How would you explain digital computation and binary math/logic to someone who does not have a mathematics or computer science background? I had about two minutes to think when my brother-in-law asked how computers work. I went with the first useful thought that came to my mind.

Explain what's counting, since counting is where all mathematics begins. Then explain positional representation of numerical values. (As an aside, I should mention this: Until zero and positional representation of values reached Europe through Arabs, clerks in Venice were sullen and bitter as they had to write shitty numbers like XMVXXXIII, instead of nice compact ones like 12829.)

After introducing arithmetic, get him to work out how arithmetic operations would work with just two numbers - 0 and 1. Supply some binary arithmetic problems. 1 + 1 = 100 . Whoa! Explain why this should not be a surprise. Then go on to how all computation is implemented as binary logic circuits. Finally, wear the expression of the Learned and say "that's how computers work".

I think I conveyed the basic idea of computational devices that use binary logic. Still, am sure there are more refined and better ways to do this, even if you have only a few minutes.