The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). It is useful in mathematical analysis, especially in integration theory. The extended real number line is denoted by R or [−∞,+∞]. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... Integration may be any of the following: Usually integration is the construction of an object, a theory, etc. ...

The extended real number line turns into a totally ordered set by defining −∞ ≤ a ≤ +∞ for all a. This order has the nice property that every subset has a supremum and an infimum: it is a complete lattice. The total order induces a topology on R. In this topology, a set U is a neighborhood of +∞ if and only if it contains a set {x : x ≥ a} for some real number a, and analogously for the neighborhoods of −∞. R is a compact Hausdorff space homeomorphic to the unit interval [0,1]. In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... Several specialized usages of the terms compact and compactness exist. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... This word should not be confused with homomorphism. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

The arithmetical operations of R can be partly extended to R as follows:

• a + ∞ = ∞ + a = ∞    if a ≠ −∞
• a − ∞ = −∞ + a = −∞    if a ≠ +∞
• a × +∞ = +∞ × a = +∞    if a > 0
• a × +∞ = +∞ × a = −∞    if a < 0
• a × −∞ = −∞ × a = −∞    if a > 0
• a × −∞ = −∞ × a = +∞    if a < 0
• a / ±∞ = 0    if −∞ < a < +∞
• ±∞ / a = ±∞    if 0 < a < +∞
• +∞ / a = −∞    if −∞ < a < 0
• −∞ / a = +∞    if −∞ < a < 0

The expressions ∞ − ∞, 0 × ±∞ and ±∞ / ±∞ are usually left undefined. Also, 1 / 0 is not defined as +∞ (because −∞ would be just as good a candidate). These rules are modeled on the laws for infinite limits. In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...

Note that with these definitions, R is not a field and not even a ring. However, it still has several convenient properties: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...

• a + (b + c) and (a + b) + c are either equal or both undefined.
• a + b and b + a are either equal or both undefined.
• a × (b × c) and (a × b) × c are either equal or both undefined.
• a × b and b × a are either equal or both undefined
• a × (b + c) and (a × b) + (a × c) are equal if both are defined.
• if a ≤ b and if both a + c and b + c are defined, then a + c ≤ b + c.
• if a ≤ b and c > 0 and both a × c and b × c are defined, then a × c ≤ b × c.

In general, all laws of arithmetic are valid in R as long as all occurring expressions are defined.

By using the intuition of limits, several functions can be naturally extended to R. For instance, one defines exp(−∞) = 0, exp(+∞) = +∞, ln(0) = −∞, ln(+∞) = ∞ etc. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... The exponential function is one of the most important functions in mathematics. ... The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ...