Notesale: Turn your study into money

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Preface
Here are my online notes for my Algebra course that I teach here at Lamar University, although I
have to admit that it’s been years since I last taught this course
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Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to
learn Algebra or needing a refresher for algebra
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However, they do assume that you’ve had some
exposure to the basics of algebra at some point prior to this
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Here are a couple of warnings to my students who may be here to get a copy of what happened on
a day that you missed
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Because I wanted to make this a fairly complete set of notes for anyone wanting to learn
algebra I have included some material that I do not usually have time to cover in class
and because this changes from semester to semester it is not noted here
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2
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Likewise, even
if I do work some of the problems in here I may work fewer problems in class than are
presented here
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Sometimes questions in class will lead down paths that are not covered here
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Sometimes a very good question gets asked in class
that leads to insights that I’ve not included here
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4
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THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!
Using these notes as a substitute for class is liable to get you in trouble
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Complex Numbers
The last topic in this section is not really related to most of what we’ve done in this chapter,
although it is somewhat related to the radicals section as we will see
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In the radicals section we noted that we won’t get a real number out of a square root of a negative
number
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Now we also saw that if a and b were both positive then
forget that restriction and do the following
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For a second let’s

9 −1 = 3 −1

Now, −1 is not a real number, but if you think about it we can do this for any square root of a
negative number
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So, even if the number isn’t a perfect square we can still always reduce the square root of a
negative number down to the square root of a positive number (which we or a calculator can deal
with) times −1
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Well the reality is that, at this level, there just isn’t any way to deal with −1 so
So, if we just had a way to deal with

instead of dealing with it we will “make it go away” so to speak by using the following definition
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This shows that, in some way, i is the only
“number” that we can square and get a negative value
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The natural question at this point is probably just why do we care about this? The answer is that,
as we will see in the next chapter, sometimes we will run across the square roots of negative

numbers and we’re going to need a way to deal with them
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So, let’s start out with some of the basic definitions and terminology for complex numbers
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When in the standard form a is called the real part of the
complex number and b is called the imaginary part of the complex number
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3 + 5i

6 − 10i

4
+i
5

16i

113

The last two probably need a little more explanation
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When the real part is zero we often will call the
complex number a purely imaginary number
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So, thinking of numbers in this light we can see that the
real numbers are simply a subset of the complex numbers
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In other words, it
is the original complex number with the sign on the imaginary part changed
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complex number
1
3+ i
2
12 − 5i
1− i
45i

conjugate
1
3− i
2
12 + 5i
1+ i
−45i

101

101

Notice that the conjugate of a real number is just itself with no changes
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We’ll start with addition and
subtraction
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Example 1 Perform the indicated operation and write the answers in standard form
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Note that the parentheses on the
first terms are only there to indicate that we’re thinking of that term as a complex number and in

general aren’t used
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Again, with one small difference, it’s probably easiest to
just think of the complex numbers as polynomials so multiply them out as you would
polynomials
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Example 2 Multiply each of the following and write the answers in standard form
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7i ( −5 + 2i ) =−35i + 14i 2

Now, this is where the small difference mentioned earlier comes into play
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The standard form for complex numbers does not have an i2 in it
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[Return to Problems]

(b) In this case we will FOIL the two numbers and we’ll need to also remember to get rid of the
i2
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8
8
5
( 4 + i )( 2 + 3i ) = + 12i + 2i + 3i 2 = + 14i + 3 ( −1) = + 14i
[Return to Problems]

(d) Here’s one final multiplication that will lead us into the next topic
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That can
and will happen on occasion
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There is a nice
general formula for this that will be convenient when it comes to discussing division of complex
numbers
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Example 3 Write each of the following in standard form
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The main idea
here however is that we want to write them in standard form
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So, we need to get the i's out of the denominator
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So, if we multiply the numerator and denominator by the conjugate of the denominator
we will be able to eliminate the i from the denominator
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(a)

( 3 − i ) ( 2 − 7i) =− 23i + 7i 2 = − 23i = 1 − 23 i
3−i
6
−1
=
−
2
2
2 + 7i ( 2 + 7i ) ( 2 − 7 i )
2 +7
53
53 53

Notice that to officially put the answer in standard form we broke up the fraction into the real and
imaginary parts
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It can be done in the same manner as the previous ones, but there is a slightly easier way
to do the problem
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6 − 9i 6 9i 3 9
= − = −
2i
2i 2i i 2

Now, we want the i out of the denominator and since there is only an i in the denominator of the
first term we will simply multiply the numerator and denominator of the first term by an i
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Let’s just take a look at what happens
when we start looking at various powers of i
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This can be a convenient fact to remember
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From the
section on radicals we know that we can do the following
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It turns out that we can actually do the same thing if one of the numbers is negative
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6 = 36 = ( −4 )( −9 ) ≠ −4 −9 = 2i )( 3i ) = i 2 = 6
6
−
(
We can summarize this up as a set of rules
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Example 4 Multiply the following and write the answer in standard form
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When
faced with them the first thing that you should always do is convert them to complex number
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So, let’s work this problem the way it should be worked
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When faced
with square roots of negative numbers the first thing that you should do is convert them to
complex numbers
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As we noted back in
the section on radicals even though 9 = 3 there are in fact two numbers that we can square to
get 9
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The same will hold for square roots of negative numbers
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As with
3i
square roots of positive numbers in this case we are really asking what did we square to get -9?
Well it’s easy enough to check that 3i is correct
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Consider the following,

( −3i )

2

= ) i 2 ==
9i 2 −9
( −3
2

and so if we square -3i we will also get -9
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However,
we will ALWAYS take the positive number for the value of the square root just as we do with the
square root of positive numbers
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math
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edu/terms

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