The application to problems of the theory of electrical circuits extending the notion of real numbers
As is well known, the mathematical theory of real numbers is based on a system of generally accepted axioms. On the set of real numbers, the operations of addition, multiplication, and also the order relation are introduced. On the multiplication operation, such axioms are introduced as the commutativity of addition, the existence of a neutral element by addition - zero, and the existence of an opposite element. The consequence of the axioms of addition is that there are only one zero in the set of real numbers. The following axioms are introduced for the operation of multiplication: commutativity of multiplication, associativity of multiplication, the existence of a neutral element by multiplication - units, the existence of an inverse element. However, this view as practice shows is not enough. The concepts of limit and infinitely small quantities appeared. In a number of applications of mathematical analysis, an extended set of real numbers is used, also called the extended number axis, which is obtained by complementing the set of real numbers and an infinite point. Infinity is the limit of a sequence of positive numbers, increasing in absolute magnitude. Returning to the section of mathematics of the theory of numbers, we consider axioms and operations on real numbers. Expand the concept of a real number. We introduce several new additional axioms which, as will be shown below, will allow us not to use the theory of limits in our calculations. This technique can significantly simplify algebraic calculations. The technique of solving the old mathematical problem of dividing real numbers by zero is proposed. It is shown that the calculation results for this method coincide with the results obtained using the theory of limits. Examples of the application of this technique to ideal sources of voltage and current are given. The author does not claim to academic rigor [1, 2, 3] of the above method and relies rather on an intuitive engineering approach and utility in use.