Introduction of Complex Numbers
By S Taylor
Many times when solving problems in algebra we obtain answers like –2,
,
,
and so on. All of these are considered real numbers.
However, not all problems can be solved with a real number answer. There are some occasions when we have to work with a different type of number called a complex number. A complex number takes on the form
where
is called the real part and
is called the imaginary part. Since we’ve already listed some examples of real numbers, let’s take a look at the imaginary part.
When we have a situation like
we have to begin dealing with imaginary numbers. We define an imaginary number
to be
.
So whenever a negative number appears in a square root, we can take the square root of that negative and call it
.
Then we are left with just
.
So our answer is
.
This is where the imaginary part of a complex number comes from.
,
,
and so on. All of these are considered real numbers.However, not all problems can be solved with a real number answer. There are some occasions when we have to work with a different type of number called a complex number. A complex number takes on the form
where
is called the real part and
is called the imaginary part. Since we’ve already listed some examples of real numbers, let’s take a look at the imaginary part.
When we have a situation like
we have to begin dealing with imaginary numbers. We define an imaginary number
to be
.
So whenever a negative number appears in a square root, we can take the square root of that negative and call it
.
Then we are left with just
.
So our answer is
.
This is where the imaginary part of a complex number comes from.1.
#1.
=
=
2.
#2.
=
=
3.
#3.
=
=
An interesting thing happens if we have
.
Since
then
.
This will be an important tool as you deal more with complex numbers. Other important facts are:
Since
and
,
raised to any power that is a multiple of 4 will also be equal to 1. In other words,
.
So if I have a problem that asks me to evaluate
,
I can rewrite it as
which will give me
.
Notice I chose to break up 38 into 36 and 2 because 36 is a multiple of 4.
4.
#4.
5.
#5.
6.
#6.
7.
#7.
If you work through additional problems like this you’ll find that when you are raising
to a power, the answer will always be
,
-1,
,
or 1.
Whenever you see a real number written simply as 4, this implies that there is not an imaginary part to this complex number.
Also, if you see an imaginary number such as the
we worked with earlier, it implies that there is no real part to the complex number.
In other words, it is possible to have a complex number that is solely a real number or solely an imaginary number. The two do not always have to appear together in order for the number to be considered a complex number.
Our need for complex numbers arises most often when working with quadratic equations and the quadratic formula. When solving a quadratic equation using the quadratic formula you will often see complex numbers as your solution. If you go to the lesson on the quadratic formula you will see an example of the use of complex numbers.
to a power, the answer will always be
,
-1,
,
or 1.
Whenever you see a real number written simply as 4, this implies that there is not an imaginary part to this complex number.
Also, if you see an imaginary number such as the
we worked with earlier, it implies that there is no real part to the complex number.
In other words, it is possible to have a complex number that is solely a real number or solely an imaginary number. The two do not always have to appear together in order for the number to be considered a complex number.
Our need for complex numbers arises most often when working with quadratic equations and the quadratic formula. When solving a quadratic equation using the quadratic formula you will often see complex numbers as your solution. If you go to the lesson on the quadratic formula you will see an example of the use of complex numbers.