How are irrational numbers best represented and processed by computers?
There’s nothing special about computers in this context. An infinite number of nonrepeating decimal places cannot be stored or manipulated on a computer, on paper, or in a human brain.
Irrational numbers can be represented in a few different ways:

A symbol that names the number, such as e or π. A computer can use symbolic computation to work with such symbols.

An algorithm that describes how to compute the number. The algorithm can only be run if it can be terminated early to produce an approximation. If the algorithm is a symbolic computation (such as sqrt(2)), it may be manipulated in that form.

A rational approximation of the number, which can be stored using any normal method such as floating point or BCD.


EDIT: Crasic points out in a comment that &quot;...system epsilon is usually well defined for computers. That is, from the perspective of future computation or manipulation within the computer system using standard built in numerical types, you should be able to state what is the smallest unit of numerical precision is for your computer system and can therefore terminate algorithm when your estimated remaining error is less than this known value.&quot;
Technically, you can do this by hand as well, but hand calculations are too slow to make it worthwhile.