Let ℝ denote the set of all real numbers. Then:
The set ℝ is a field, meaning that addition and multiplication are defined and have the usual properties.
The field ℝ is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
if x ≥ y then x + z ≥ y + z;
if x ≥ 0 and y ≥ 0 then xy ≥ 0.
The order is Dedekind-complete; that is: every non-empty subset S of ℝ with an upper bound in ℝ has a least upper bound (also called supremum) in ℝ.
The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.