Question

State true or false:
Rational numbers from following real numbers
 $ - 8.0, $ $ \sqrt 5 , $ $ \dfrac{5}{7}, $ $ - \sqrt {18} , $ $ \sqrt {32} , $ $ 4.28, $ $ \pi , $ $ 3, $ $ \dfrac{8}{{15}}, $ $ 0.075 $ are $ - 8.0, $ $ \dfrac{5}{7}, $ $ 4.28, $ $ 3, $ $ \dfrac{8}{{15}}, $ $ 0.075 $
(A) True
(B) False

Answer 1

Hint: Rational numbers are the numbers that can be represented as a ratio of two integers. i.e. if we take $ \mathbb{Q} $ as a set of rational numbers, then, $ \mathbb{Q} = \{ \dfrac{p}{q}:p,q \in \mathbb{Z},q \ne 0\} $ . Check which of the above values satisfy the condition of rational numbers to solve the question.

Complete step-by-step answer:
Rational numbers are the numbers that can be represented as the ratio of two integers.
Let us denote the set of rational numbers as $ \mathbb{Q} $
Then by definition of rational numbers we can write
 $ \mathbb{Q} = \{ \dfrac{p}{q}:p,q \in \mathbb{Z},q \ne 0\} $
The numbers that cannot be written in this form are called irrational numbers. Irrational numbers are square roots of primes and some constants like $ \pi , $ $ e $ etc.
Clearly, $ \sqrt 5 $ is an irrational number.
Product of a rational number and an irrational number is also an irrational number.
Therefore, $ - \sqrt {18} = - 3\sqrt 2 $ is an irrational number.
And using the same logic, we can say that $ \sqrt {32} = 4\sqrt 2 $ is also an irrational number.
 $ \pi $ is an irrational number as well.
By the definition given above, clearly, $ \dfrac{5}{7} $ and $ \dfrac{8}{{15}} $ are rational numbers.
Now, $ 4.28 = \dfrac{{428}}{{100}} = \dfrac{{107}}{{25}} $ can be written as the ratio of two integers. Hence it is a rational number.
Similarly, $ 0.075 = \dfrac{{75}}{{1000}} = \dfrac{3}{{40}} $ is also a rational number.
If in the definition, $ \mathbb{Q} = \{ \dfrac{p}{q}:p,q \in \mathbb{Z},q \ne 0\} $ we put $ q = 1 $ then we get a set of integers. That means, every integer is also a rational number.
Therefore, $ - 8.0 $ and $ 3 $ are rational numbers.
Therefore, the statement given above is true.
Therefore, from the above explanation, the correct answer is, option (A) True
So, the correct answer is “Option A”.

Note: Rational numbers can be recurring and non-recurring decimals. So do not get confused by that. For example, $ 0.5 $ is a recurring decimal and it is a rational number as it can be written as $ 0.5 = \dfrac{1}{2} $ . And, $ 0.33333.... $ is a non-recurring number, but it is also a rational number, as it can be written as $ 0.33333.... = \dfrac{1}{3} $ .