Complex numbers

From ancient times all the numbers were considered valid. All because first of all people were interested in natural numbers. New, complex numbers, at that time could not be at the thought; even negative numbers were then considered false. In the 16th century, in connection with the study of cubic equations, scientists needed to extract square roots from negative numbers-under the square root sign was a negative number. It turned out that the path to these roots leads through an impossible operation of extracting a square root from a negative number. In 1545, Gerolamo Cardano proposed to introduce the numbers of a new nature. Cardano called such quantities "purely negative", considered them useless and tried not to use. Leibniz called complex numbers "a freak from the world of ideas". And indeed: with the help of such numbers it is impossible to express either the result of measuring any value, or the change itself. In 1572 the book of the Italian algebraist R. Bombelli "Algebra" was published, in which the first rules of arithmetic operations on such numbers were established, up to the extraction of cubic roots from them. Over time, complex numbers lost their supernatural, although their full recognition occurred only in the 19th century. The term "imaginary numbers" was introduced in 1637 by the French mathematician and philosopher Rene Descartes, and in 1777 Leonard Euler proposed using the first letter of the French word imaginaire (imaginary) to denote an imaginary number (imaginary unit).

Thus, around 1800 the invention of Cardano and Bombelli became a new type of pure - number i, the square of which is -1. The term "complex numbers" was introduced by Gauss in 1831, and in the late 18th and early 19th centuries. a geometric interpretation of complex numbers was given. To do this, we used the coordinate system introduced by Descartes. Complex numbers remained for mathematicians only the subject of abstract manipulations until the 19th century, until surveyor Casper Wessel (1745-1818) from Denmark first gave a geometric representation of complex numbers. Then the Swiss mathematician Jean Argan gave a geometric interpretation of the complex number on the plane (1806) and introduced the term modulus of the complex number. Karl Gauss first proposed to represent the complex number z = a + i*b by the point M(a, b) on the coordinate plane. Later it turned out that it is even more convenient to represent the number not by the point M, but in the form of a vector OM going to this point from the origin. With this interpretation, the same operations on vectors correspond to the addition and subtraction of complex numbers. A vector can be defined not only by its coordinates a and b but also by the length r and the angle ф, which it forms with the positive direction of the abscissa axis: z = r*(соsф +i*sinф), where z is the so-called trigonometric form of the complex number. The number r is called the modulus of the complex number z. The number ? is called the argument z.

An important relation for complex numbers was obtained by the English mathematician A. Moivre. He derived the rules for raising and extracting the root n -th power of complex numbers, which are widely used in trigonometry (the Moivre formula).

Complex numbers may seem strange, but they are a magical means for understanding physics. The problems of heat, light, sound, vibration, elasticity, gravity, magnetism, electricity and fluid flow all retreat before this complex weapon in two dimensions.