Affinely Extended Real Numbers

The set R union {+infty,-infty} obtained by adjoining two improper elements to the set R of real numbers is normally called the set of (affinely) extended real numbers. Although the notation for this set is not completely standardized, R^_ is commonly used. The set may also be written in interval notation as [-infty,+infty]. With an appropriate topology, R^_ is the two-point compactification (or affine closure) of R. The improper elements, the affine infinities +infty and -infty, correspond to ideal points of the number line. Note that these improper elements are not real numbers, and that this system of extended real numbers is not a field.

Instead of writing +infty, many authors write simply infty. However, the compound symbol +infty will be used here to represent the positive improper element of R^_, allowing the individual symbol infty to be used unambiguously to represent the unsigned improper element of R^*, the one-point compactification (or projective closure) of R.

A very important property of R^_, which R lacks, is that every subset S of R^_ has an infimum (greatest lower bound) and a supremum (least upper bound). In particular, supemptyset=-infty and, if S is unbounded above, then supS=+infty. Similarly, infemptyset=+infty and, if S is unbounded below, then infS=-infty.

Order relations can be extended from R to R^_, and arithmetic operations can be partially extended. For x in R^_,

-infty<x<+infty if x!=+/-infty,-infty<+infty
x+(+infty)=+infty+x=+infty if x!=-infty
x+(-infty)=-infty+x=-infty if x!=+infty
x·(+/-infty)=+/-infty·x=+/-infty if x>0
x·(+/-infty)=+/-infty·x=∓infty if x<0
x/(+/-infty)=0 if x!=+/-infty
|x/0|=+infty if x!=0.

However, the expressions +infty+(-infty), -infty+(+infty), and 0/0 are undefined.

The above statements which define results of arithmetic operations on R^_ may be considered as abbreviations of statements about determinate limit forms. For example, -(+infty)=-infty may be considered as an abbreviation for "If x increases without bound, then -x decreases without bound." Most descriptions of R^_ also make a statement concerning the products of the improper elements and 0, but there is no consensus as to what that statement should be. Some authors (e.g., Kolmogorov 1995, p. 193) state that, like +infty+(-infty) and -infty+(+infty), 0·(+/-infty) and +/-infty·0 should be undefined, presumably because of the indeterminate status of the corresponding limit forms. Other authors (such as McShane 1983, p. 2) accept 0·(+/-infty)=+/-infty·0=0, at least as a convention which is useful in certain contexts.

Many results for other operations and functions can be obtained by considering determinate limit forms. For example, a partial extension of the function f(x,y)=x^y can be obtained for x,y in R^_ as

(+infty)^y={0 if y<0; +infty if y>0
x^(+infty)={0 if 0<x<1; +infty if x>1
x^(-infty)={+infty if 0<x<1; 0 if x>1.

The functions e^x and ln|x| can be fully extended to R^_, with


Some other important functions (e.g., tanh(+/-infty)=+/-1 and tan^(-1)(+/-infty)=+/-pi/2) can be extended to R^_, while others (e.g., sinx, cosx) cannot. Evaluations of expressions involving +infty and -infty, derived by considering determinate limit forms, are routinely used by computer algebra languages such as the Wolfram Language when performing simplifications.

Floating-point arithmetic, with its two signed infinities, is intended to approximate arithmetic on R^_ (Goldberg 1991, pp. 21-22).

See also

Compactification, Indeterminate, Limit, Projectively Extended Real Numbers, R, R--, R-+, Real Number, Set Closure

This entry contributed by David W. Cantrell

Explore with Wolfram|Alpha


Goldberg, D. "What Every Computer Scientist Should Know About Floating-Point Arithmetic." ACM Comput. Surv. 23, 5-48, March 1991., N. A. "Infinity." Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia," 2nd ed., Vol. 3. (Managing Ed. M. Hazewinkel). Dordrecht, Netherlands: Reidel, 1995.McShane, E. J. Unified Integration. Orlando, FL: Academic Press, p. 2, 1983.

Referenced on Wolfram|Alpha

Affinely Extended Real Numbers

Cite this as:

Cantrell, David W. "Affinely Extended Real Numbers." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications