(a) Answer The set of irrational numbers is closed under addition. Explanation The set of irrational numbers is closed under addition, since the sum of two irrational numbers is an irrational number. (b) To determine To explain: The set of irrational numbers is closed under multiplication or not. Answer The set of irrational numbers is not closed under multiplication. Explanation The definition of irrational numbers is: An irrational number is a number which cannot be expressed as a fraction p q for any integers p and q . Irrational numbers have decimal expansions that neither terminate nor become periodic. Thus, from the above definition, The set of irrational numbers is not closed under multiplication, since 2 ⋅ 8 = 16 = 4 and 4 is not in the set.

Algebra and Trigonometry: Structure and Method, Book 2

2000th Edition

ISBN: 9780395977255

Author: MCDOUGAL LITTEL

Publisher: McDougal Littell

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Question

Chapter 1.3, Problem 32WE

(a)

To determine

To explain:

The set of irrational numbers is closed under addition or not.

(a)

Expert Solution

Answer to Problem 32WE

The set of irrational numbers is closed under addition.

Explanation of Solution

The set of irrational numbers is closed under addition, since the sum of two irrational numbers is an irrational number.

(b)

To determine

To explain:

The set of irrational numbers is closed under multiplication or not.

(b)

Expert Solution

Answer to Problem 32WE

The set of irrational numbers is not closed under multiplication.

Explanation of Solution

The definition of irrational numbers is:

An irrational number is a number which cannot be expressed as a fraction pq for any integers p and q .

Irrational numbers have decimal expansions that neither terminate nor become periodic.

Thus, from the above definition,

The set of irrational numbers is not closed under multiplication, since 2⋅8=16=4 and 4 is not in the set.

Chapter 1.3, Problem 31WE

Chapter 1.3, Problem 33WE

Chapter 1 Solutions

Algebra and Trigonometry: Structure and Method, Book 2

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