Video Transcript
Let 𝜔 be an 𝑛th root of unity. Which of the following is the
correct relationship between 𝜔 to the power negative one and 𝜔? Is it (A) 𝜔 to the power negative
one is negative 𝜔? (B) 𝜔 to the power negative one is
the complex conjugate of 𝜔. (C) 𝜔 to the power negative one is
actually 𝜔. Or (D) 𝜔 to the power negative one
is the negative complex conjugate of 𝜔. Express 𝜔 to the power negative
one in terms of positive powers of 𝜔.
We’re told that 𝜔, which is a
complex number, is an 𝑛th root of unity. And recall that if complex number
𝜔 is an 𝑛th root of unity, then it’s a solution to the equation 𝜔 to the power 𝑛
is equal to one. That’s for a positive integer
𝑛. And there are two parts to this
question. The first is what’s the correct
relationship between 𝜔 to the power negative one and 𝜔, and we’re then asked to
express 𝜔 to the power negative one in terms of positive powers of 𝜔. So let’s look first at the first
part.
We know that 𝜔 is an 𝑛th root of
unity. And recall it from de Moivre’s
theorem, we can express these roots in exponential form; that is, 𝜔 is 𝑒 to the
𝑖𝜃, where 𝜃 is two 𝑘𝜋 over 𝑛 and 𝑘 takes values from zero to 𝑛 minus
one. Since we want to know the
relationship between 𝜔 and 𝜔 to the power negative one, we can also express 𝜔 to
the power negative one in exponential form. 𝜔 to the power negative one is
equal to 𝑒 to the power negative 𝑖𝜃, where 𝑒 is Euler’s number. And that’s equal to 𝑒 to the power
𝑖 times negative 𝜃. And now, if we look at this in
trigonometric form, we have 𝜔 to the power negative one is the cos of negative 𝜃
plus 𝑖 times the sin of negative 𝜃.
But now we can use the fact that for
an angle 𝜃, the cos of negative 𝜃 is the cos of 𝜃 and the sin of negative 𝜃 is
negative the sin of 𝜃 so that 𝜔 to the power negative one is the cos of 𝜃 minus
𝑖 sin 𝜃 in trigonometric form. And this is simply the complex
conjugate of 𝜔 since 𝜔 is cos 𝜃 plus 𝑖 sin 𝜃 in trigonometric form. So in fact, we have 𝜔 to the power
negative one is the complex conjugate of 𝜔; that is, option (B) 𝜔 to the negative
one is equal to the complex conjugate of 𝜔.
We can check that none of the other
options apply. So, for example, option (A) says
that 𝜔 to the negative one is equal to negative 𝜔. But we can see, in fact, from the
exponential form that this is not true, so we can discount option (A). Option (C) is that 𝜔 to the
negative one is equal to 𝜔. This again is not true because we
have 𝑒 to the negative 𝑖𝜃 against 𝑒 to the 𝑖𝜃, so these are not equal and we
can discount option (C). And option (D) says that 𝜔 to the
negative one is the negative complex conjugate of 𝜔. But the negative complex conjugate
of 𝜔 is negative cos 𝜃 minus 𝑖 sin 𝜃, which is negative one times cos 𝜃 plus 𝑖
sin 𝜃, which is not equal to 𝜔 to the power negative one. So we can discount option
(D).
Now, for the second part of our
question, we’re asked to express 𝜔 to the power negative one in terms of positive
powers of 𝜔. We could look again at the
exponential form, but let’s instead look at what we mean by 𝜔 to the power negative
one in terms of powers or exponents. We know that the laws of exponents
for real numbers tell us that 𝑎 to the power negative one is one over 𝑎 if 𝑎 is a
real number. And in fact, this applies also to
complex numbers. So in fact, 𝜔 to the power
negative one is equal to one over 𝜔. And, of course, 𝜔 is never zero
since zero is never a root of unity.
But let’s look again at our equation
𝜔 to the power 𝑛 is equal to one. If we divide through by 𝜔, we have
𝜔 to the power 𝑛 over 𝜔 is one over 𝜔. But that’s equal to 𝜔 to the
negative one. And now if we were to write this
out in full, we have 𝜔 to the power 𝑛 over 𝜔 is 𝜔 times itself 𝑛 times over
𝜔. And we can simple this by canceling
the 𝜔 in the denominator with one on the top. So now, in our numerator, we have
𝜔 times itself 𝑛 minus one times and one in the denominator, which is actually 𝜔
to the power 𝑛 minus one so that 𝜔 to the power negative one is 𝜔 to the power 𝑛
minus one.
Remember that 𝑛 is a positive
integer so that for any 𝑛 greater than one, we have 𝜔 to the negative one in terms
of a positive power of 𝜔. And if 𝑛 is equal to one, we have
𝑛 minus one equal to zero. And in this case, in our equation,
𝜔 to the power 𝑛 is equal to one; that’s the 𝑛th root of unity. That means 𝜔 is the first root of
unity, which is one. And so if 𝜔 is an 𝑛th root of
unity, 𝜔 to the power negative one is the complex conjugate of 𝜔, that’s option
(B), and 𝜔 to the power negative one in terms of positive powers of 𝜔 is 𝜔 to the
power 𝑛 minus one.
