Unit circle and R 2 π Roots of unity and Z n Section 1 – Introduction and Examples Instructor: Yifan Yang Fall 2006 Instructor: Yifan Yang Section 1 – Introduction and Examples
Unit circle and R 2 π Roots of unity and Z n Outline Unit circle and R 2 π 1 Algebra on the unit circle Addition on R 2 π Isomorphism between R 2 π and U Roots of unity and Z n 2 Roots of unity Isomorphism between Z n and U n Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Outline Unit circle and R 2 π 1 Algebra on the unit circle Addition on R 2 π Isomorphism between R 2 π and U Roots of unity and Z n 2 Roots of unity Isomorphism between Z n and U n Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Algebra on the unit circle Let U = { z ∈ C : | z | = 1 } . Observe that if z 1 , z 2 ∈ U , then | z 1 z 2 | = | z 1 || z 2 | = 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 = e i θ for some θ ∈ R . For a given z ∈ U , the quantity θ with z = e i θ is not uniquely determined. In fact, we have e i θ = e i θ + 2 n π i for all n ∈ Z . That is, under the mapping θ �→ e i θ , the point θ looks just like θ + 2 n π . In mathematics, we describe the above observation in terms of equivalence relation. Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U Outline Unit circle and R 2 π 1 Algebra on the unit circle Addition on R 2 π Isomorphism between R 2 π and U Roots of unity and Z n 2 Roots of unity Isomorphism between Z n and U n Instructor: Yifan Yang Section 1 – Introduction and Examples
Algebra on the unit circle Unit circle and R 2 π Addition on R 2 π Roots of unity and Z n Isomorphism between R 2 π and U The set R 2 π of equivalence classes Define an equivalence relation ∼ 2 π on R by a ∼ 2 π b ⇔ a − b = 2 n π for some integer n . Let R 2 π denote the set of all equivalence classes. We can define addition + 2 π on R 2 π by θ 1 + 2 π ¯ ¯ θ 2 = θ 1 + θ 2 , 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 θ 1 from the first equivalence class, pick another element θ 2 from the second equivalence class, and then designate the sum of the two equivalence classes to be the class containing θ 1 + θ 2 . Instructor: Yifan Yang Section 1 – Introduction and Examples
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