How to handle division by zero in a language that doesn't support exceptions?

In combinatorial game theory, the star game is fuzzy with zero: not less or greater, but not equal either.

The zero game is (iirc) the game where neither the left nor the right player have any moves, so whoever's turn it is loses.

The star game (*) is iirc the game where both players have the move to move to the zero game, so whoever starts wins.

But the star game is fuzzy with the zero game, less than the 1 game, and more than the -1 game.

Iirc.

In order to work with infinitesimals in any reasonable way, you have to define how they work. There is a reasonable way to extend the real numbers in a larger field containing infinitesimals. Such a field is non-Archimedean and has very surprising properties: for example, you can construct a geometry in which Euclid's fifth postulate is false, and yet the angles on a triangle still always add up to 180. These fields also allow you to formalize your intuition about the way infinitesimals like "dx" and "dy" in calculus and analysis work: you can take an equation and multiply both sides by dx, for example. This is ordinarily meaningless, since the term dy/dx is ordinarily just notation that does not actually signify a fraction. In non-standard analysis, it IS a fraction. I think it's a shame that analysis classes in college don't use non-Archimedean fields because the proofs are so much simpler.

However, even in these fields which admit infinitesimals, there is no definition for x/0. You can prove it for yourself.

Suppose x = 10/0.

    0*x = 10
    (0+0)*x = 10
    0*x + 0*x = 10
    10 + 10 = 10
    20 = 10

P.S. Although infinitesimals exist, non-standard analysis still does not permit you to, say, plug ∞ into an equation, because there are many different infinite numbers. However, other systems (projective geometry) admit ∞ as a value.