Complex numbers

Sometimes we find linear equations that have no solution in the set of real numbers, such as:

x² + 4 = 0

x² = - 4

x = √ (-4)

If we define the expression "√ (-1)" as "I", such that i = √ (-1), we have:

x = √ (-4) = √4√ (-1) = 2i

This type of number is called imaginary numbers. Therefore, with this type of number, we can obtain the solutions of equations that previously “did not have it”.

Example: Solve the equation x² + 2x + 5 = 0

x = [- b ± √ (b² - 4ac)] / 2a

x = [- 2 ± √ (2² - 4 · 1 · 5)] / 2 (1) =

x = [- 2 ± √ (4 - 20)] / 2 =

x = [- 2 ±
√ (- 16)] / 2 =

x = [- 2 ± 4i] / 2

From here we get:

  • x1 = (- 2 + 4i) / 2 = -1 + 2i
  • x2 = (- 2 - 4i) / 2 = -1 - 2i

Let us observe that both solutions are made up of two numbers: one real and the other imaginary. This pair is called complex numbers.

Definition

Complex numbersA complex number is a pair order of numbers (real, imaginary). These can be represented with the letter z, as follows:

z = (a, b)

Where the first component is called real part and it is written  Re; the second is called imaginary part and it is written Im.

  • a is the real part Re (z).
  • b is the imaginary part Im (z).

Complex numbers in binomial form

To the number a + bi we call it complex number in binomial form, where a is the real part y b is the imaginary part.

  • Yes b = 0, the complex number is reduced to a real one since a + 0i = a.
  • Yes a = 0, the complex number is reduced to bi and it is said to be a pure imaginary number.

Conjugated complex numbers

Complex numbers z = a + bi y z = a - bi called conjugated.

Example: z1 = (4, -5) is the conjugate of z2 = (4, 5).

 Opposite complex numbers

Complex numbers a + bi y - (a + bi) called opposites.

Example: z3 = 5 + 7i is the opposite az4 = - 5 - 7i.

 Equality of complex numbers

Two complex numbers are the same when they have the same real component and the same imaginary component.

The set of all complex numbers is designated by C:

C = {a + bi / a, b ∈ R}

Complex numbers2

Graphic representation

Let the complex number (a, b) be:

Representation of complex numbers1. We draw rectangular coordinate axes and place the component of each complex number in it:

  • The first component (which is the real part) is located on the horizontal axis (called real axis).
  • The second component (which is the imaginary part) is located on the vertical axis (called imaginary axis).

2. The point of intersection of the parallels to the axes by said components is the image of the complex number.

Example: Represent the following complex number: z = - 4, 2.

Representation of complex numbers1