Question

Is \[\pi \] a rational number? Justify your answer.

Answer 1

Hint: Any number in the form \[\dfrac{p}{q}\] where p and q are both integers and q is not equal to 0 is called a rational number it must be clear that p and q individually cannot be any type of fraction or decimal number.
Complete step by step answer:
\[\pi \] itself is a non-terminating, non-recurring, Which clearly means we cannot find a ratio of 2 numbers p and q in the form \[\dfrac{p}{q}\] . It has a never ending value after \[3.14................\] . Thus we cannot conclude any ratio the number exists on our real number line but it certainly does not exist in any other number line. \[\pi \] has an approximation value in rational number as \[\dfrac{{22}}{7} = 3.1428571.....\]
But in reality it has a value of 3.14159265… We also have another rational number which after conversion matches till the \[{6^{th}}\] place of the decimal the number is
\[\dfrac{{355}}{{133}} = 3.1415929.....\] but in any case both numbers cannot be treated as an alternative of \[\pi \] in rational numbers.
Therefore \[\pi \] is irrational.

Note: It must be very much clear that \[\pi \approx \dfrac{{22}}{7}\] . There is no substitute of \[\pi \] in a rational number. Therefore all the values we put in place of \[\pi \] to get an answer for questions like area or circumference of a circle we get approximate values ones we put the value of \[\pi \] .