# the sum, the difference, and the product of two rational numbers are rational numbers, whether the product of two irrational numbers necessarily irrational, to find about the sum.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.1, Problem 81E
To determine

## To explain: the sum, the difference, and the product of two rational numbers are rational numbers, whether the product of two irrational numbers necessarily irrational, to find about the sum.

Expert Solution

### Explanation of Solution

Suppose that

pq,rs are two rational numbers. Then p,q,r,s are integers and

q0,s0

ps,qr,qs are integers and qs0

Since the sum of integers is an integer, then ps+qr is an integer

ps+qrqs is a rational number

The sum of rational numbers is rational

Similarly, psqr is an integer, so psqrqs is rational, I.e., pqrs is rational

The product of rational numbers is a rational number

2,18 are irrational

But

218=36=6isarationalnumber

The sum of irrational numbers is irrational

The product of irrational numbers is not necessarily irrational.

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