 The real numbers are closed under addition, subtraction, multiplication, and division by all real numbers except 0. They have the further feature that there are no “gaps”: for any bounded set of real numbers, there is a least upper bound, and for any continuous function from real numbers to real numbers, if the function is negative at one point and positive at another, there must be some point at which it is exactly equal to 0. For further discussion, see the entry on  Dedekind’s contributions to the foundations of mathematics. 