Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete. Here, "completely characterized" … See more

Previous properties do not distinguish real numbers from rational numbers. This distinction is provided by Dedekind completeness, which states that every set of real numbers … See more

A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete (in the sense of See more

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. … See more

The real numbers form an ordered field. Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations, addition and multiplication, and a total order that have the following properties. See more

The most common way of describing a real number is via its decimal representation, a sequence of decimal digits each representing the product of an integer between zero … See more

The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers {1, 2, 3, 4, ...} and the set of all real numbers are See more