Real number

In mathematics, the real numbers may be described informally as numbers with an infinite decimal representation, such as 2.4871773339.... The real numbers include the rational numbers, such as 42 and −23/129, and the irrational numbers, such as π and the square root of 2. They can also be visualized, or represented, as points along an infinitely long number line.

A rigorous definition of the real numbers was one of the most important developments of 19th century mathematics. Indeed, several equivalent definitions were developed. Popular approaches that are still used nowadays include

• the real numbers as equivalence classes of Cauchy sequences of rational numbers,
• the real numbers as Dedekind cuts, a more sophisticated version of "decimal representation", and
• the real numbers as the unique complete Archimedean ordered field.

The name real numbers arose to distinguish them from what were then called imaginary numbers (and now complex numbers).