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:

- x
_{1}= (- 2 + 4i) / 2 = -1 + 2i - x
_{2}= (- 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**

A 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**,**a + 0i = a**. - Yes
**a = 0**,**bi**and it is said to be a pure imaginary number.

**Conjugated complex numbers**

Complex numbers **z = a + b****i** y **z**** ****= a - b****i** called **conjugated**.

**Example**: z_{1} = (4, -5) is the conjugate of z_{2} = (4, 5).

** ****Opposite complex numbers**

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

**Example**: z_{3} = 5 + 7i is the opposite az_{4} = - 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**}*

**Graphic representation**

Let the complex number (a, b) be:

1. 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.