2.3 Real Numbers

Real numbers are a mathematical abstraction commonly used when modelling real-world phenomenon. Real numbers are an extension of the rational numbers [64]. The set of reals is denoted by   .

Each real number can be specified by a converging infinite sequence of rational numbers [26]. The limit of the sequence is the value of the real number.

The real numbers have a partial square root operator as did the rationals. Although the square root operator is defined for all non-negative real numbers, it is not defined for any negative real numbers. There is a natural homomorphism from the rationals to the reals, defined by:

The rationals are envisioned as being a subset of the reals because of this natural homomorphism. Addition and multiplication are associative and commutative over the reals and jointly satisfy the distributive law. Both operators have inverses, as they did with the rationals. The real number system is preferred over the rational system by mathematicians because many popular operations are closed over the reals. The set of real numbers not in are called the irrationals and are denoted by   ; it is these numbers that allow many common operators to be closed over the reals.

Common Practice

Modelling phenomenon with real numbers is overkill in most cases. Efficiently computing with real numbers directly is quite difficult. In some cases, operations involving real numbers are not computable [42, 60]. Many computational difficulties can be overcome by using a suitable representation for real numbers [69]. This will be discussed at this chapter's end. Even when numerical computation using reals is desirable, symbolic computation can sometimes be used instead.

Jeff Tupper

March 1996