Let's consider the equation** x² - 2x + 5 = 0**:

We know that the number does not belong to the set of real numbers, as there is no number that squared **-1**. For the above equation to have a solution, we have to extend the set of real numbers to get a new set, called **set of complex numbers** and represented by** **.

The number was called the imaginary unit and created the number **i**, so that:

i² = -1 |

Soon,

i = |

So the solutions of the equation **x² - 2x + 5 = 0** in are **1 - 2i** and **1 + 2i**.

## Algebraic form of a complex number

Every complex number **z** can be written in the form:

z = a + bi, with The, B |

This form is called the algebraic form of the complex number. Note that a complex number in this format has two parts:

We indicate:**Re (z) = aIm (z) = b**

**Examples**

**z = 3 + 5i****Re (z) = 3**and**Im (z) = 5****z = -7 + 18i****Re (z) = -7**and**Im (z) = 18****z = 53 - 25i****Re (z) = 53**and**Im (z) = -25**

- If the real part of the complex number is null then the number is
**pure imaginary**.

**Example: z = 3i ** **Re (z) = 0 **and** Im (z) = 3**

**Example**

Determine the value of **k** so the complex number **z = (k - 4) + 3i** be pure imaginary:

*Resolution*

For the number to be pure imaginary, the real part must be null:

k - 4 = 0 **k = 4**

- If the imaginary part of the complex number is null then the number is
**real**.

**Example: z = 10** **Re (z) = 10** and **Im (z) = 0**

**Example**

Determine the value of **k** so the complex number **z = 3+ (k² - 4) i** be a real number:

*Resolution*

For the number to be real, the imaginary part must be null:

k² - 4 = 0 k² = 4 **k = -2** or **k = 2**

We can associate any complex number **z = a + bi** to a point on the Argand-Gauss plane. On the abscissa axis (real axis), the real part is represented, and on the ordinate axis (imaginary axis), the immigration part of the complex number. The point **P** it's the **affix** or **geometric image of** **z**.

**Example**

Represent in the Argand-Gauss plane the complex numbers:

*Resolution*

Each complex will be a point on the plane whose abscissa is the real part and the ordinate is the imaginary part:

Note that real numbers are located on the real axis, just as pure imaginary numbers are on the imaginary axis.

**Note:** The order relationship field is not defined for the field of complex numbers, that is, there is no complex number larger or smaller than another.