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Unit roots/Properties of unit roots

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In this chapter, we will look at the basic properties of the root of unity.

An example

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Example 1 It is given , prove that

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Prove From the given equation, we can show that :

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Therefore,

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

The roots of unity

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We have make use of an important observation, namely , in the examples above. Numbers that satisfy the equation:

are called the nth roots of unity or the unit roots. From the knowledge of algebra, the following formula:

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:

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Note that .

On the other hand, the roots of unity are the solution of the equation:

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Moreover:

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Therefore, are all roots of the equation:

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The cube roots of unity

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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:

,
,
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We usually write . Then:

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Therefore, the cube roots of unity can also be written as . The cube root of unity has the following properties:

  1. They have a unit modulus: .
  2. are the roots of the equation .
  3. are the roots of the equation .
  4. . So, the cube roots of unity still have the form of if we let .
  5. 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.
  6. , .

General properties of roots of unity

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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:

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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:

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Proof From the multiplication rule of complex number:

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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 :

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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:

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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:

Proof When is a multiple of , for any integer , so:

When is not a multiple of , . Then:

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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:

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Therefore, if we exclude , the nth roots of unity are the roots of the equation:

.

Examples

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Example 4 Find the fifth roots of unity.
Solution It can be proved that:

,
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Therefore,

,
,

by corollary 4 of property 2,

,

by corollary 5 of property 2,

,
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Example 5 Find the sixth roots of unity in terms of .
Solution

,
,
,
,
,
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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:

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A similar but more familiar sum is:

,

which can be computed by summing the binomial expansions:

for (note that these are the square root of unity). The sum is

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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,

For the sum in this example, property 3 for the cube roots of unity may be useful.
Solution Summing the binomial expansions:

for yields

.

By property 3, the coefficient of every first of three terms equals 3 and all other terms vanish. Therefore,

.