Question

Why is $\pi$ irrational and \[\dfrac{{22}}{7}\] is rational ?

Answer 1

Hint: Here in this question we have to give the reason why the Pi is called irrational and \[\dfrac{{22}}{7}\] is rational. As we know about the definition of the rational and irrational number, it is defined as the rational number will be in the form of \[\dfrac{p}{q}\], where \[q \ne 0\] but irrational number cannot be written in the form of \[\dfrac{p}{q}\]. So by using these definitions we are going to know the reason.

Complete step by step solution:
In mathematics we have different kinds of numbers namely, natural number, whole number, integers, rational numbers, irrational numbers and real numbers.
Natural numbers - Contain all counting numbers which start from 1.
Example: All numbers such as 1, 2, 3, 4, 5, 6,…
Whole Numbers - Collection of zero and natural numbers.
Example: All numbers including 0 such as 0, 1, 2, 3, 4, 5, 6,…
Integers- The collective result of whole numbers and negative of all natural numbers.
Example: \[ - \infty , \cdot \cdot \cdot 0,1,2,3, \cdot \cdot \cdot + \infty \]
Rational Numbers- Numbers that can be written in the form of \[\dfrac{p}{q}\] where \[q \ne 0\]
Example: 3, -7, -100, \[\dfrac{1}{2}\], \[\dfrac{5}{3}\], 0.16, 0.4666 etc
Irrational Numbers- All the numbers which are not rational and cannot be written in the form of \[\dfrac{p}{q}\]
Example: \[\sqrt 2 \], \[\pi \], \[\sqrt 3 \], \[2\sqrt 2 \] and \[ - \sqrt {45} \] etc
Real numbers: Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”.
Example: 0, -7, \[\sqrt 2 \], \[\dfrac{5}{3}\]..etc.,
Now we will consider the given question.
The Pi is irrational. The value of Pi is given as \[\pi = 3.142..\] . The value of Pi will continue, there is no end for the number. This cannot be written in the form of \[\dfrac{p}{q}\].
Therefore Pi is irrational.
The \[\dfrac{{22}}{7}\] is rational. This is in the form of \[\dfrac{p}{q}\], where \[q \ne 0\]. Since it is in the form \[\dfrac{p}{q}\]. So it is called rational.

Note: The Pi and \[\dfrac{{22}}{7}\] are the same, but the way of representation is different. So to distinguish these between numbers we must know the definition. The numbers are classified based on the representation of the number. One term is represented in the form of a fraction and another term is represented as a non terminating decimal.