The field of all rational and irrational numbers is called the real numbers,
or simply the "reals," and denoted . The set of real numbers is also called the continuum,
denoted .
The set of reals is called Reals
in the Wolfram Language, and a number
can be tested to see if it is a member
of the reals using the command Element[x, Reals], and expressions
that are real numbers have the Head
of Real.

The real numbers can be extended with the addition of the imaginary numberi, equal to . Numbers of the form , where and
are both real, are called complex numbers, which
also form a field. Another extension which includes both
the real numbers and the infinite ordinal numbers
of Georg Cantor is the surreal numbers.

"Plouffe's Inverter" includes a huge database of 54 million real numbers which are algebraically related to fundamental mathematical constants
and functions.

Almost all real numbers are lexicons, meaning that they do not obey probability laws such as the law of large numbers (Gruber 1991; Calude and Zamfirescu 1998; Trott 2004, p. 69).