Question

Is the fraction $\dfrac{\pi }{2}$ rational or irrational? Please explain.

Answer 1

Hint: We know that $\pi $ is not a rational number as it cannot be written in the form of $\dfrac{p}{q}$, where q is not equal to zero. Therefore, we have to find if we divide an irrational number from a rational number what we get. We should keep in mind that $\pi $ is not equal to $\dfrac{{22}}{7}$.

Complete step-by-step solution:
A rational number is any number that can be expressed as a fraction of two integers provided that the denominator is not zero.
Mathematically, a rational number can be expressed as
$ \Rightarrow \dfrac{p}{q}\,,\,q \ne 0$
Examples are: $0,1, - 2,0.\overline 3 ,\sqrt 4 ,\dfrac{1}{5},0.6,7\dfrac{8}{9}$
An irrational number is any number that cannot be expressed as a fraction of two integers where the denominator is not zero.
Examples are \[\pi ,e,\sqrt 2 , - \sqrt 3 ,\dfrac{4}{0},\dfrac{{\sqrt 5 }}{3},1.3578239412762....\]
We know that $\pi $ is an irrational number because its approximation is $\dfrac{{22}}{7}$ which is not the exact value of $\pi $ .
Next, we know that the quotient of irrational numbers can be rational or irrational.
Here is the case where we have $\dfrac{\pi }{2}$, an irrational number divided by a rational number giving a quotient that is irrational.
Thus, $\dfrac{\pi }{2}$ is an irrational number.

Note: A rational number is expressible in the form $\dfrac{p}{q}$ for integers p, q with \[q \ne 0\].Any real number that cannot be expressed in this form is called irrational. The number $\pi $ is an irrational number, so cannot be expressed as a fraction, though there are some famous rational approximations to it, namely $\dfrac{{22}}{7}$ and $\dfrac{{355}}{{113}}$.Since π is irrational, it follows that $\dfrac{\pi }{2}$ is also irrational.