Download Complex Numbers - Miami Dade College

MIAMI-DADE COLLEGE, HIALEAH CAMPUS
Project V-Coach, Title IV
Basic Study of Imaginary & Complex Numbers
Compiled and Edited by:
Jean N. Alcime, STEM Specialist & Math Professor
V-Coach Title IV & the Department of Mathematics
Also Edited by:
Ms. Ivonne Cruz, Administrative Supervisor
Mr. Jared Blanchette, Math Professor
Definition of Imaginary numbers
Complex numbers are fundamentally based in the understanding of imaginary numbers.
To fully comprehend complex numbers, you need to thoroughly grasp the concept of
imaginary numbers.
You might be wondering why you cannot take the square root of a negative number. In
fact, you have been told that the square root of any negative number, say x, is not real.
Applying the definition of the multiplication of sign, a number multiplies by itself
produces a positive result; whether the number is positive or negative.
Furthermore, when multiplying two numbers with the same sign, the result will always
be positive. The rationale of this factual/proven definition is true in the basis of the
definition of square root itself. With that in mind, you can truly say that the radicand (a
number or quantity from which a root is to be extracted) of a square root must be
positive in order for the root to be real.
For example: √
√
Radicand
Notice from the above example, 5 is the squares root of 25 where
is the
radicand. It should now be clear that the result of the square of a number is positive.
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Now let’s try to take the square root of negative radicand, say √
. As previously
mentioned when 5 or -5 is squared, the result is 25. That proves the fact that the
radicand cannot be negative. However; we perceive the square root of a negative
number as imaginary. This leads to the following definition:
√
The next example demonstrates how to work with a square root containing a negative
radicand.
Example of square root with negative radicand:
√(
√
The result
)
√
√
is an imaginary number, where indicates imaginary.
The following shows the guidelines when working with imaginary numbers:
√
(√
Definition of
)
))
((
(√
)
Square both sides of the above equation.
From above, multiply by -1
From the definition of
we multiply -1 by -1
Simplifying imaginary numbers for n≥4
When simplifying an imaginary number, do the following:
a. If
where is a positive integer and indicates imaginary
number ( ), Divide
b.
c.
d.
e.
If the remainder is 0, then the answer is 1.
If the remainder is 1, then the answer is .
If the remainder is 2, then answer is -1
If the remainder is 3, then the answer -
Example a:
Solution
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, we perform the division , the remainder is 3; therefore the answer – .
Since
Example b:
Again we divide 57 by 4; the remainder is 1; therefore, the answer is
Based on the examples above, the quotient of the fraction has no significance in terms
of obtaining the result. In fact, we only care about the remainder. The remainder should
always be between 0 and 3 where 0 and 3 are included. The remainder always tells you
the answer.
Now consider example b, after we divided 57 by 4, we found the quotient to be 14 with
a remainder of 0. Since the remainder is 0, it follows that the answer is 1.
Example c:
Solution:
. The quotient is 13 and the remainder is 0; therefore
.
Working with Complex Number
A number written in the form
part and
is the imaginary part.
is called a complex number where
is the real
Having a complex number written in the form
is said to be in the standard form.
You might have studied different types of numbers and their definitions in a previous
class or prior to reading this booklet. Recall that the set of Rational (natural, whole,
integer) and irrational numbers make up the set of the real numbers.
Adding and subtracting complex numbers
Complex numbers can be added and subtracted. When adding complex numbers you
add the real parts and the imaginary parts separately.
Example 1: (
)
(
)
Solution:
(
)
(
)
(
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)
(
)
Eliminate the parenthesis and
combine like terms
Example 2: (
)
(
)
Solution
(
)
(
)
(
Example 3: (
)
(
)
(
)
)
Solution
(
)
(
)
(
)
(
)
Multiplying Complex Numbers
Complex numbers are multiplied just like you would if you were to multiply a polynomial,
with the exception that
.
Example 4:
(
)
Solution
Applying the distributive property, we end up with the following:
(
)
(
)
Notice that before getting to the final answer, the complex number is written in
descending power, and the final answer is written in the standard form.
Example 5: (
)(11+6i)
Solution
Here we are applying the FOIL method
(
)(
)
( ) (
)
( ) ( )
(
( ) (
)
( ) ( )
)
As we do when we rationalize radical expressions, we apply the similar method with
complex numbers. That is; complex numbers, say
and
are conjugates.
That is the conjugate of a complex number
is
.
Definition of Product of Conjugates
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(
)(
)
; When we apply the FOIL method, we find that the product of
a complex number and its conjugate is the sum of the real numbers
.
Example 6: (
)(
)
Solution
(
)(
)
( ) ( )
( ) (
(
)
)
( ) ( )
( ) (
)
=13
)(
)
Example 6 demonstrates the fact that(
. Thus, the product of a
complex number and its conjugate is the sum of the square of the real part and the
square of the coefficient of the imaginary part.
Dividing complex numbers
Example 7: Write in standard form:
Solution
To write in the standard form, you need to multiply the numerator and the denominator
by the conjugate of the denominator. The denominator should always be a real
number.
Notice that in the above example the fraction was separated into two fractions so that
the result would be in standard form.
Example 8: Write in standard form:
Solution
(
)
(
)
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(
)
(
)
Quadratic equations involving complex numbers:
Example 9 solve for x:
Solution:
Re-write the quadratic equation in the standard form
√(
( ) ( ) (
)
applying the quadratic formula:
√
√
(
)
Factor by the greatest common divisor (6)
The 6 is cancelled out, we left with
The answer is *
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√
+