Question

Is \[\dfrac{7}{6}\] an irrational number?

Answer 1

Hint: We are going to use the concepts of rational and irrational numbers to solve this problem. We will define both rational and irrational numbers with some examples and will also discuss some special cases too. By definitions we will get the required solution.

Complete step by step answer:
In mathematics, numbers are classified into rational numbers and irrational numbers.
Numbers which can be represented in the form of \[\dfrac{p}{q},q \ne 0\] are called Rational numbers. Here, the numbers \[p{\text{ and }}q\] are co-primes. And numbers which can’t be represented in \[\dfrac{p}{q}\] form are called irrational numbers.
Some examples of rational numbers are \[\dfrac{6}{{11}}\] and \[ - 30.2\] and \[67\].
Irrational numbers are non-terminating non-recurring decimal numbers.
For example, take the number \[12.479347.....\]
It is a non-terminating non-recurring decimal number. So, this is an irrational number.
Some examples of irrational numbers are \[\pi = 3.141592......\] and \[\sqrt 2 = 1.41421......\]
Now, in the question, we need to find whether \[\dfrac{7}{6}\] is irrational or not.
So, when you do actual division, you get its value as \[1.166666.....\]
So, we got a non-terminating recurring decimal number. That means, \[\dfrac{7}{6}\] is not an irrational number.
So, \[\dfrac{7}{6}\] is a rational number.

Note:
A rational number can be written in a fractional form, where the denominator is not equal to zero. If the denominator is equal to zero, the fraction is not defined. And the value \[\pi = 3.141592......\] can be written as \[\pi = \dfrac{{22}}{7}\] but still, it is not a rational number. It is an irrational number.
In the given question we can directly say that it is a rational number as it is represented in \[\dfrac{p}{q}\] form and $q \ne 0$.