How do you find out if a number is an irrational number?

Answer 1

If a number is real but it is not rational then it is irrational. So the question can be converted to how to find out if a number is rational.

If the number can be expressed as a ratio of two integers, then it is rational and if not, it is not.

There is a decimal equivalent to this rule but it is harder to apply. If a number can be expressed as a terminating decimal or one in which a finite string of numbers repeats itself endlessly, the number is rational. The difficulty lies with determining when the repeating string will start. Euler's number, e, is irrational but is starts of with 2.718281828 so, if you stopped there you might think that it had gone into a recurring sequence of 1828. It has not. The next few digits are 4590. On the other hand a rational number such as 1/n might not start repeating until the nth digit.