Question

Classify the following numbers as rational or irrational.
 $ 2\pi $

Answer 1

Hint: To classify whether a given number is a rational or irrational number. We see what kind of a number is given, whether the given number is single or a product of numbers. As in this problem, a given number is a product of two numbers. So, we discuss whether numbers that form a given number are rational or irrational. Then according to that we will say that the given number is rational or irrational.

Complete step-by-step answer:
Given number is $ 2\pi $
To discuss whether a given number is rational or irrational.
We clearly see that the given number is a product of two numbers $ 2 $ and $ \pi $ .
Here, number $ 2 $ is a rational number as it can be written as in the form of $ \dfrac{p}{q}\,\,with\,\,q \ne 0 $ .
But the number $ \pi $ can also be written as $ \dfrac{{22}}{7} $ .
On dividing $ Numerator(22)\,\,by\,\,deno\min ator\,(7) $ we will get number = $ 3.148571428571..... $
Clearly we see that the number obtained by dividing $ 22\,\,by\,\,7 $ is a non-terminating decimal expression. In which digits are not repeated.
So, we can say that the number $ \dfrac{{22}}{7}\,\,or\,\,\pi $ is an irrational number.
And we know that the product of a rational and irrational number is an irrational number.
Therefore, from above we can say that $ 2\pi $ is an irrational number as it can be written as a product of rational numbers which is $ 2 $ and irrational numbers which is $ \pi $ .
So, the correct answer is “ irrational number.”.

Note: We generally we say that numbers that car be written in the form of $ \dfrac{p}{q} $ can be called as rational, but here $ \dfrac{{22}}{7} $ even can be written as $ \dfrac{p}{q} $ is irrational as its decimal expression has non terminating non repeated numbers or digits. But for convenience of calculation we take the value of $ \pi $ as $ \dfrac{{22}}{7} $ looks rational but actually not.