  The arithmetic of natural numbers has a natural and simple encoding within
sets, and the simplest set whose structure is not that of any natural number
extends this set-theoretic representation to positive and negative integers.
The operation that implements addition when applied to sets that represent
natural numbers yields both addition and subtraction when used with the sets
that encode integers. The encoding of the integers naturally extends beyond
them and identifies sets that encode arithmetic expressions and rational
numbers. The sets that encode arithmetic expressions naturally specify the set
operations that should be performed to evaluate those expressions. The natural
encoding of rational numbers within sets expresses each rational number as a
novel order-preserving form of continued fraction, which provides a new
efficient algorithm for finding rational approximations to irrational numbers.
It also arranges all rational numbers within a tree that shows which numbers
are constituents of which others. The part of this tree containing the positive
rationals coincides with the well-known Stern-Brocot tree, which it extends to
all rationals, including zero and negative numbers, introducing new non-trivial
symmetries. These symmetries overlap with each other and form a group
isomorphic to the group of symmetries of an equilateral triangle.
