The German mathematician Dirichlet Peter Gustav Lejeune (02.13.1805 - 05.05.1859) is known as the founder of the principle named after him. But besides the theory traditionally explained by the example of “hares and cages”, there are many works on mathematical analysis and number theory on the account of a foreign corresponding member of the Petersburg Academy of Sciences, a member of the Royal Society of London, the Paris Academy of Sciences, the Berlin Academy of Sciences, a professor at the University of Berlin and Gottingen .
He not only introduced the well-known principle into mathematics, Dirichlet was also able to prove the theorem on an infinitely large number of primes that exist in any arithmetic progression from integers with a certain condition. And this condition is that its first term and the difference are mutually prime numbers.
He thoroughly studied the law of distribution of prime numbers, which are characteristic of arithmetic progressions. Dirichlet introduced functional series that have a special form, he was able in the part of mathematical analysis for the first time to precisely formulate and study the concept of conditional convergence and establish a sign of convergence of the series, give rigorous proof of the possibility of expanding a Fourier series function that has a finite number of both maxima and minima . In his works, Dirichlet did not ignore questions of mechanics and mathematical physics (the Dirichlet principle for the theory of harmonic functions).
The uniqueness of the method developed by the German scientist lies in its obvious simplicity, which allows you to study the Dirichlet principle in elementary school. A universal tool for solving a wide range of problems, which is used both to prove simple theorems in geometry and to solve complex logical and mathematical problems.
The availability and simplicity of the method made it possible to use a graphically method for explaining it. The complex and slightly confusing expression formulating the Dirichlet principle is: “For a set of N elements, divided into a certain number of disjoint parts - n (no common elements), provided N> n, at least one part will contain more than one element". They decided to rephrase it successfully, for this purpose, for the sake of clarity, we had to replace N with “rabbits”, and n with “cages”, and the abstruse expression took the form: “Provided that there are at least one more rabbit than cells, there are always at least there would be one cage into which two or more hares will get. "
This method of logical reasoning is still called the opposite, it is widely known as the Dirichlet principle. The tasks that are solved when using it are very diverse. Without going into a detailed description of the solution, the Dirichlet principle is applied with equal success both to prove simple geometric and logical problems, and forms the basis of inferences when considering problems of higher mathematics.
Proponents of the use of this method argues that the main difficulty in using the method is to determine which data falls under the definition of “hares” and which should be considered as “cells”.
In the problem of a straight line and a triangle lying in the same plane, if necessary, prove that it cannot intersect three sides at once, one condition is used as a limitation - the straight line does not pass through any height of the triangle. We consider the heights of the triangle as “hares,” and “half-cells” are two half-planes that lie on both sides of a straight line. Obviously, at least two heights will be in one of the half-planes, respectively, the segment that they limit, the line does not stop, which was required to prove.
The Dirichlet principle is also simply and succinctly used in the logical problem of ambassadors and pennants. Ambassadors of various states are located at the round table, but the flags of their countries are located around the perimeter so that each ambassador is next to the symbol of a foreign country. It is necessary to prove the existence of such a situation when at least two flags will be near the representatives of the respective countries. If we take the ambassadors as “hares,” and “cells” indicate the remaining positions when the table rotates (there will already be less by one), then the problem comes to its own decision.
These two examples are presented to show how complicated problems are easily solved using the method developed by the German mathematician.