Otherwise, starting from a set of axioms, it is possible to derive the usual properties of arithmetic, even including long multiplication and division. The proofs, though simple, are tricky.

This Demonstration shows two proofs by example that for any two positive integers and , , that is, a negative times a negative is a positive. The proofs can be adapted to a proper algebraic proof with letters replacing the variable numbers.

The statements on the right explain the steps taken. Those statements are either axioms or previously proved statements (i.e., theorems). For example, the fact that multiplication by zero is zero is not an axiom; it takes six steps to prove it.

The axioms needed for rational or real numbers are in this table (see the related links):

It is also necessary to talk about closure and to prove the uniqueness of the identities and of inverses, so the discussion here is not complete. The abbreviation is used to avoid some clutter. Division is not used in this Demonstration.