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|
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:
Re (z) = a
Im (z) = b
- 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
Determine the value of k so the complex number z = (k - 4) + 3i be pure imaginary:
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
Determine the value of k so the complex number z = 3+ (k² - 4) i be a real number:
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.
Represent in the Argand-Gauss plane the complex numbers:
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