Chapter 6.6, Problem 10OE

To determine

whether the given number is rational or irrational

Answer to Problem 10OE

It is an irrational number

Explanation of Solution

Given information :

Given number is $\frac{\pi }{\pi +1}$

Rational number are those numbers which can be represent in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$

Value of $\pi$ is ( $3.14159\dots$ ) . And therefore have no exact value. We sometimes use 3.14 or $\frac{22}{7}$ as the values of $\pi$ . But there is no exact value of pie as in the value of $\pi$ decimal places goes to infinity. Hence we can conclude that the given number $\frac{\pi }{\pi +1}$ cannot be represented in the form $\frac{p}{q}$ . Thus it is irrational.