A real number is any element of the set R, which is the union of the set of rational numbers and the set of irrational numbers. In mathematical expressions, unknown or unspecified real numbers are usually represented by lowercase italic letters u through z. The set R gives rise to other sets such as the set of imaginary numbers and the set of complex numbers. The idea of a real number (and what makes it "real") is primarily of interest to theoreticians. Abstract mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.
If x and z are real numbers such that x < z, then there always exists a real number y such that x < y < z. The set of reals is "dense" in the same sense as the set of irrationals. Both sets are nondenumerable. There are more real numbers than is possible to list, even by implication.
The set R is sometimes called the continuum because it is intuitive to think of the elements of R as corresponding one-to-one with the points on a geometric line. This notion, first proposed by Georg Cantor who also noted the difference between the cardinalities (sizes) of the sets of rational and irrational numbers, is called the Continuum Hypothesis. This hypothesis can be either affirmed or denied without causing contradictions in theoretical mathematics.
See an introduction to real numbers and subsets: