Explanation Given information: ℂ is the set of all ordered pairs ( a , b ) of real numbers a and b . Equality, addition, and multiplication are defined in ℂ by ( a , b ) = ( c , d ) if and only if a = c and b = d ( a , b ) + ( c , d ) = ( a + c , b + d ) ( a , b ) ( c , d ) = ( a c − b d , a d + b c ) The elements of ℂ are called complex numbers. Formula used: i) S is a subset of a field F that contains at least two elements and satisfies both of the following conditions: x ∈ S and y ∈ S imply, x − y ∈ S , and x ∈ S and y ≠ 0 ∈ S imply, x y − 1 ∈ S . This S is called a subfield of F . ii) ( a , b ) − 1 = ( a a 2 + b 2 , − b a 2 + b 2 ) Proof: Given that, R ' = { ( a , 0 ) | a ∈ ℝ } Let ℂ be the set of all ordered pairs ( a , b ) of real numbers a and b . In particular if b = 0 implies that ( a , 0 ) ∈ ℂ for all real numbers a . Thus, R ' is a subset of ℂ that contains at least two elements. Let ( x , 0 ) ∈ R ' and ( y , 0 ) ∈ R ' with x , y ∈ ℝ Consider ( x , 0 ) − ( y , 0 ) = ( x , 0 ) + ( − ( y , 0 ) ) By using definition of addition in complex number, ⇒ ( x , 0 ) − ( y , 0 ) = ( x − y , 0 ) By using, ℝ is a field

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

See similar textbooks

Videos

Question

Chapter 7.2, Problem 17E

To determine

To prove: R'={(a, 0)|a∈ℝ} is a subfield of ℂ.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.2, Problem 16E

Chapter 7.2, Problem 18E

Students have asked these similar questions

Find the perimeter and area

Assume {u1, U2, us} spans R³. Select the best statement. A. {U1, U2, us, u4} spans R³ unless u is the zero vector. B. {U1, U2, us, u4} always spans R³. C. {U1, U2, us, u4} spans R³ unless u is a scalar multiple of another vector in the set. D. We do not have sufficient information to determine if {u₁, u2, 43, 114} spans R³. OE. {U1, U2, 3, 4} never spans R³. F. none of the above

Assume {u1, U2, 13, 14} spans R³. Select the best statement. A. {U1, U2, u3} never spans R³ since it is a proper subset of a spanning set. B. {U1, U2, u3} spans R³ unless one of the vectors is the zero vector. C. {u1, U2, us} spans R³ unless one of the vectors is a scalar multiple of another vector in the set. D. {U1, U2, us} always spans R³. E. {U1, U2, u3} may, but does not have to, span R³. F. none of the above

Chapter 7 Solutions

Elements Of Modern Algebra

Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Prove that is irrational. (That is, prove there...Ch. 7.1 - Prove that is irrational. Ch. 7.1 - Prove that if is a prime integer, then is...Ch. 7.1 - Prove that if a is rational and b is irrational,...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Prove that if is an irrational number, then is...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Give counterexamples for the following...Ch. 7.1 - Let S be a nonempty subset of an order field F....Ch. 7.1 - Prove that if F is an ordered field with F+ as its...Ch. 7.1 - If F is an ordered field, prove that F contains a...Ch. 7.1 - Prove that any ordered field must contain a...Ch. 7.1 - If and are positive real numbers, prove that...Ch. 7.1 - Prove that if and are real numbers such that ,...Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Prob. 2TFE Ch. 7.2 - Prob. 3TFE Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Prob. 5TFE Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Prob. 7TFE Ch. 7.2 - Prob. 1E Ch. 7.2 - Prob. 2E Ch. 7.2 - Prob. 3E Ch. 7.2 - Prob. 4E Ch. 7.2 - Prob. 5E Ch. 7.2 - Prob. 6E Ch. 7.2 - Prob. 7E Ch. 7.2 - Prob. 8E Ch. 7.2 - Prob. 9E Ch. 7.2 - Prob. 10E Ch. 7.2 - Prob. 11E Ch. 7.2 - Prob. 12E Ch. 7.2 - Prob. 13E Ch. 7.2 - Prob. 14E Ch. 7.2 - Prob. 15E Ch. 7.2 - Prob. 16E Ch. 7.2 - Prob. 17E Ch. 7.2 - Prob. 18E Ch. 7.2 - Prob. 19E Ch. 7.2 - Prob. 20E Ch. 7.2 - Prob. 21E Ch. 7.2 - Prob. 22E Ch. 7.2 - Prob. 23E Ch. 7.2 - Prob. 24E Ch. 7.2 - Prob. 25E Ch. 7.2 - Prob. 26E Ch. 7.2 - Prob. 27E Ch. 7.2 - Prob. 28E Ch. 7.2 - Prob. 29E Ch. 7.2 - Prob. 30E Ch. 7.2 - Prob. 31E Ch. 7.2 - Prob. 32E Ch. 7.2 - Prob. 33E Ch. 7.2 - Prob. 34E Ch. 7.2 - Prob. 35E Ch. 7.2 - Prob. 36E Ch. 7.2 - Prob. 37E Ch. 7.2 - Prob. 38E Ch. 7.2 - Prob. 39E Ch. 7.2 - Prob. 40E Ch. 7.2 - Exercise are stated using the notation in the...Ch. 7.2 - Prob. 42E Ch. 7.2 - Prob. 43E Ch. 7.2 - Prob. 44E Ch. 7.2 - Prob. 45E Ch. 7.2 - Prob. 46E Ch. 7.2 - Prob. 47E Ch. 7.2 - Prob. 48E Ch. 7.2 - Prob. 49E Ch. 7.2 - Prob. 50E Ch. 7.2 - An element in a ring is idempotent if . Prove...Ch. 7.2 - Prove that a finite ring R with unity and no zero...Ch. 7.3 - True or False Label each of the following...Ch. 7.3 - Prob. 2TFE Ch. 7.3 - Prob. 3TFE Ch. 7.3 - Prob. 4TFE Ch. 7.3 - Prob. 1E Ch. 7.3 - Find each of the following products. Write each...Ch. 7.3 - Prob. 3E Ch. 7.3 - Show that the n distinct n th roots of 1 are...Ch. 7.3 - Prob. 5E Ch. 7.3 - Prob. 6E Ch. 7.3 - Prob. 7E Ch. 7.3 - Prob. 8E Ch. 7.3 - Prob. 9E Ch. 7.3 - Prob. 10E Ch. 7.3 - Prob. 11E Ch. 7.3 - Prob. 12E Ch. 7.3 - Prob. 13E Ch. 7.3 - Prob. 14E Ch. 7.3 - Prove that the group in Exercise is cyclic, with ...Ch. 7.3 - Prob. 16E Ch. 7.3 - Prob. 17E Ch. 7.3 - Prob. 18E Ch. 7.3 - Prob. 19E Ch. 7.3 - Prob. 20E Ch. 7.3 - Prob. 21E Ch. 7.3 - Prob. 22E Ch. 7.3 - Prove that the set of all complex numbers that...Ch. 7.3 - Prob. 24E Ch. 7.3 - Prob. 25E Ch. 7.3 - Prob. 26E Ch. 7.3 - Prob. 27E Ch. 7.3 - Prob. 28E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

To prove: R ' = { ( a , 0 ) | a ∈ ℝ } is a subfield of ℂ .

Solution Summary: The author explains that C is the set of all ordered pairs (a,b) of real numbers a and b.

Videos

To prove: R'={(a, 0)|a∈ℝ} is a subfield of ℂ.

Want to see the full answer?

Chapter 7 Solutions