Since the XVI century, the practical needs of mathematics have experienced the need to expand the notion of numbers from natural to real. Initially, for the counting of objects, natural numbers were used. Then the mathematicians were forced to resort to division operations, which led to the concept of fractional numbers. Later, the inability to perform certain operations, leading only to positive results, led mathematics to use zero and negative numbers; and finally, due to the need to extract roots, a notion such as irrational numbers was formed.
All of the above operations can be performed with a set of real numbers. However, there remain operations that can not be performed on this set, for example, extracting a square root from a negative number. Hence, the expansion of concepts has not reached its limit and there is a need for further development of the concepts of numbers, new numbers, different from real ones, are required.
Solving problems using the apparatus of complex numbers greatly facilitated many complex mathematical problems. Therefore, over time, complex numbers have become more and more important tools in mathematics and its applications. First and foremost, they deeply embedded in the theory of algebraic equations, radically facilitating their study.
After, in the nineteenth century, a clear geometric description of complex numbers appeared with the help of points and vectors on the plane, it was possible to reduce to the complex numbers and equations for them many problems of electro-radio technology, applied trigonometry, cartography and natural sciences.
The end of the twentieth century was marked by the rapid development of many areas of science and technology. Computing technology has become an indispensable tool for solving problems using the plane of complex numbers. In addition, there are a huge number of automatic control systems that use complex planes to support their work. In particular, there are closed stable systems in which, according to the theorem proved by AM Lyapunov, the roots of the characteristic equation must lie in the left half-plane of the complex plane of the roots.
Later this theorem not only found application in solving similar problems, but also served as an impetus for the development of more efficient algorithms. One of them is an algorithm for localizing the roots of a polynomial in the domain of complex numbers.
The problem of calculating or at least localizing the roots of a polynomial along with numerical methods of linear algebra has an important applied value. Thus, for example, the question of the stability of the equilibrium position of a dynamical system under very general propositions reduces to the question of whether all the roots of the characteristic equation of a linearized system are located in the complex plane to the left of the imaginary axis. This explains the continuing interest of mathematicians and engineers in stable polynomials.
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