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Complex numbers


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) = a
Im (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.

Next: Equality of Complex Numbers