Chapter 6.6, Problem 3WE

a.

To determine

Find whether the number $\pi +\frac{1}{\pi }$ is rational or irrational.

Answer to Problem 3WE

The number $\pi +\frac{1}{\pi }$ is irrational.

Explanation of Solution

Given information: The given number is $\pi +\frac{1}{\pi }$ .

Calculation: The given number $\pi +\frac{1}{\pi }$ is irrational because the number $\pi +\frac{1}{\pi }$ is a non-repeating and non-terminated number which cannot be expressed as the ratio of two integers in the form $\frac{p}{q}$ where $q\ne 0$ . Hence the number $\pi +\frac{1}{\pi }$ is irrational.

b.

To determine

Find whether the number $\pi \cdot \frac{1}{\pi }$ is rational or irrational.

Answer to Problem 3WE

The number $\pi \cdot \frac{1}{\pi }$ is rational.

Explanation of Solution

Given information: The given number is $\pi \cdot \frac{1}{\pi }$ .

Calculation: The given number $\pi \cdot \frac{1}{\pi }$ is rational because the number $\pi \cdot \frac{1}{\pi }$ is a terminated decimal number when written its simplified form which can be expressed as the ratio of two integers in the form $\frac{p}{q}$ where $q\ne 0$ . Hence the number $\pi \cdot \frac{1}{\pi }$ is rational.