Section 1 Introduction and Examples Instructor: Yifan Yang Fall - - PowerPoint PPT Presentation

Unit circle and R2π Roots of unity and Zn

Section 1 – Introduction and Examples

Instructor: Yifan Yang Fall 2006

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn

Outline

1

Unit circle and R2π Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

2

Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Outline

1

Unit circle and R2π Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

2

Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Algebra on the unit circle

Let U = {z ∈ C : |z| = 1}. Observe that if z1, z2 ∈ U, then |z1z2| = |z1||z2| = 1. Thus, the product of two elements of U is again in U. We say that U is closed under multiplication. Now recall that every element z in C can be written as z = eiθ for some θ ∈ R. For a given z ∈ U, the quantity θ with z = eiθ is not uniquely determined. In fact, we have eiθ = eiθ+2nπi for all n ∈ Z. That is, under the mapping θ → eiθ, the point θ looks just like θ + 2nπ. In mathematics, we describe the above observation in terms of equivalence relation.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Outline

1

Unit circle and R2π Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

2

Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

The set R2π of equivalence classes

Define an equivalence relation ∼2π on R by a ∼2π b ⇔ a − b = 2nπ for some integer n. Let R2π denote the set of all equivalence classes. We can define addition +2π on R2π by ¯ θ1 +2π ¯ θ2 = θ1 + θ2, where the + on the right is the usual addition on R. That is, to find the sum of two equivalence classes, we pick one element θ1 from the first equivalence class, pick another element θ2 from the second equivalence class, and then designate the sum of the two equivalence classes to be the class containing θ1 + θ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

The set R2π of equivalence classes

Define an equivalence relation ∼2π on R by a ∼2π b ⇔ a − b = 2nπ for some integer n. Let R2π denote the set of all equivalence classes. We can define addition +2π on R2π by ¯ θ1 +2π ¯ θ2 = θ1 + θ2, where the + on the right is the usual addition on R. That is, to find the sum of two equivalence classes, we pick one element θ1 from the first equivalence class, pick another element θ2 from the second equivalence class, and then designate the sum of the two equivalence classes to be the class containing θ1 + θ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

The set R2π of equivalence classes

Define an equivalence relation ∼2π on R by a ∼2π b ⇔ a − b = 2nπ for some integer n. Let R2π denote the set of all equivalence classes. We can define addition +2π on R2π by ¯ θ1 +2π ¯ θ2 = θ1 + θ2, where the + on the right is the usual addition on R. That is, to find the sum of two equivalence classes, we pick one element θ1 from the first equivalence class, pick another element θ2 from the second equivalence class, and then designate the sum of the two equivalence classes to be the class containing θ1 + θ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

The set R2π of equivalence classes

Define an equivalence relation ∼2π on R by a ∼2π b ⇔ a − b = 2nπ for some integer n. Let R2π denote the set of all equivalence classes. We can define addition +2π on R2π by ¯ θ1 +2π ¯ θ2 = θ1 + θ2, where the + on the right is the usual addition on R. That is, to find the sum of two equivalence classes, we pick one element θ1 from the first equivalence class, pick another element θ2 from the second equivalence class, and then designate the sum of the two equivalence classes to be the class containing θ1 + θ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

The set R2π of equivalence classes

Define an equivalence relation ∼2π on R by a ∼2π b ⇔ a − b = 2nπ for some integer n. Let R2π denote the set of all equivalence classes. We can define addition +2π on R2π by ¯ θ1 +2π ¯ θ2 = θ1 + θ2, where the + on the right is the usual addition on R. That is, to find the sum of two equivalence classes, we pick one element θ1 from the first equivalence class, pick another element θ2 from the second equivalence class, and then designate the sum of the two equivalence classes to be the class containing θ1 + θ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

The set R2π of equivalence classes

Define an equivalence relation ∼2π on R by a ∼2π b ⇔ a − b = 2nπ for some integer n. Let R2π denote the set of all equivalence classes. We can define addition +2π on R2π by ¯ θ1 +2π ¯ θ2 = θ1 + θ2, where the + on the right is the usual addition on R. That is, to find the sum of two equivalence classes, we pick one element θ1 from the first equivalence class, pick another element θ2 from the second equivalence class, and then designate the sum of the two equivalence classes to be the class containing θ1 + θ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Addition on R2π

Question Is the addition +2π well-defined? That is, if we pick elements from equivalence classes in two different ways, will we get the same result? To be more precise, let θ1 and θ′

1

be two elements in the first equivalence class, θ2 and θ′

2 be

two elements in the second class. Will θ1 + θ2 = θ′

1 + θ′ 2?

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Addition on R2π

Question Is the addition +2π well-defined? That is, if we pick elements from equivalence classes in two different ways, will we get the same result? To be more precise, let θ1 and θ′

1

be two elements in the first equivalence class, θ2 and θ′

2 be

two elements in the second class. Will θ1 + θ2 = θ′

1 + θ′ 2?

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Addition on R2π

Question Is the addition +2π well-defined? That is, if we pick elements from equivalence classes in two different ways, will we get the same result? To be more precise, let θ1 and θ′

1

be two elements in the first equivalence class, θ2 and θ′

2 be

two elements in the second class. Will θ1 + θ2 = θ′

1 + θ′ 2?

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Addition on R2π

Question Is the addition +2π well-defined? That is, if we pick elements from equivalence classes in two different ways, will we get the same result? To be more precise, let θ1 and θ′

1

be two elements in the first equivalence class, θ2 and θ′

2 be

two elements in the second class. Will θ1 + θ2 = θ′

1 + θ′ 2?

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Addition on R2π

Question Is the addition +2π well-defined? That is, if we pick elements from equivalence classes in two different ways, will we get the same result? To be more precise, let θ1 and θ′

1

be two elements in the first equivalence class, θ2 and θ′

2 be

two elements in the second class. Will θ1 + θ2 = θ′

1 + θ′ 2?

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

We have θ1 ∼ θ′

1

⇔ θ1 − θ′

1 = 2mπ for some integer m,

and θ2 ∼ θ′

2

⇔ θ2 − θ′

2 = 2nπ for some integer n,

It follows that (θ1 + θ2) − (θ′

1 + θ′ 2) = 2(m + n)π.

Therefore, θ1 + θ2 = θ′

1 + θ′ 2,

and +2π is well-defined.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

We have θ1 ∼ θ′

1

⇔ θ1 − θ′

1 = 2mπ for some integer m,

and θ2 ∼ θ′

2

⇔ θ2 − θ′

2 = 2nπ for some integer n,

It follows that (θ1 + θ2) − (θ′

1 + θ′ 2) = 2(m + n)π.

Therefore, θ1 + θ2 = θ′

1 + θ′ 2,

and +2π is well-defined.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

We have θ1 ∼ θ′

1

⇔ θ1 − θ′

1 = 2mπ for some integer m,

and θ2 ∼ θ′

2

⇔ θ2 − θ′

2 = 2nπ for some integer n,

It follows that (θ1 + θ2) − (θ′

1 + θ′ 2) = 2(m + n)π.

Therefore, θ1 + θ2 = θ′

1 + θ′ 2,

and +2π is well-defined.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

We have θ1 ∼ θ′

1

⇔ θ1 − θ′

1 = 2mπ for some integer m,

and θ2 ∼ θ′

2

⇔ θ2 − θ′

2 = 2nπ for some integer n,

It follows that (θ1 + θ2) − (θ′

1 + θ′ 2) = 2(m + n)π.

Therefore, θ1 + θ2 = θ′

1 + θ′ 2,

and +2π is well-defined.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Outline

1

Unit circle and R2π Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

2

Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Define a map φ : R2π → U by φ(¯ θ) = eiθ. This is well-defined since every element in an equivalence class takes the same value of eiθ. We claim φ have the following properties.

1

The map φ is one-to-one.

2

The map φ is onto.

3

The map φ satisfies φ(¯ θ1 +2π ¯ θ2) = φ(¯ θ1) · φ(¯ θ2).

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Define a map φ : R2π → U by φ(¯ θ) = eiθ. This is well-defined since every element in an equivalence class takes the same value of eiθ. We claim φ have the following properties.

1

The map φ is one-to-one.

2

The map φ is onto.

3

The map φ satisfies φ(¯ θ1 +2π ¯ θ2) = φ(¯ θ1) · φ(¯ θ2).

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Define a map φ : R2π → U by φ(¯ θ) = eiθ. This is well-defined since every element in an equivalence class takes the same value of eiθ. We claim φ have the following properties.

1

The map φ is one-to-one.

2

The map φ is onto.

3

The map φ satisfies φ(¯ θ1 +2π ¯ θ2) = φ(¯ θ1) · φ(¯ θ2).

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Define a map φ : R2π → U by φ(¯ θ) = eiθ. This is well-defined since every element in an equivalence class takes the same value of eiθ. We claim φ have the following properties.

1

The map φ is one-to-one.

2

The map φ is onto.

3

The map φ satisfies φ(¯ θ1 +2π ¯ θ2) = φ(¯ θ1) · φ(¯ θ2).

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Define a map φ : R2π → U by φ(¯ θ) = eiθ. This is well-defined since every element in an equivalence class takes the same value of eiθ. We claim φ have the following properties.

1

The map φ is one-to-one.

2

The map φ is onto.

3

The map φ satisfies φ(¯ θ1 +2π ¯ θ2) = φ(¯ θ1) · φ(¯ θ2).

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Define a map φ : R2π → U by φ(¯ θ) = eiθ. This is well-defined since every element in an equivalence class takes the same value of eiθ. We claim φ have the following properties.

1

The map φ is one-to-one.

2

The map φ is onto.

3

The map φ satisfies φ(¯ θ1 +2π ¯ θ2) = φ(¯ θ1) · φ(¯ θ2).

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Remark Properties 1 and 2 state that R2π and U have the same

• cardinality. Property 3 states that the set R2π with operation

+2π and the set U with multiplication · have the same algebraic

• structure. In mathematics, we say (R2π, +2π) and (U, ·) are

isomorphic and φ is an isomorphism. Isomorphic algebraic structures share many common algebraic properties. In the above example, (R2π, +2π) has an element ¯ 0 such that ¯ 0 + ¯ θ = ¯ θ for all ¯ θ ∈ R2π. The corresponding element in U is 1, which satisfies 1 · z = z for all z ∈ U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Remark Properties 1 and 2 state that R2π and U have the same

• cardinality. Property 3 states that the set R2π with operation

+2π and the set U with multiplication · have the same algebraic

• structure. In mathematics, we say (R2π, +2π) and (U, ·) are

isomorphic and φ is an isomorphism. Isomorphic algebraic structures share many common algebraic properties. In the above example, (R2π, +2π) has an element ¯ 0 such that ¯ 0 + ¯ θ = ¯ θ for all ¯ θ ∈ R2π. The corresponding element in U is 1, which satisfies 1 · z = z for all z ∈ U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Remark Properties 1 and 2 state that R2π and U have the same

• cardinality. Property 3 states that the set R2π with operation

+2π and the set U with multiplication · have the same algebraic

• structure. In mathematics, we say (R2π, +2π) and (U, ·) are

isomorphic and φ is an isomorphism. Isomorphic algebraic structures share many common algebraic properties. In the above example, (R2π, +2π) has an element ¯ 0 such that ¯ 0 + ¯ θ = ¯ θ for all ¯ θ ∈ R2π. The corresponding element in U is 1, which satisfies 1 · z = z for all z ∈ U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Remark Properties 1 and 2 state that R2π and U have the same

• cardinality. Property 3 states that the set R2π with operation

+2π and the set U with multiplication · have the same algebraic

• structure. In mathematics, we say (R2π, +2π) and (U, ·) are

isomorphic and φ is an isomorphism. Isomorphic algebraic structures share many common algebraic properties. In the above example, (R2π, +2π) has an element ¯ 0 such that ¯ 0 + ¯ θ = ¯ θ for all ¯ θ ∈ R2π. The corresponding element in U is 1, which satisfies 1 · z = z for all z ∈ U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Remark Properties 1 and 2 state that R2π and U have the same

• cardinality. Property 3 states that the set R2π with operation

+2π and the set U with multiplication · have the same algebraic

• structure. In mathematics, we say (R2π, +2π) and (U, ·) are

isomorphic and φ is an isomorphism. Isomorphic algebraic structures share many common algebraic properties. In the above example, (R2π, +2π) has an element ¯ 0 such that ¯ 0 + ¯ θ = ¯ θ for all ¯ θ ∈ R2π. The corresponding element in U is 1, which satisfies 1 · z = z for all z ∈ U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Remark Properties 1 and 2 state that R2π and U have the same

• cardinality. Property 3 states that the set R2π with operation

+2π and the set U with multiplication · have the same algebraic

• structure. In mathematics, we say (R2π, +2π) and (U, ·) are

isomorphic and φ is an isomorphism. Isomorphic algebraic structures share many common algebraic properties. In the above example, (R2π, +2π) has an element ¯ 0 such that ¯ 0 + ¯ θ = ¯ θ for all ¯ θ ∈ R2π. The corresponding element in U is 1, which satisfies 1 · z = z for all z ∈ U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Proof of Claim 1. If φ(¯ θ1) = φ(¯ θ2), then eiθ1 = eiθ2. This happens only when θ1 − θ2 = 2nπ for some integer n. Therefore, θ1 ∼2π θ2 and ¯ θ1 = ¯ θ2. Proof of Claim 2. For all z ∈ U, choose θ = Im log z. Then φ(¯ θ) = z. Proof of Claim 3. It follows immediately from ei(θ1+θ2) = eiθ1 · eiθ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Proof of Claim 1. If φ(¯ θ1) = φ(¯ θ2), then eiθ1 = eiθ2. This happens only when θ1 − θ2 = 2nπ for some integer n. Therefore, θ1 ∼2π θ2 and ¯ θ1 = ¯ θ2. Proof of Claim 2. For all z ∈ U, choose θ = Im log z. Then φ(¯ θ) = z. Proof of Claim 3. It follows immediately from ei(θ1+θ2) = eiθ1 · eiθ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

Isomorphism between R2π and U

Proof of Claim 1. If φ(¯ θ1) = φ(¯ θ2), then eiθ1 = eiθ2. This happens only when θ1 − θ2 = 2nπ for some integer n. Therefore, θ1 ∼2π θ2 and ¯ θ1 = ¯ θ2. Proof of Claim 2. For all z ∈ U, choose θ = Im log z. Then φ(¯ θ) = z. Proof of Claim 3. It follows immediately from ei(θ1+θ2) = eiθ1 · eiθ2.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Outline

1

Unit circle and R2π Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

2

Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Roots of unity

Definition A complex number z satisfying zn = 1 is called an nth root of

• unity. We denote by Un the set of all nth roots of unity.

Properties of Un

1

Let ζ = e2πi/n. Then Un = {1 = ζ0, ζ1, . . . , ζn−1}.

2

The set Un is closed under multiplication.

3

The value of ζk depends only on the residue class of k modulo n. (That is, if k and m are in the same residue class modulo n, then ζk = ζm.)

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Roots of unity

Definition A complex number z satisfying zn = 1 is called an nth root of

• unity. We denote by Un the set of all nth roots of unity.

Properties of Un

1

Let ζ = e2πi/n. Then Un = {1 = ζ0, ζ1, . . . , ζn−1}.

2

The set Un is closed under multiplication.

3

The value of ζk depends only on the residue class of k modulo n. (That is, if k and m are in the same residue class modulo n, then ζk = ζm.)

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Roots of unity

Definition A complex number z satisfying zn = 1 is called an nth root of

• unity. We denote by Un the set of all nth roots of unity.

Properties of Un

1

Let ζ = e2πi/n. Then Un = {1 = ζ0, ζ1, . . . , ζn−1}.

2

The set Un is closed under multiplication.

3

The value of ζk depends only on the residue class of k modulo n. (That is, if k and m are in the same residue class modulo n, then ζk = ζm.)

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Roots of unity

Definition A complex number z satisfying zn = 1 is called an nth root of

• unity. We denote by Un the set of all nth roots of unity.

Properties of Un

1

Let ζ = e2πi/n. Then Un = {1 = ζ0, ζ1, . . . , ζn−1}.

2

The set Un is closed under multiplication.

3

The value of ζk depends only on the residue class of k modulo n. (That is, if k and m are in the same residue class modulo n, then ζk = ζm.)

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Roots of unity

Definition A complex number z satisfying zn = 1 is called an nth root of

• unity. We denote by Un the set of all nth roots of unity.

Properties of Un

1

Let ζ = e2πi/n. Then Un = {1 = ζ0, ζ1, . . . , ζn−1}.

2

The set Un is closed under multiplication.

3

The value of ζk depends only on the residue class of k modulo n. (That is, if k and m are in the same residue class modulo n, then ζk = ζm.)

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Outline

1

Unit circle and R2π Algebra on the unit circle Addition on R2π Isomorphism between R2π and U

2

Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Isomorphism between Zn and Un

The last property shows that the map φ : Zn → Un defined by φ(¯ k) = ζk is well-defined. In fact, we have Theorem The function φ : Zn → Un is an isomorphism between (Zn, +n) and (Un, ·). That is,

1

φ is one-to-one,

2

φ is onto,

3

φ(¯ k +n ¯ m) = φ(¯ k) · φ( ¯ m). Proof. Similar to that of R2π and U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Isomorphism between Zn and Un

The last property shows that the map φ : Zn → Un defined by φ(¯ k) = ζk is well-defined. In fact, we have Theorem The function φ : Zn → Un is an isomorphism between (Zn, +n) and (Un, ·). That is,

1

φ is one-to-one,

2

φ is onto,

3

φ(¯ k +n ¯ m) = φ(¯ k) · φ( ¯ m). Proof. Similar to that of R2π and U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Isomorphism between Zn and Un

The last property shows that the map φ : Zn → Un defined by φ(¯ k) = ζk is well-defined. In fact, we have Theorem The function φ : Zn → Un is an isomorphism between (Zn, +n) and (Un, ·). That is,

1

φ is one-to-one,

2

φ is onto,

3

φ(¯ k +n ¯ m) = φ(¯ k) · φ( ¯ m). Proof. Similar to that of R2π and U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Isomorphism between Zn and Un

The last property shows that the map φ : Zn → Un defined by φ(¯ k) = ζk is well-defined. In fact, we have Theorem The function φ : Zn → Un is an isomorphism between (Zn, +n) and (Un, ·). That is,

1

φ is one-to-one,

2

φ is onto,

3

φ(¯ k +n ¯ m) = φ(¯ k) · φ( ¯ m). Proof. Similar to that of R2π and U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Isomorphism between Zn and Un

The last property shows that the map φ : Zn → Un defined by φ(¯ k) = ζk is well-defined. In fact, we have Theorem The function φ : Zn → Un is an isomorphism between (Zn, +n) and (Un, ·). That is,

1

φ is one-to-one,

2

φ is onto,

3

φ(¯ k +n ¯ m) = φ(¯ k) · φ( ¯ m). Proof. Similar to that of R2π and U.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Remark

Remark For a given n, there may be more than one isomorphisms between Zn and Un. For instance, for n = 8 and a = 1, 3, 5, 7, the functions φa : Z8 → U8 defined by φa(¯ k) = ζak are all isomorphisms.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Remark

Remark For a given n, there may be more than one isomorphisms between Zn and Un. For instance, for n = 8 and a = 1, 3, 5, 7, the functions φa : Z8 → U8 defined by φa(¯ k) = ζak are all isomorphisms.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

In-class exercise

Let ζ = e2πi/8 be an 8th root of unity.

1

Prove that for a = 3, the function φ3 : Z8 → U8 defined by φ3(¯ k) = ζ3k is one-to-one, onto, and satisfies φ3(¯ k + ¯ m) = φ3(¯ k) · φ3( ¯ m).

2

Prove that for a = 2, the function φ2 : Z8 → U8 defined by φ2(¯ k) = ζ2k is not one-to-one nor onto.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Homework

Homework Do Problems 16, 18, 30, 32, 34, 36, 37 of Section 1. Note that our definition of R2π is different from that given in the

• textbook. You should answer the problems using our definition.

Instructor: Yifan Yang Section 1 – Introduction and Examples

Unit circle and R2π Roots of unity and Zn Roots of unity Isomorphism between Zn and Un

Homework

Homework Do Problems 16, 18, 30, 32, 34, 36, 37 of Section 1. Note that our definition of R2π is different from that given in the

• textbook. You should answer the problems using our definition.

Instructor: Yifan Yang Section 1 – Introduction and Examples