The set of real numbers ℝ can be best understood as all the finite and infinite decimal fractions. ℝ is the first known uncountable set. The term "real" comes from the Latin word "realis," meaning "actual" or "real."

By the 3rd century, it became clear that rational numbers were not sufficient for representing distances in geometry. A comprehensive theory of real numbers emerged in the 19th century with the development of mathematical analysis.

The sets [U:2115], [U:2124], [U:211A], [U:211D], [U:2102] are often written in bold font (for example, in the works of Nicolas Bourbaki) or as double-struck letters on blackboards and manuscripts. American mathematicians introduced the usage of double-struck font in printed texts, and it became popularized with the famous textbook on complex analysis by Gunning and Rossi, published in 1965.