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Powers and Roots
In this section we’re going to take a look at a really nice
way of quickly computing integer powers and roots of complex numbers.
We’ll start with integer powers of 
since they are easy enough. If n
is an integer then,
There really isn’t too much to do with powers other than
working a quick example.
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Example 1 Compute
.
Solution
Of course we could just do this by multiplying the number
out, but this would be time consuming and prone to mistakes. Instead we can convert to exponential form
and then use (1)
to quickly get the answer.
Here is the exponential form of .
Note that we used the principle value of the argument for
the exponential form, although we didn’t have to.
Now, use (1) to quickly do the
computation.
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So, there really isn’t too much to integer powers of a
complex number.
Note that if then we have,
and if we take the last two terms and convert to polar form
we arrive at a formula that is called de
Moivre’s formula.
We now need to move onto computing roots of complex
numbers. We’ll start this off “simple”
by finding the nth roots of
unity. The nth roots of
unity for are the distinct solutions to the equation,
Clearly (hopefully) is one of the solutions. We want to determine if there are any other
solutions. To do this we will use the
fact from the previous sections that states that if and only if
So, let’s start by converting both sides of the equation to
complex form and then computing the power on the left side. Doing this gives,
So, according to the fact these will be equal provided,
Now, r is a
positive integer (by assumption of the exponential/polar form) and so solving
gives,
The solutions to the equation are then,
Recall from our discussion on
the polar form (and hence the exponential form) that these points will lie on
the circle of radius r. So, our points will lie on the unit circle
and they will be equally spaced on the unit circle at every radians.
Note this also tells us that there n
distinct roots corresponding to since we will get back to where we started
once we reach
Therefore there are n
nth roots of unity and they are given by,
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(2)
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There is a simpler notation that is often used to denote nth
roots of unity. First define,
then the nth roots of unity are,
Or, more simply the nth roots of unity are,
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(4)
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where is defined in (3).
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Example 2 Compute
the nth roots of unity for n
= 2, 3, and 4.
Solution
We’ll start with n
= 2.
This gives,
So, for n = 2 we
have -1, and 1 as the nth roots of unity. This should be too surprising as all we
were doing was solving the equation
and we all know that -1 and 1 are the two solutions.
While the result for n
= 2 may not be that surprising that for n
= 3 may be somewhat surprising. In
this case we are really solving
and in the world of real numbers we know that the solution
to this is z = 1. However, from the work above we know that
there are 3 nth roots of unity in this case. The problem here is that the remaining two
are complex solutions and so are usually not thought about when solving for
real solution to this equation which is generally what we wanted up to this
point.
So, let’s go ahead and find the nth roots of
unity for n = 3.
This gives,
I’ll leave it to you to check that if you cube the last
two values you will in fact get 1.
Finally, let’s go through n = 4. We’ll do this one
much quicker than the previous cases.
This gives,
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Now, let’s move on to more general roots. First let’s get some notation out of the
way. We’ll define to be any number that will satisfy the
equation
To find the values of
we’ll need to solve this equation and we can
do that in the same way that we found the nth roots of unity. So, if and (note can be any value of the argument, but we
usually use the principle value) we have,
So, this tells us that,
The distinct solutions to (5) are
then,
So, we can see that just as there were n nth roots of unity there are also n nth roots of .
Finally, we can again simplify the notation up a
little. If a is any of the nth roots of then all the roots can be written as,
where is defined in (3).
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Example 3 Compute
all values of the following.
(a)
(b)
Solution
(a) The first
thing to do is write down the exponential form of the complex number we’re
taking the root of.
So, if we use we can use (6) to
write down the roots.
Plugging in for k
gives,
I’ll leave it to you to check that if you square both of
these will get 2i.
(b) Here’s the
expon |