Numbers - basic mathematical objects needed for different calculation and settlement.The collection of natural, whole, rational and irrational numerical values forms a set of so-called real numbers.But there is still quite unusual category - complex numbers, Rene Descartes defined as "imaginary quantities."And one of the leading mathematicians of the eighteenth century Leonhard Euler proposed to designate them the letter i from the French word imaginare (allegedly).What is the complex numbers?
So called expressions of the form a + bi, where a and b are real numbers, and i is an index of a particular digital value whose square is -1.Operations with complex numbers are performed by the same rules as the various mathematical operations with polynomials.This category does not express mathematical results of any measurements or calculations.To do this is quite enough real numbers.Why, then, do we need them?
complex numbers as a mathematical concept is needed because of the fact that some equations with real coefficients have solutions in the field of "ordinary" numbers.Consequently, the decision to expand the scope of inequalities became necessary to introduce a new mathematical categories.Complex numbers of predominantly abstract theoretical value, allow to solve such equations as x2 + 1 = 0. It should be noted that, despite its apparent formality, this category of numbers quite active and is widely used, for example, for a variety of practical problemstheory of elasticity, electrical engineering, aerodynamics and fluid mechanics, nuclear physics and other scientific disciplines.
module and argument of a complex number used in the construction schedules.This notation is called trigonometric.In addition, the geometrical interpretation of the numbers has further expanded their scope.It became possible to use them for different mapping algorithms.
Mathematics has come a long way from the simple natural numbers to complex integrated systems and their functions.On this theme, you can write a separate tutorial.Here we look at just a few moments of the evolutionary theory of numbers to make it clear all the historical and scientific background of the emergence of the mathematical categories.
Greek mathematician considered "real" only natural number that can be used to count anything.Already in the second millennium BC.e.the ancient Egyptians and Babylonians in a variety of practical calculations actively used fractions.Another important milestone in the development of mathematics was the appearance of negative numbers in ancient China for two hundred years BC.They are also used by the ancient Greek mathematician Diophantus, who knew the rules of simple operations on them.With negative numbers became possible to describe the various changes in values, not only in the positive plane.
In the seventh century AD, it was well established that the square roots of positive numbers always have two values - in addition to positive and negative yet.From the last square root conventional algebraic methods of that time considered impossible: there is no such value of x to x2 = ─ 9. For a long time it did not matter.It was only in the sixteenth century, when there were and have been actively studied cubic equations, it became necessary to extract the square root of a negative number, as in the formula for the solution of these expressions contains not only the cube, but also the square roots.
This formula smoothly, if the equation is not more than one real root.In the case of the presence in the equation of three real roots for their healing it gets the number with a negative value.It turns out that the road to recovery runs through the three roots impossible from the standpoint of mathematics at the time the operation.
For an explanation of the resulting paradox J. Italian algebraists. Cardano was asked to introduce a new category of the unusual nature of the numbers, which are called complex.I wonder what he Cardano considered them useless and did everything to avoid using them as proposed mathematical categories.But in 1572 there was another Italian book algebraist Bombelli, which were detailed rules for operations on complex numbers.
Throughout the seventeenth century continued the discussion of the mathematical nature of these numbers and their geometrical interpretation capabilities.Also gradually developed and perfected the technique of working with them.And at the turn of the 17th and 18th centuries it was created the general theory of complex numbers.A huge contribution to the development and improvement of the theory of functions of complex variables was made by the Russian and Soviet scientists.Muskhelishvili studied its application to the problems of the theory of elasticity, Keldysh and Lavrent'ev have been used in the field of complex numbers hydro- and aerodynamics, and Vladimir Bogolyubov - in quantum field theory.
