Question

Classify that 11.2132435465 is a rational or irrational number.

Answer 1

Hint: A rational number is always in the form of $\dfrac{p}{q},q \ne 0$. For example, $\dfrac{3}{7}$ and an irrational number is a real number that can NOT be made by dividing two integers (an integer has no fractional part) example $\dfrac{5}{0}$ is an irrational number, with the denominator as zero.

Complete step-by-step answer:
Rational number: A rational number is always in the form of $\dfrac{p}{q},q \ne 0$, On division of p by q, two main things happen:
Either the remainder becomes 0 for example $\dfrac{7}{8} = 0.875$, which we can write in the form of p/q as $\dfrac{{875}}{{1000}}$, in this the remainder becomes 0 after some steps, the decimal expansion terminates after a finite step.
Or the remainder never becomes 0 and we get a non terminating string of remainders. For example $\dfrac{1}{7}$ because the decimal expansion of it is $0.142857142.....$
This expansion never ends or in other words we have a repeating block of digits in the quotient and the remainder cannot be 0.
The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.
Irrational number: The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational. Examples: π is an irrational number that has value 3.142… and is a never-ending and non-repeating number and values in root as they cannot be simplified like$\sqrt 2 $.
So, according to the above definition, we can conclude that 11.2132435465 is a rational number because
It can be written as $\dfrac{{112132435465}}{{10000000000}}$ and it is a terminating rational number.

Therefore the given number is a rational number.

Note: Key Differences between Rational and Irrational Numbers:
An irrational number is a number which cannot be expressed in a ratio of two integers.
In rational numbers, both numerator and denominator are whole numbers, where the denominator is not equal to zero.
While an irrational number cannot be written in a fraction.