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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 7.2, Problem 1E
Textbook Problem

Perform the computations in exercises 1 12 and express the results in standard form a + b i .

( 2 3 i ) ( 1 + 4 i )

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Chapter 7 Solutions

Elements Of Modern Algebra
Ch. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Find the decimal representation for each of the...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 - 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 - Label each of the following statements as either...Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Label each of the following statements as either...Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Label each of the following statements as either...Ch. 7.2 - Perform the computations in exercises and express...Ch. 7.2 - Perform the computations in exercises and express...Ch. 7.2 - Perform the computations in exercises and express...Ch. 7.2 - Perform the computations in exercises and express...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Perform the computations in exercises 112 and...Ch. 7.2 - Find two sqaure roots of each given...Ch. 7.2 - With addition as given in definition 7.5, prove...Ch. 7.2 - With addition and multiplication as in definition...Ch. 7.2 - Show that in=im for all integers n, where...Ch. 7.2 - 17. With given in Definition 7.5, prove that is a...Ch. 7.2 - Let B={ bib } be a subset of with the usual...Ch. 7.2 - Let be the mapping defined for in standard form...Ch. 7.2 - It follows from Exercise 19 that z1+z2=z1+z2 and...Ch. 7.2 - Show that satisfies the equation where and...Ch. 7.2 - Assume that : is an isomorphism and (a)=a for all...Ch. 7.2 - Perform the computations in Exercises 2334 and...Ch. 7.2 - Perform the computations in Exercises and...Ch. 7.2 - Perform the computations in Exercises 2334 and...Ch. 7.2 - Perform the computations in Exercises and...Ch. 7.2 - Perform the computations in Exercises and express...Ch. 7.2 - Perform the computations in Exercises and express...Ch. 7.2 - Perform the computations in Exercises 2334 and...Ch. 7.2 - Perform the computations in Exercises 2334 and...Ch. 7.2 - Perform the computations in Exercises and express...Ch. 7.2 - Perform the computations in Exercises 2334 and...Ch. 7.2 - Perform the computations in Exercises 2334 and...Ch. 7.2 - Perform the computations in Exercises 2334 and...Ch. 7.2 - With addition as given in Example 3 of this...Ch. 7.2 - Prove that multiplication in the set H of Example...Ch. 7.2 - With addition and multiplication as defined in...Ch. 7.2 - Exercise 38-42 are stated using the notation in...Ch. 7.2 - Exercise 38-42 are stated using the notation in...Ch. 7.2 - Exercise are stated using the notation in the...Ch. 7.2 - Exercise are stated using the notation in the...Ch. 7.2 - Exercise are stated using the notation in the...Ch. 7.2 - (See Exercise 42.) For arbitrary q=x+yi+zj+wk in...Ch. 7.2 - Let . Prove implies or . Ch. 7.2 - Show that the equation has an infinite number of...Ch. 7.2 - With as defined in Example 3, Prove that the...Ch. 7.2 - Assume that C=(a,b,0,0)a,b is a subring of the...Ch. 7.2 - Suppose the mapping is defined on the set of...Ch. 7.2 - Let S be the subset of M2() given by S={ [ xyyx...Ch. 7.2 - Let be the group of nonzero quaternions with the...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 - True or False Label each of the following...Ch. 7.3 - True or False Label each of the following...Ch. 7.3 - True or False Label each of the following...Ch. 7.3 - Graph each of the following complex numbers, and...Ch. 7.3 - Find each of the following products. Write each...Ch. 7.3 - Use De Moivres Theorem to find the value of each...Ch. 7.3 - Show that the n distinct n th roots of 1 are...Ch. 7.3 - If , show that the distinct th roots of are given...Ch. 7.3 - Find the indicated roots of in standard form ,...Ch. 7.3 - Find all the indicated roots of the given number....Ch. 7.3 - Find all complex numbers that are solutions of the...Ch. 7.3 - If =cos2n+isin2n, and u is any n th root of z show...Ch. 7.3 - Prove that for a fixed value of , the set of all...Ch. 7.3 - In Exercises , take to be the group in Exercise...Ch. 7.3 - In Exercises , take to be the group in Exercise...Ch. 7.3 - In Exercises 1114, take Un to be the group in...Ch. 7.3 - In Exercises , take to be the group in Exercise...Ch. 7.3 - Prove that the group in Exercise is cyclic, with ...Ch. 7.3 - Any generator of the group in Exercise 10 is...Ch. 7.3 - Find all primitive sixth roots of . Find all...Ch. 7.3 - Let k=cos2kn+isin2kn be a primitive n th root of...Ch. 7.3 - Prove that the set of all roots of 1 forms a group...Ch. 7.3 - Prove that the sum of all the distinct th roots of...Ch. 7.3 - Prove that the product of all the distinct th...Ch. 7.3 - Prove that the product of all the distinct n th...Ch. 7.3 - Prove that the set of all complex numbers that...Ch. 7.3 - Prove that if is a nonzero complex number in...Ch. 7.3 - Prove that if n is a positive integer and...Ch. 7.3 - Prove that if and are complex numbers in...Ch. 7.3 - Let u be an nth root of unity. Show that u1 is...Ch. 7.3 - In the ordered field , absolute value is defined...

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