Question

Which of the following is a rational number.
[a] $\sqrt{5}$
[b] $\pi $
[c] $0.101001000100001\ldots $
[d] $0.853853853\ldots $

Answer 1

Hint: Use the fact that every rational number can be expressed in the form of $\dfrac{p}{q}$, where p and q are integers, and q is non-zero. Use the fact that rational numbers have terminating decimal representation or have non-terminating recurring decimal representation.

Complete step-by-step answer:

Rational Numbers: Numbers which can be expressed in the form of $\dfrac{p}{q}$, where p and q are integers, and q is non- zero are called rational numbers, e.g. $1.3,1$ etc
Rational numbers have terminating, or non-terminating recurring decimal representations whereas irrational numbers have non-terminating and non-recurring decimal representations
If p is not a perfect square, then $\sqrt{p}$ is irrational.
Since 5 is not a perfect square, we have $\sqrt{5}$ is irrational.
Since $\pi =3.141592653\ldots $ has non-terminating, non-recurring decimal representation, we have $\pi $ is not rational.
Since 0.101001000100001… has non-terminating non-recurring decimal representation, it is not rational.
$0.853853853\ldots =0.\overline{853}$ , it has non-terminating recurring decimal representation and hence is rational.
Hence option [d] is correct.
Note: [1] Alternatively let x = $0.853853853\ldots =0.\overline{853}$
We have $1000x=853.\overline{853}$
Subtracting, we get
$999x=853$
Dividing both sides by 999 we get
$\begin{align}
  & \dfrac{999x}{999}=\dfrac{853}{999} \\
 & \Rightarrow x=\dfrac{853}{999} \\
\end{align}$
Hence $0.853853853\ldots =\dfrac{853}{999}$, which is of the form $\dfrac{p}{q}$, where p = 853 and q =999 are integers and $q\ne 0$.
Hence 0.853853…. is rational.

Note: Square root of a natural number which is not a perfect square is not rational.
This can be proved by proving that if the square root of a number which is not a perfect square is rational, then the square root of a prime is rational, which is not possible. Start by prime factorisation of that number and arrive at a conclusion.