Download Imaginary and Complex Numbers

Imaginary Numbers!!! What, numbers may not be real????
We have learned about the real number system. Real numbers are
rational and irrational. A rational number is an integer, a
terminating or repeating decimal, or a ratio of two numbers. An
irrational number is a non-terminating, non-repeating decimal. An
example of an irrational number is  . No, no, not apple or cherry,
but the ratio of the circumference of a circle to its diameter. Man,
and more recently woman, has tried for centuries to determine if
 would repeat or terminate. No luck so far.
But what do we do with all of the other numbers?  4 has no
solution. OR, does it??? Dun, dun, dunnnnnnnn!!!!
A new system of numbers was developed to work with these
numbers. The system was developed by a great mathematician:
Ozzy Osbourne, who used imaginary numbers as a prelude to the
song Crazy Train. (Just kidding! Seriously, just kidding!!!)
Imaginary numbers are used to represent solutions to problems
where no solution exists, kind of like the way Kirk was able to
overcome the kobayashi maru no-win scenario on Star Trek.
Imaginary numbers have been around for a very long time.  4
does have a solution! It is  2i . i is used to represent imaginary
numbers. In fact, every number can be written using imaginary
numbers. A complex number can be written in the form a  bi .
So, any integer can be expressed as a complex number. The
integer 5 can be written as 5  0i .
Imaginary numbers are arranged on the basis of 4.
i0 1
i1  i
i 2  1
i 3  i
This is a recursive pattern that continues forever……..
i 75  i 3  i HOW DID THIS HAPPEN????????
75
Divide the exponent by 4. So
 18 R3 . So, we equate i 75 to i 3
4
, which, in turn, is -1. This works for every exponent. Divide by
4, check the remainder, and equate to that value. If there is no
remainder, then we have i 0 , which, as everyone knows, is 1.
We add and subtract imaginary numbers the same way we add and
subtract polynomials, we collect like terms. 7  2i  3  5i can be
simplified to 4  7i .
Multiply complex numbers the same way that we multiply
binomials together. Really, we perform the operation the same
way. The outcome, however, is different.
(5  3i)(4  6i)
20  30i  12i  18i 2
20  18i  18i 2
Hold on, we are not done!! Remember that i 2  1 .
So…
20  18i  18i 2
20  18i  (18)
20  18i  18
38  18i
Now, we are finished with the problem.
Let’s try another one:
(7  4i)(3  5i)
Good job!!
Hold on, wait a second, division is in the house!
We can’t have an imaginary number in our denominator. It is a
major no-no. So, we must multiply the complex denominator by
its, wait for it, CONJUGATE!!!!!!!
What is a conjugate you may ask?(Go ahead, ask)
A conjugate is occurs when you reverse the sign on the imaginary
part of a complex number. The product of a complex number and
its conjugate will produce a real number. Therefore, we can call
a  bi and a  bi complex conjugates.
An example: the conjugate of 5  3i is 5  3i .
Another example: the conjugate of  7  2i is  7  2i .
Let’s try to simplify the following expression:
4  3i
2  5i
First, we need to determine the conjugate of the denominator:
2  5i .
Now, we rationalize the denominator in the following manner:
4  3i 2  5i

2  5i 2  5i
8  20i  6i  15i 2
4  25i 2
8  26i  15
4  25
 7  26i
29
What happened to the two middle terms in the denominator? Why
does this happen?
Try:
7  5i
3  4i
Hey, what about determining the reciprocal of a complex number?
It is possible, you know.
Determine the reciprocal of 8  5i .
What is the conjugate of 8  5i ? Of course, it is 8  5i .
8  5i
1
8  5i
1
8  5i

8  5i 8  5i
8  5i
64  25i 2
8  5i
64  25
8  5i
89
Try finding the reciprocal of  7  6i
Hey, here is a crazy idea. Can we determine the absolute value of
a complex number? What do you think? Is it possible?
ANYTHING is possible.
In order to determine the absolute value of a complex number, we
use the formula: a  bi  a 2  b 2 .
WOW!
What is the absolute value of 8  5i ?
8  5i
(8) 2  (5) 2
64  25
89
Try to determine the absolute value of  2  2i .
We can even graph a complex number on the complex plane!!
Could this get any better? Amazing!!
The way we do this is we let the x-axis represent the Real axis and
the y-axis represent the Imaginary axis. So, to plot 8  5i , we
would go 8 to the right and 5 down. Fantastic!!
Well, if we can plot one point, we surely can plot two, or more
points. And, if we plot two points, then we can determine the
distance between the points.
On a graph, plot 3  2i and 6  2i . Determine their horizontal
distance between the points and their vertical distance between the
points. The horizontal distance is 3, 6-3, and their vertical distance
is 4, 2i-(-2i). Now, use the Pythagorean Theorem to determine the
distance: 32  4 2  c 2 c = 5
How about finding the midpoint between the two points? We take
the average of the real numbers and the average of the imaginary
3  6 2i  2i 
9 
,
numbers: 
 . Therefore the midpoint is  ,0  .
 2
2

2

Determine the distance and the midpoint between 6  5i and  4  8i