In this chapter, we will look at the basic properties of the root of unity.
Example 1
It is given
, prove that
.
Prove
From the given equation, we can show that
:
.
Therefore,
![{\displaystyle a^{2012}=a^{670\times 3+2}=(a^{3})^{670}+a^{2}=a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738fee58bf4f1d97e30e5d456943d2955f3371ed)
![{\displaystyle ({\frac {1}{a}})^{2012}=a^{-671\times 3+1}=(a^{3})^{-671}+a=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/683dc49f006416c65269a067bfdfe88c5ac4e188)
![{\displaystyle a^{1987}=a^{662\times 3+1}=(a^{3})^{662}+a=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dff4f14342d2c2a6e34b60f0627d9fef104110b)
![{\displaystyle ({\frac {1}{a}})^{1987}=a^{-663\times 3+2}=(a^{3})^{-663}+a^{2}=a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b79af36d5c78ff7b1032d9c17edddc4428ff28)
So, both sides of the equation equal
.
QED
Moreover, we can calculate the value of each side:
.
In fact, we can obtain a more general result:
Example 2
Given
, and
is a natural number. Evaluate
.
Solution
![{\displaystyle a^{n}+({\frac {1}{a}})^{n}={\begin{cases}-1&{\text{if }}n{\text{ is not a multiple of 3}}\\2&{\text{if }}n{\text{ is a multiple of 3}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3816846911d54b689f28a80eb8eaf5bdcfc6d81a)
We have make use of an important observation, namely
, in the examples above. Numbers that satisfy the equation:
![{\displaystyle a^{n}=1,\quad n{\text{ is a natural number}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb5c3f610d6dd941ceef0f1263cc6c9d2430196)
are called the nth roots of unity or the unit roots. From the knowledge of algebra, the following formula:
![{\displaystyle \epsilon _{k}=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}},\quad k{\text{ is a natural number}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff075a93c2ad08b5ec1575a1ea9f61ba44f99808)
always gives a root of unity. When
,
takes distinct values, and when
takes other values,
equals one of the values
. Moreover, as a polynomial equation of degree
, the equation has exactly
roots. Therefore, ALL roots of unity are:
.
Note that
.
On the other hand, the roots of unity are the solution of the equation:
.
Moreover:
.
Therefore,
are all roots of the equation:
.
The cube roots of unity is a good starting point in our study of the properties of unit roots.
Example 3
The cube roots of unity are:
,
,
.
We usually write
. Then:
.
Therefore, the cube roots of unity can also be written as
.
The cube root of unity has the following properties:
- They have a unit modulus:
.
are the roots of the equation
.
are the roots of the equation
.
. So, the cube roots of unity still have the form of
if we let
.
- On the complex plane, the roots of unity are at the vertices of the regular triangle inscribed in the unit circle, with one vertex at 1.
,
.
![{\displaystyle 1^{n}+\omega ^{n}+(\omega ^{2})^{n}={\begin{cases}0&{\text{if }}n{\text{ is not a multiple of 3}}\\3&{\text{if }}n{\text{ is a multiple of 3}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61f07a7f2f771f9d4c9b2b5dc48c4f7d7c9b90f8)
After looking at the properties of the cube roots of unity, we are ready to study the general properties of the nth roots of unity.
Property 1
The nth roots of unity have a unit modulus, that is:
.
Proof It follows from the polar form of the unit roots.
Property 2
The product of two unit roots is also a unit root. Specifically, if
and
are integers, then:
.
Proof From the multiplication rule of complex number:
.
This is a very important property of the roots of unity, from which a series of corollary can be derived:
Corollary 1
.
Proof
.
Now, since
, multiplying its inverse on both sides yields
.
Corollary 2 For any integer
:
.
Proof When
is positive,
.
When
, non-zero complex number raised to the power of 0 is 1, so
.
When
is negative,
is positive, so
.
Corollary 3 If
is the remainder when
is divided by
, then
.
Proof Let
where
is an integer and
, then:
.
Corollary 4
.
Any root of unity can be expressed as a power of
.
We may ask the following question: is there any other root of unity
such that any root of unity can be expressed as a power of
?
In fact we have seen such an example when we studied the cube root of unity. A unit root with such property is called a primitive root.
Corollary 5 The conjugate of a unit root is also a unit root.
Proof From the property of complex numbers
and
,
Corollary 6
.
Proof
.
Property 3 Let
be an integer, then:
![{\displaystyle 1+\epsilon _{1}^{m}+\epsilon _{2}^{m}+\cdots +\epsilon _{n-1}^{m}={\begin{cases}0&{\text{if }}m{\text{ is not a multiple of }}n\\n&{\text{if }}m{\text{ is a multiple of }}n\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5a4d43688e6ab0a3041faf47dbb9403d0caf45)
Proof When
is a multiple of
,
for any integer
, so:
![{\displaystyle 1+\epsilon _{1}^{m}+\epsilon _{2}^{m}+\cdots +\epsilon _{n-1}^{m}=\overbrace {1+1+\cdots +1} ^{n{\text{ times}}}=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3119d9874ed1cdc77fe62db7186cacde3047a5ac)
When
is not a multiple of
,
. Then:
.
Corollary 7 If
, the sum of all unit roots is zero:
.
Proof Take
. Alternatively, the sum of roots of the equation
is zero.
Corollary 8 If
and
, then
.
Proof Since
,
is not a multiple of
. Then:
.
Therefore, if we exclude
, the nth roots of unity
are the roots of the equation:
.
Example 4 Find the fifth roots of unity.
Solution It can be proved that:
,
.
Therefore,
,
,
by corollary 4 of property 2,
,
by corollary 5 of property 2,
,
.
Example 5 Find the sixth roots of unity in terms of
.
Solution
,
,
,
,
,
.
Example 6 Evaluate:
,
where
is the greatest multiple of 3 not exceeding
.
Analysis The expression is the sum of every first of three consecutive binomial coefficients:
.
A similar but more familiar sum is:
,
which can be computed by summing the binomial expansions:
![{\displaystyle (1+x)^{n}={\tbinom {0}{n}}+{\tbinom {1}{n}}x+{\tbinom {2}{n}}x^{2}+\cdots +{\tbinom {n}{n}}x^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab42c061ba86ea195cc35741cbe433a0fe1e0d6)
for
(note that these are the square root of unity). The sum is
.
The value
(the coefficient of
) equals zero when
is odd, but equals two when
is even. (Note also that this follows from Property 3 for the square roots of unity.) Therefore,
![{\displaystyle 2{\tbinom {0}{n}}+2{\tbinom {2}{n}}+2{\tbinom {4}{n}}+\cdots =2^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b50b4e12d12940e5250667bb0cdcb53777a5e0d7)
![{\displaystyle {\tbinom {0}{n}}+{\tbinom {2}{n}}+{\tbinom {4}{n}}+\cdots =2^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e677ec05726539792b97884740481c334d73f670)
For the sum in this example, property 3 for the cube roots of unity may be useful.
Solution Summing the binomial expansions:
![{\displaystyle (1+x)^{n}={\tbinom {0}{n}}+{\tbinom {1}{n}}x+{\tbinom {2}{n}}x^{2}+\cdots +{\tbinom {n}{n}}x^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab42c061ba86ea195cc35741cbe433a0fe1e0d6)
for
yields
.
By property 3, the coefficient of every first of three terms equals 3 and all other terms vanish. Therefore,
![{\displaystyle 3{\tbinom {0}{n}}+3{\tbinom {3}{n}}+3{\tbinom {6}{n}}+\cdots =2^{n}+(-\omega ^{2})^{n}+(-\omega )^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de32acfcc9dd9498a2b738a599d2aa358ee86ded)
![{\displaystyle =2^{n}+(\cos {\frac {\pi }{3}}+i\sin {\frac {\pi }{3}})^{n}+(\cos {\frac {-\pi }{3}}+i\sin {\frac {-\pi }{3}})^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e06dbe0d5c92a25c5c1a27c28c86258a76d0d6c)
![{\displaystyle =2^{n}+2\cos {\frac {n\pi }{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a42ccf70db0de469e3cf933f1b1d5238fae497b6)
.