Imaginary and complex numbers
Real numbers. Complex numbers.
Consider the pure quadratic equation:
where a – a known value. Its solution may be presented as:
Here the three cases are possible:
1). 
If
a =
0 , then
x =
0.

2). 
If
a
is a positive number, then its square root has two values: one positive and one negative; for example, the equation
x
^{
2
}
= 25 has the two roots: 5 and –5.
This is often written as the root with double sign before: 
3). 
If
a –
a negative number,
then the equation has no solution among known us positive and negative numbers,
because the second power of any number is a
nonnegative
number (think
over this!). But, if we wish to receive solutions of the equation
x
^{
2
}
=
a
also
at negative values
of
a
,
we are obliged to introduce the new kind numbers –
imaginary numbers.
So, a number is
imaginary
,
if its second power is a negative number.
According to this definition of imaginary numbers we can define an
imaginary unit
as:

Then, for the equation
we receive the two imaginary roots: Substituting both these roots into our equation we’ll receive the identity. Check it, please!
In contrast to imaginary numbers all the rest numbers (positive and negative, integers and fractional, rational and irrational ones) are called real numbers. A sum of a real and an imaginary number is called a complex number , and marked as:
a + b i ,
where a, b – real numbers, i – an imaginary unit.
In more details about complex numbers see the section “Complex numbers” .
E x a m p l e s of complex numbers: 3 + 4 i , 7 – 13.6 i , 0 + 25 i = 25 i , 2 + i .