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Complex numbers

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The complex numbers make up a group of figures resulting from the sum between a real number and a type number imaginary . A real number, according to the definition, is one that can be expressed by a whole number (4, 15, 2686) or decimal (1.25; 38.1236; 29854.152). Instead, an imaginary number is one whose square is negative. The concept of imaginary number was developed by Leonhard Euler in 1777 , when he granted v-1 the name of i (from "imaginary" ).

The notion of a complex number appears before the impossibility of real numbers to encompass the roots of even order of the set of negative numbers. Complex numbers can therefore reflect all roots of polynomials , something that real numbers are not in a position to do.

Thanks to this particularity, complex numbers are used in various fields of mathematics, in physics and in engineering . Because of their ability to represent electric current and electromagnetic waves, to name a case, they are frequently used in the electronics and the telecommunications . And the so-called complex analysis, that is, the theory of functions of this type, is considered one of the richest facets of mathematics.

It should be noted that the body of each real number It is made up of ordered pairs (a, b ). The first component (to ) is the real part, while the second component (b ) is the imaginary part. The pure imaginary numbers they are those that are only formed by the imaginary part (therefore, a = 0 ).

Complex numbers make up the so-called complex body (C ). When the real component a is identified with the corresponding complex (a, 0 ), the body of these real numbers (R ) becomes a sub-body of C . On the other hand, C make up a space vector two-dimensional about R . This shows that complex numbers do not support the possibility of maintaining an order, unlike real numbers.

History of complex numbers

Already since the first century BC, some Greek mathematicians, such as Herón de Alejandría, began to outline the concept of complex numbers, before difficulties to build a pyramid. However, it was only in the 16th century that they began to occupy an important place for science; At that time, a group of people were looking for formulas to obtain the estate Exact polynomials of grades 2 and 3.

First, his interest was to find the real roots of the aforementioned equations; however, they also had to face the roots of negative numbers. The famous philosopher, mathematician and physical Descartes was of French origin who created the term of imaginary numbers in the seventeenth century, and just over 100 years later the concept of complexes would be accepted. However, it was necessary for Gauss, a German scientist, to rediscover him some time later so that he would receive the attention he deserved.

The complex plane

To interpret complex numbers geometrically it is necessary to use a flat complex. In the case of its sum, it can be related to that of the vectors, while its multiplication can be expressed by polar coordinates, with the following characteristics:

* the magnitude of your product is the multiplication of the magnitudes of the terms;

* the angle that goes from the axis Actual product results from the sum of the angles of the terms.

When representing the positions of the poles and zeros of a function in a complex plane, the so-called Argand diagrams are often used.

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