Mathematics

Complex numbers

Complex numbers are those that form a group of digits that result from the addition made between a real number and an imaginary number. It is important to know that the real number is the number that can be expressed through a whole number, for example, 5, 28, 21; and the imaginary number is the number whose square is presented negatively. They are represented by two numbers placed in parentheses (x and y).

What are complex numbers?

Complex numbers are entities of the branch of mathematics that are represented through a pair of real numbers, the first one that is called x and represents the real part, and the second one, called y, that represents the imaginary part.

Which are the complex numbers ?

They are numbers composed by all the extension of real numbers that conform the minimum algebraically closed body, this means that they are formed by all those numbers that can be expressed through the whole numbers. Real numbers also include all the numbers known as complex numbers, which include all the polynomial roots.

What are complex numbers for?

Real numbers are incapable of encompassing all the roots of the set of negative numbers, a characteristic that can be performed by complex numbers. This particularity allows complex numbers to be used in different fields of mathematics, engineering and mathematical physics. This is because they have the ability to represent electric current and different electromagnetic waves. They are widely used in electronics and also in telecommunications.

They are used for different algebraic works, in pure mathematics, in the solution of differential equations, in the branch of aerodynamics, hydrodynamics and in electromagnetism. They are essential in quantum mechanics.

Characteristics

Among the main characteristics that complex numbers have; we can mention the following:

History

The first notion of people trying to use imaginary numbers, dates back to the first century. The first scholar who made the first concepts of complex numbers was Heron of Alexandria, and he started facing the difficulties that arose when he tried to build a pyramid.

Once negative numbers were “invented,” mathematicians tried to find a number that, squared, could be equal to a negative one. Not finding an answer, they gave up. In the year 1500, speculations were again made about the square roots of negative numbers.

Formulas for solving polynomial equations of 3° and 4° degree were discovered at that time, and it was concluded that some work with square roots of negative numbers would be required. In 1545, the first great work with imaginary numbers was produced. Descartes, an important philosopher, mathematician and physicist, was the one who created the term imaginary number in the XVII silo and many years later, the concept of complex number would be formed.

How complex numbers are represented?

Complex numbers can be represented in the complex plane. The real part of the complex is represented in the abscissa axis and the imaginary part must be placed in the ordinate axis. In the complex plane, each complex number z = a + bi is assigned the coordinate point P (a, b), which is called the affix of the complex number. Any complex number can be represented as a vector OP, being O the origin of coordinates and P the affix of the complex.

Properties

The complete numbers have different properties, which are detailed below.

Transitive property

If z1=z2 and z2=z3 then z1=z3

Properties of the sum

The sum of two complex numbers z1=a + bi and z2=c + di is defined as

(a + bi) + (c + di) = (a + c) + (b + d) i

Among the properties of the sum we have the following:

Multiplication properties

The product of two complex numbers z1= a + bi and z2= c + di is defined as

(a + bi)⋅(c + di)=(ab – bd)+(ad + bc) i

Among the properties of multiplication we have the following:

Operations

The operations that can be performed using complex numbers are as follows:

Examples

Addition:

(-3 + 3i) + (7 – 2i) = 4 + i

Subtraction

(5 + 3i) – (3 – i) = 2 + 4i

Written by Gabriela Briceño V.
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