Shamir's Secret Share
Computers are unable to handle real numbers, especially irrational numbers.

Computers can handle a fraction of the rational numbers (for being limited in size in a binary representation), and without some special tricks it can't handle the entire field $\mathbb{Q}$.

So you're down to integer arithmetic anyway (if you consider floats: rounding and cancellation can really mess things up). However, addition and multiplication over integers are just a ring structure and not a field (multiplicative inverse elements do not exist). So if you need to invert multiplication you need to use the finite structure.