Why does pi never end?

Answer 1

Pi is an irrational number; it can't be represented as a fraction of two integers. It has been proved that the majority of real numbers are irrational. The proof that pi is irrational was found in 1770; it's slightly too complicated to put in this answer, but if you search with Google for pi irrational proof then you will find several different proofs.

Answer 2

pi is called a transcendental number; it is not a rational number and so therefore cannot be written as a 'terminating' decimal (a decimal that ends).