Question

The following real numbers have decimal expansions as given below. In each case decide whether they are rational or not. If they are rational or not. If they are rational, and of them $\dfrac{{\text{p}}}{{\text{q}}}$, what can you say about the prime factors of q?
A.43.123456789
B.0.120120012000120000……...

Answer 1

Hint: Remember that the numbers are rational if they can be represented in the form of $\dfrac{{\text{p}}}{{\text{q}}}$ otherwise they are irrational.

Complete step-by-step answer:
Let us analyze each number one by one
First no is 43.123456789, As we can see that there are certain no of digits after decimal.
So it can be represented in form $\dfrac{{\text{p}}}{{\text{q}}}$.
It is a terminating number and the denominator, i.e., ‘q’ can have prime factors 2 and 5 only.
The second number is 0.120120012000120000……..
Here the prime factor of denominator Q will has a value which is not equal to
2 or 5.
So, it is an irrational number as it is non-terminating and non-repeating.

Note: Try to remember that rational numbers are non-terminating and non-repeating. Hence they cannot be represented in the form $\dfrac{{\rm{p}}}{{\rm{q}}}$.Additionally, it is also important to know about different classifications of numbers. The broadest classification starts from a number being a complex number or a real number. Then these are further classified into different categories like rational, irrational, integer, fractional and so on.