Imaginary and complex numbers
Imaginary numbers. Imaginary unit. Imaginary roots.
Real numbers. Complex numbers.
Consider the pure quadratic equation:x 2 = a ,
where a – a known value. Its
solution may be presented as:
Here the three cases are possible:
||If a = 0 , then x = 0.
|| If a is a positive number, then its square root has two values: one positive and one negative; for example, the equation
x2 = 25 has the two roots: 5 and –5.
This is often written as the root with double sign before:
||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 non-negative number (think
over this!). But, if we wish to receive solutions of the equation x2 = 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 equationx2 = – 25
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.