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.
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.
. 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,
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 withwe 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 −1 so instead of dealing with it we will
“make it go away” so to speak by using the following definition.
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 −1 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,
i2 = −1
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
a
+ bi
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 a + bi is the complex
number a − bi . 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.
complex
number conjugate
12
– 5i 12 + 5i
1
– i 1
+ 5i
45i – 45i
101 101
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.
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.
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
discussion division of complex numbers.
(a +
bi)(a
− bi)
= a2 − abi
+ abi −
b2i2 = a2 + b2
So, when we
multiply a complex number by its conjugate we get a real number given by,
(a +
bi)(a
− bi)
= a2 + b2
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.
i1 = i i1 = i
i2 = – 1 i2 =
– 1
i3 = i · i2
= – i i3
= – i
i4 = (i2)2 = (–
1)2 = 1 i4
= 1
i5 = i · i4
= i i5
= i
i6 = i2 · i4 = (–
1)(1) = – 1 i6 =
– 1
i7 = i · i6 =
– i i7
= – i
i8 = (i4)2 = (1)2 = 1 i8 =
1
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,
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
= 3
there are in fact two numbers that we can square to get 9. We can square both 3
and – 3.
= 3
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 −9
= 3i . 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.
(3i)2 = 9i2
= −9
However, that is
not the only possibility. Consider the following,
(−3i)2 = (−3)2 i2
= 9i2 = −9
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.
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