 - 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.
| LAST UPDATED: |
02 Sep 2001
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