Question

Is \[\pi \] an irrational number? Why?

Answer 1

Hint: We solve this problem by assuming that \[\pi \] is a rational number, then, we check whether our assumption is right or wrong from the definition and properties of rational numbers.
We have the definition of a rational number as any number which can be represented as \[\dfrac{p}{q}\] such that \[q\ne 0\] is called a rational number.
We have the property of a rational number that the numbers after the decimal point in a rational number are either finite or infinite and in infinite cases the numbers get repeated after some digits.

Complete step by step answer:
We are asked to find whether \[\pi \] is irrational or not.
Let us assume that the value \[\pi \] is a rational number.
We know that any number which can be represented as \[\dfrac{p}{q}\] such that \[q\ne 0\] is called a rational number.
By using the above definition let us assume that
\[\Rightarrow \pi =\dfrac{p}{q}......equation(i)\]
We know that the numbers after the decimal point in the value of \[\pi \] is neither fixed nor repeating.
But we know that the property of rational numbers that the digits after the decimal point are either fixed or repeating.
By using the above two conditions we can say that the equation (i) doesn’t hold because in LHS the digits after decimal point are neither fixed nor repeating but in RHS the digits after the decimal point are either fixed or repeating which are not equal in any case.
So, we can say that our assumption is wrong.

Therefore, we can conclude that \[\pi \] is not a rational number which makes it an irrational number.

Note: Students may do mistake by considering the value of \[\pi \] as
\[\begin{align}
  & \pi =3.14 \\
 & \pi =\dfrac{22}{7} \\
\end{align}\]
Here, the above values of \[\pi \] are not exact values.
They are assumed to be almost equal to \[\pi \] and are considered for calculation purposes.
But the fact is that \[\pi \] is an universal irrational number.