 1, 2…) true:
- first and last numbers are 1;
- second and penultimate - n;
- the third number is equal to the triangular number (the number of circles that can be arranged in an equilateral triangle, i.e. 1, 3, 6, 10): T -1 =n (n - 1) / 2.
- The fourth number is tetrahedral, i.e. it is a pyramid with a triangle at the base.
In addition, relatively recently, in 1972, another property of Pascal's triangle was established. In order for himto find out, you need to write the elements of this scheme in the form of a table with a row shift by 2 positions. Then note the numbers divisible by the line number. It turns out that the number of the column in which all the numbers are highlighted is a prime number.
The same trick can be done in another way. To do this, in Pascal's triangle, the numbers are replaced by the remainders of their division by the row number in the table. Then the lines are arranged in the resulting triangle so that the next one starts 2 columns to the right from the first element of the previous one. Then the columns with numbers that are prime numbers will consist only of zeros, and those with composite numbers will contain at least one zero.
Connection with Newton's binomial
As you know, this is the name of the formula for the expansion into terms of an integer non-negative power of the sum of two variables, which has the form:
The coefficients present in them are equal to C m =n! / (m! (n - m)!), where m is the ordinal number in row n of Pascal's triangle. In other words, having this table at hand, you can easily raise any numbers to a power, having previously decomposed them into two terms.
Thus, Pascal's triangle and Newton's binomial are closely related.
Math Wonders
A close examination of Pascal's triangle reveals that:
- the sum of all numbers in the line withserial number n (counting from 0) is 2;
- if the lines are left aligned, then the sums of numbers that are located along the diagonals of Pascal's 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 Pascal's 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 in terms of solving complex problems, is obtained by coloring the even numbers of the Pascal image in one color, and the odd numbers in another.
The Sierpinski triangle can be built in another way:
- in the shaded Pascal scheme, the middle triangle is repainted in a different color, which is formed by connecting the midpoints of the sides of the original one;
- do exactly the same with three unpainted ones located in the corners;
- if the procedure is continued indefinitely, then 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 they, as shown by the latestresearch is best suited for mathematical modeling of clouds, plants, river deltas, and the universe itself.
Several interesting tasks
Where is Pascal's triangle used? Examples of problems that can be solved with its help are quite diverse and belong to different fields of science. Let's take a look at some of the more interesting ones.
Problem 1. Some large city surrounded by a fortress wall has only one entrance gate. At the first intersection, the main road splits into two. The same happens on any other. 210 people enter the city. At each of the intersections they meet, they are divided in half. How many people will be found at each intersection when it will no longer be possible to share. Her answer is line 10 of Pascal's triangle (the coefficient formula is presented above), where the numbers 210 are located on both sides of the vertical axis.
Task 2. There are 7 names of colors. You need to make a bouquet of 3 flowers. It is required to find out in how many different ways this can be done. This problem is from the field of combinatorics. To solve it, we again use Pascal's 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. Its famous triangle, when used correctly, can become a real lifesaver for solving many problems, especially from the fieldcombinatorics. In addition, it can be used to solve numerous mysteries related to fractals.
