The numbers generally used in mathematics, scientific work, and everyday life are the real numbers. They can be pictured as points of a real line, with the integers equally spaced along the line and a real number b to the right of a real number a if a < b. The set of real numbers is usually denoted by ℝ. As a set, ℝ is uncountable.

Every real number is either rational or irrational, and every real number has an expression as a decimal representation.

The set of real numbers forms an ordered field. The completeness axiom of the real numbers then uniquely characterizes ℝ. See order (real numbers).

The real number line