Question

Say true or false: 0.120120012000120000... is a rational number.
(a) True
(b) False

Answer 1

Hint: Observe that the given number is terminating or non terminating and repeating or non repeating. Use the definition of rational and irrational numbers to check if the given number is a rational number or not.

Complete step-by-step answer:

We have to check if the given number 0.120120012000... is a rational number or not.
To do so, we will use the definition of rational and irrational numbers.
We know that any rational number is of the form $\dfrac{p}{q}$, where p and q are integers and $q\ne 0$.
We know that irrational numbers are all those real numbers that are not rational numbers. All the non terminating non repeating numbers are irrational numbers.
We know that a non terminating number is a decimal number that goes on endlessly with an infinite number of digits.
Thus, we observe that 0.12012001200012... is a non terminating number.
We know that a non terminating, non repeating number is a decimal number that continues endlessly, with no group of digits repeating endlessly.
Thus, we observe that 0.12012001200012... is a non terminating, non repeating number.
As, we observe that 0.12012001200012... is a non terminating, non repeating number, it’s an irrational number.
Hence, the given statement is incorrect, which is option (b).

Note: We can’t solve this question without knowing the difference between rational and irrational numbers. Keep in mind that we can write all the non terminating, repeating numbers as rational numbers.