Question

Check whether the number \[\pi +3\] is irrational or not.

Answer 1

Hint: We solve this problem in a contradiction method. First we assume that the number \[\pi +3\] is a rational number. We use the definition of rational number that is a number which can be represented in the form \[\dfrac{p}{q}\] such that \[p,q\] are integers and \[q\ne 0\] then we can say that the number is a rational number. By using the definition we solve the number where we can get that our assumption is true or not knowing that \['\pi '\] is an irrational number.

Complete step-by-step answer:
We are asked to find whether \[\pi +3\] is irrational or not.
Let us assume that \[\pi +3\] is a rational number.
We know that the definition of rational number as a number which can be represented in the form \[\dfrac{p}{q}\] such that \[p,q\] are integers and \[q\ne 0\]
By using the definition of rational number we can assume that
\[\Rightarrow \pi +3=\dfrac{p}{q}\]
Now, by subtracting with 3 on both sides we get
\[\begin{align}
  & \Rightarrow \pi +3-3=\dfrac{p}{q}-3 \\
 & \Rightarrow \pi =\dfrac{p-3q}{q}.......equation(i) \\
\end{align}\]
We know that the numbers \[p,q\] are integers
So, we know that by adding or subtracting or multiplying the integers they always be integers.
So, let us assume an integer \['k'\] such that
\[\Rightarrow p-3q=k\]
By substituting this value in equation (i) we get
\[\Rightarrow \pi =\dfrac{k}{q}\]
We know that \['\pi '\] is an irrational number.
Here, we can see that the number on RHS \[\dfrac{k}{q}\] is a rational number.
We know that a rational will never be equal to an irrational number
So, here, we can say that there is a contradiction of our assumption which is wrong.
Therefore, \[\pi +3\] is not a rational number which is an irrational number.

Note: We can have the shortcut explanation of the above question.
We know that from the definition of irrational number that adding any number to an irrational number gives the irrational number only.
We know that \['\pi '\] is an irrational number.
So, from the definition we can say that adding 3 to \['\pi '\] will also give an irrational number.
Therefore, \[\pi +3\] is an irrational number.