Complex numbers. The meaning and evolution of "imaginary quantities"

Numbers are the basic mathematical objects needed for various computations and calculations. The totality of natural, integer, rational and irrational numerical values forms a set of so-called real numbers. But there is also a rather unusual category - complex numbers, defined by Rene Descartes as "imaginary values". And one of the leading mathematicians of the eighteenth century, Leonard Euler proposed to denote them by the letter i from the French word imaginare (imaginary). What are complex numbers?

So-called expressions of the form a + bi, in which a and b are real numbers, and i is a digital indicator of a special value, the square of which is -1. Operations on complex numbers are carried out by the same rules as various mathematical operations on polynomials. This mathematical category does not express the results of any measurements or calculations. To do this, it is enough to have real numbers. For what then do they really need?

Complex numbers, as a mathematical concept, are necessary because certain equations with real coefficients do not have solutions in the region of "ordinary" numbers. Consequently, to expand the scope of solving inequalities , it became necessary to introduce a new mathematical category. Complex numbers, which have mainly an abstract theoretical value, allow solving equations such as x2 + 1 = 0. It should be noted that, despite all its seeming formality, this category of numbers is quite actively and widely used, for example, to solve various practical Tasks of the theory of elasticity, electrical engineering, aerodynamics and hydromechanics, atomic physics and other scientific disciplines.

The module and the argument of a complex number are used when constructing graphs. This form of writing is called trigonometric. In addition, the geometric interpretation of these numbers further widened the scope of their application. It became possible to use them for various cartographic calculations.

Mathematics has come a long way from simple natural numbers to complex complex systems and their functions. On this topic you can write a separate textbook. Here we consider only some evolutionary moments of number theory, so that all the historical and scientific prerequisites for the emergence of a given mathematical category become clear.

Ancient Greek mathematicians considered "real" exclusively natural numbers, which can be used to count anything. Already in the second millennium BC. E. Ancient Egyptians and Babylonians in various practical calculations actively used fractions. The next important milestone in the development of mathematics was the appearance of negative numbers in Ancient China two hundred years before our era. They were also used by the ancient Greek mathematician Diophantus, who knew the rules of the simplest operations on them. With the help of negative numbers it became possible to describe various changes in quantities not only in the positive plane.

In the seventh century of our era it was precisely established that the square roots of positive numbers always have two meanings - except positive, also negative. From the latter it was impossible to extract the square root by the usual algebraic methods of that time: there is no such value of x that x ² = 9. For a long time this did not have much significance. And only in the sixteenth century, when cubical equations appeared and began to be actively studied, it became necessary to extract the square root of the negative numbers, since the formula for solving these expressions contains not only cubic, but also square roots.

Such a formula is flawless if the equation has at most one real root. In the case of the presence of three real roots in the equation, when they were cured, a number with a negative value was obtained. So it turned out that the way to extract the three roots lies through an operation impossible from the standpoint of the mathematics of that time.

To explain the resulting paradox, the Italian algebraist J. Cardano was asked to introduce a new category of numbers of unusual nature, which were called complex ones. It is interesting that Cardano himself considered them useless and in every possible way tried to avoid using the same mathematical category proposed by him. But already in 1572 appeared the book of another Italian algebraist Bombelli, where the rules of operations on complex numbers were laid out in detail.

During the entire seventeenth century, the mathematical nature of these numbers and the possibilities of their geometric interpretation continued to be discussed. Also, the technique of working with them was gradually developed and improved. And at the turn of the 17th and 18th centuries a general theory of complex numbers was created. A huge contribution to the development and improvement of the theory of functions of complex variables was introduced by Russian and Soviet scientists. NI Muskhelishvili was engaged in its application to the problems of the theory of elasticity, Keldysh and Lavrentyev found application to complex numbers in the field of hydro- and aerodynamics, and Vladimirov and Bogolyubov in quantum field theory.