Complex numbers

The complex numbers are those that form a group of digits resulting from the addition being performed between a real number and an imaginary number . It is important to know that the real number is the number that can be expressed by means of a whole number , for example, 5, 28, 21; and the imaginary number is the number whose square is in negative form. They are represented by two numbers that are enclosed in parentheses (x and y).

What are complex numbers

They are composed of the entire extension of the real numbers that make up the minimum algebraically closed body , this means that they are formed by all those numbers that can be expressed by means of the integers . The real numbers also include all the numbers known by the name of complex numbers which include all the roots of the polynomials .

What are complex numbers for?

Real numbers are unable to cover all the roots of the set of negative numbers , a characteristic that complex numbers can do. This particularity allows complex numbers to be used in different fields of mathematics , engineering, and mathematical physics . This, because they have the ability to represent the electric current and the different electromagnetic waves . They are widely used in electronics and also in the field of telecommunications . They are used for different algebraic works, in pure mathematics, in solvingdifferential equations , in the branch of aerodynamics , hydrodynamics and electromagnetism . They are essential in quantum mechanics .

features

Among the main characteristics that complex numbers possess, we can mention the following:

• In mathematics they constitute a body .
• They are considered as points in the complex plane .
• Contains real numbers and imaginary numbers
• The imaginary unit of complex numbers is recognized by the letter i .
• They are represented by the letter
• The first component is represented by the letter a , and belongs to the real part , the second component is represented by the letter b , and corresponds to the imaginary part .
• They are not able to maintain an order like the real numbers .

History

The earliest notion of people trying to use imaginary numbers dates back to the 1st century. The first scholar who made the first concepts of complex numbers was Heron of Alexandria , and he began before the difficulties that arose when he tried to build a pyramid . Once negative numbers were “made up,” mathematicians tried to find a number that, squared, could be equal to a negative one . Not finding an answer, they gave up. In 1500, speculations about the square roots of negative numbers were re-formulated . The formulas for solving polynomial equations were discovered at that time3rd and 4th graders, 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 of imaginary number in the seventeenth century 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 on the abscissa axis and the imaginary part must be placed on 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 complex number affix. Every complex number can be represented as a vector OP, where O is the origin of the coordinates and P is the affix of the complex.

Properties

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:

• Closing or closing property for the sum:  For z1, z2 ∈ C we have that z1 + z2∈C
• Commutative property :  For any z1, z2 ∈ C it is true that: z1 + z2 = z2 + z1
• Associative property :  For any z1, z2, z3∈C it is true that: (z1 + z2) + z3 = z1 + (z2 + z3)
• Existence of the neutral element for the sum:  0 + 0i, abbreviated by 0, is the neutral element for the sum.
• Existence of the additive or opposite inverse:  Every complex number z has a unique additive inverse, denoted by −z.

Multiplication properties

We define the product of two complex numbers z1 = a + bi and z2 = c + di as

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

Among the properties of multiplication we have the following:

• Closing property for multiplication:  For z1, z2∈C we have that z1⋅ z2 ∈ C
• Commutative property:  For any z1, z2 ∈ C it is true that: z1⋅z2 = z2⋅z1
• Associative property:  For any z1, z2, z3∈C it is true that: (z1⋅z2) ⋅z3 = z1⋅ (z2⋅z3)
• Existence of the neutral element for multiplication:  1 + 0i, abbreviated by 1, is the neutral element for multiplication.
• Existence of the multiplicative or reciprocal inverse:  Every complex number z, other than 0, has a single multiplicative inverse, denoted by z − 1
• Distributive property :  For any z1, z2, z3∈C it is true that: z1⋅ (z2 + z3) = z1⋅z2 + z1⋅z3

Operations

The operations that can be performed using complex numbers are the following:

• Add complex numbers.
• Subtract complex numbers.
• Multiply complex numbers.
• Find conjugates of complex numbers
• Divide complex numbers.

Examples

Sum:

(-3 + 3i) + (7 – 2i) = 4 + i

Subtraction

(5 + 3i) – (3 – i) = 2 + 4i