How can you approximate an irrational number?

Answer 1

There is a whole branch of mathematics that deals with the question and it is not possible to answer the question wholly here.

For some irrationals, such as roots of polynomials, it is often simple to use the Newton-Raphson method. For other irrationals, such as e, or pi or phi, there are convergent series that can be used.

For example, e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ... The sum of the first 5 ters is accurate to 99.6%.

or

pi = 4/1 - 4/3 + 4/5 - 4/7 + ... This series is excruciatingly slow but there are much faster alternatives.