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. We also
won’t need the material here all that often in the remainder of this course,
but there are a couple of sections in which we will need this and so it’s best
to get it out of the way at this point.
In the radicals section we noted that we won’t get a real
number out of a square root of a negative number. For instance 
isn’t a real number since there is no real
number that we can square and get a NEGATIVE 9.
Now we also saw that if a
and b were both positive then . For a second let’s forget that restriction
and do the following.
Now, is not a real number, but if you think about
it we can do this for any square root of a negative number. For instance,
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 .
So, if we just had a way to deal with we could actually deal with square roots of
negative numbers. Well the reality is
that, at this level, there just isn’t any way to deal with so instead of dealing with it we will “make it
go away” so to speak by using the following definition.
Note that if we square both sides of this we get,
It will be important to remember this later on. This shows that, in some way, i is the only “number” that we can
square and get a negative value.
Using this definition all the square roots above become,
These are all examples of complex numbers.
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. So, to deal with them we will need to discuss
complex numbers.
So, let’s start out with some of the basic definitions and
terminology for complex numbers. The standard form of a complex number is
where a and b are real numbers and they can be
anything, positive, negative, zero, integers, fractions, decimals, it doesn’t
matter. 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.
Here are some examples of complex numbers.
The last two probably need a little more explanation. It is completely possible that a or b
could be zero and so in 16i the real
part is zero. When the real part is zero
we often will call the complex number a purely
imaginary number. In the last
example (113) the imaginary part is zero and we actually have a real
number. So, thinking of numbers in this
light we can see that the real numbers are simply a subset of the complex
numbers.
The conjugate of
the complex number is the complex number . In other words, it is the original complex
number with the sign on the imaginary part changed. Here are some examples of complex numbers and
their conjugates.
Notice that the conjugate of a real number is just itself
with no changes.
Now we need to discuss the basic operations for complex
numbers. We’ll start with addition and
subtraction. The easiest way to think of
adding and/or subtracting complex numbers is to think of each complex number as
a polynomial and do the addition and subtraction in the same way that we add or
subtract polynomials.
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Example 1 Perform
the indicated operation and write the answers in standard form.
(a)
(b)
(c)
Solution
There really isn’t much to do here other than add or
subtract. 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.
(a)
(b)
(c)
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Next let’s take a look at multiplication. 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.
The one difference will come in the final step as we’ll see.
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Example 2 Multiply
each of the following and write the answers in standard form.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
(a) So all that we need to do is distribute the 7i through the parenthesis.
Now, this is where the small difference mentioned earlier
comes into play. This number is NOT in
standard form. The standard form for
complex numbers does not have an i2
in it. This however is not a problem
provided we recall that
Using this we get,
We also rearranged the order so that the real part is
listed first.
[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.
[Return to Problems]
(c) Same thing with this one.
[Return to Problems]
(d) Here’s one final multiplication that will lead us into the
next topic.
Don’t get excited about it when the product of two complex
numbers is a real number. That can and
will happen on occasion.
[Return to Problems]
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In the final part of the previous example we multiplied a
number by its conjugate. There is a nice
general formula for this that will be convenient when it comes to discussing
division of complex numbers.
So, when we multiply a complex number by its conjugate we
get a real number given by,
Now, we gave this formula with the comment that it will be convenient
when it came to dividing complex numbers so let’s look at a couple of examples.
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Example 3 Write
each of the following in standard form.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
So, in each case we are really looking at the division of
two complex numbers. The main idea
here however is that we want to write them in standard form. Standard form does not allow for any i's to be in the denominator. So, we need to get the i's out of the denominator.
This is actually fairly simple if we recall that a complex
number times its conjugate is a real number.
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.
Now that we’ve figured out how to do these let’s go ahead
and work the problems.
(a)
Notice that to officially put the answer in standard form
we broke up the fraction into the real and imaginary parts.
[Return to Problems]
(b)
[Return to Problems]
(c)
[Return to Problems]
(d) This one is a little different from the previous ones since
the denominator is a pure imaginary number.
It can be done in the same manner as the previous ones, but there is a
slightly easier way to do the problem.
First, break up the fraction as follows.
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.
[Return to Problems]
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The next topic that we want to discuss here is powers of i.
Let’s just take a look at what happens when we start looking at various
powers of i.
Can you see the pattern?
All powers if i can be reduced
down to one of four possible answers and they repeat every four powers. This can be a convenient fact to remember.
We next need to address an issue on dealing with square
roots of negative numbers. From the
section on radicals we know that we can do the following.
In other words, we can break up products under a square root
into a product of square roots provided both numbers are positive.
It turns out that we can actually do the same thing if one of the numbers is negative. For instance,
However, if BOTH numbers are negative this won’t work
anymore as the following shows.
We can summarize this up as a set of rules. If a
and b are both positive numbers then,
Why is this important enough to worry about? Consider the following example.
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Example 4 Multiply
the following and write the answer in standard form.
Solution
If we where to multiply this out in its present form we
would get,
Now, if we were not being careful we would probably
combine the two roots in the final term into one which can’t be done!
So, there is a general rule of thumb in dealing with
square roots of negative numbers. When
faced with them the first thing that you should always do is convert them to
complex number. If we follow this rule
we will always get the correct answer.
So, let’s work this problem the way it should be worked.
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The rule of thumb given in the previous example is important
enough to make again. When faced with
square roots of negative numbers the first thing that you should do is convert
them to complex numbers.
There is one final topic that we need to touch on before
leaving this section. As we noted back
in the section on radicals even though there are in fact two numbers that we can
square to get 9. We can square both 3
and -3.
The same will hold for square roots of negative
numbers. As we saw earlier . As with 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.
However, that is not the only possibility. Consider the following,
and so if we square -3i
we will also get -9. So, when taking the
square root of a negative number there are really two numbers that we can
square to get the number under the radical.
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.