Question

circumference of circle =2πr, so we can write π as circumference/diameter, which we know will be a rational number, so how can we say that π is an irrational number?

Answer 1

π, the ratio of a circle’s circumference to its diameter, is an irrational number, which means it can’t be written as a fraction a/b, where a and b are integers. That means that, unlike decimals like 1/4, or 0.25, or repeating decimals, like 1/3 or 0.33333333, it neither terminates nor repeats. It goes on forever without repeating itself: 3.1415926. . . . ad infinitum.

eg:-

Suppose you are given a circle whose circumference is either $C = 10.123456789 12345678 1234567 12345678 12345678923456789 1orC = 10.123456789 12345678 1234567 12345678 123456789 23456789 2$. How would you go about measuring the circumference to determine which of the two equations is true?

In practice you cannot even measure an object in real life to such accuracy that you can give its size an exact finite decimal expansion with the certainty that it cannot be any other finite decimal. The alternative is to use mathematics to determine what Cshould be, ideally, for some given D. If you start with a finite decimal D and do the mathematics correctly, however, you will never reach the last digit of C, so you'll never be able to use the "add zeros after the decimal" trick.