Shamir's Secret Share I'm not quite sure the benefits of working over a prime modulus in Shamir's secret share- but doesn't limiting the numbers you pull from make the secret easier to guess? Instead of being over the real numbers, with infinite elements, limiting yourself to some prime limits the possibilities-albeit that number can get very large. 
Isn't this bad? What benefits are there to working over a prime that justify this choice?
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.