Mathematics
Constructing the Integers from Peano Axioms
Quick fact
In the standard construction, the integer 0 is defined as the equivalence class of the pair (0,0), and the integer -1 is the equivalence class of (0,1), using a clever rule that treats pairs as differences.
Why this is interesting
You've counted 1, 2, 3... but where does 0 come from? And how do you get -3 when you only have the counting numbers? Surprise: integers are not given—they are built.