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The group of rational points on the unit circle.º A point in the complex plane with both coordinates rational is called a rational point. The set of rational points on the unit circle will be denoted by C(Q): C(Q) = {a + bi EC | a, b E Q, and a? + b? = 1}. (a) Show that, with the usual multiplication of complex numbers, C(Q) is a group. (b) Is i e C(Q)? What is o(i)? (c) Let x = +i. Is x E C(Q)? Find x², x³, and r-1.

The group of rational points on the unit circle.º A point in the complex plane with both coordinates rational is called a rational point. The set of rational points on the unit circle will be denoted by C(Q): C(Q) = {a + bi EC | a, b E Q, and a? + b? = 1}. (a) Show that, with the usual multiplication of complex numbers, C(Q) is a group. (b) Is i e C(Q)? What is o(i)? (c) Let x = +i. Is x E C(Q)? Find x², x³, and r-1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The group of rational points on the unit circle.9 A point in the complex plane with both coordinates rational is called a rational point. The set of rational points on the unit circle will be denoted by C(Q): C(Q) = {a + bi C | a, b € Q, and a² + b2 = 1}. (a) Show that, with the usual multiplication of complex numbers, C(Q) is a group. (b) Is i e C(Q)? What is o(i)? (c) Let x = +i. Is x E C(Q)? Find x², x³, and x-1. (d) A triple (a, b, c), where a, b, and c are integers, c + 0, and a²+b² = c² is called a Pythagorean triple. Given an element of C(Q), can you construct a Pythagorean triple? Given a Pythagorean triple, can you construct an element of C(Q)? Under this correspondence, when do two different Pythagorean triples give the same element of C(Q)? (e) In C(Q), what is the order of 3/5 + 4/5i? Make a conjecture. 3
Transcribed Image Text:The group of rational points on the unit circle.9 A point in the complex plane with both coordinates rational is called a rational point. The set of rational points on the unit circle will be denoted by C(Q): C(Q) = {a + bi C | a, b € Q, and a² + b2 = 1}. (a) Show that, with the usual multiplication of complex numbers, C(Q) is a group. (b) Is i e C(Q)? What is o(i)? (c) Let x = +i. Is x E C(Q)? Find x², x³, and x-1. (d) A triple (a, b, c), where a, b, and c are integers, c + 0, and a²+b² = c² is called a Pythagorean triple. Given an element of C(Q), can you construct a Pythagorean triple? Given a Pythagorean triple, can you construct an element of C(Q)? Under this correspondence, when do two different Pythagorean triples give the same element of C(Q)? (e) In C(Q), what is the order of 3/5 + 4/5i? Make a conjecture. 3
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