Extended real number line

# Extended real number line

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the affinely extended real number system is obtained from the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

system R by adding two elements: +∞ and −∞ (read as positive infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

and negative infinity respectively). The projective extended real number system adds a single object, ∞ (infinity) and makes no distinction between "positive" or "negative" infinity. These new elements are not real numbers. It is useful in describing various limiting behavior
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

s in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, especially in the theory of measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

and integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

.
Discussion
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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the affinely extended real number system is obtained from the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

system R by adding two elements: +∞ and −∞ (read as positive infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

and negative infinity respectively). The projective extended real number system adds a single object, ∞ (infinity) and makes no distinction between "positive" or "negative" infinity. These new elements are not real numbers. It is useful in describing various limiting behavior
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

s in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, especially in the theory of measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

and integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

. The affinely extended real number system is denoted or [−∞, +∞].

When the meaning is clear from context, the symbol +∞ is often written simply as ∞.

### Limits

We often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) gets "very big" in some sense. For example, consider the function

The graph of this function has a horizontal asymptote
Asymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

at f(x) = 0. Geometrically, as we move farther and farther to the right along the x-axis, the value of 1/x2 approaches 0. This limiting behavior is similar to the limit of a function
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

at a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

, except that there is no real number to which x approaches.

By adjoining the elements +∞ and −∞ to R, we allow a formulation of a "limit at infinity" with topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

properties similar to those for R.

To make things completely formal, the Cauchy sequences definition of R allows us to define +∞ as the set of all sequences of rationals which, for any K>0, from some point on exceed K. We can define −∞ similarly.

### Measure and integration

In measure theory, it is often useful to allow sets which have infinite measure and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, in assigning a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

to R that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering infinite integrals, such as

the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as

Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and the dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...

would not make sense.

## Order and topological properties

The affinely extended real number system turns into a totally ordered set by defining −∞ ≤ a ≤ +∞ for all a. This order has the desirable property that every subset has a supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

and an infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

: it is a complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...

.

This induces the order topology
Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...

on . 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 −∞. is a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

to the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...

[0, 1]. Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric which is an extension of the ordinary metric on R.

With this topology the specially defined limits
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

for x tending to +∞ and −∞, and the specially defined concepts of limits equal to +∞ and −∞, reduce to the general topological definitions of limits.

## Arithmetic operations

The arithmetic operations of R can be partially extended to as follows:

Here, "a + ∞" means both "a + (+∞)" and "a − (−∞)", and "a − ∞" means both "a − (+∞)" and "a + (−∞)".

The expressions ∞ − ∞, 0 × (±∞) and ±∞ / ±∞ (called indeterminate form
Indeterminate form
In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution...

s) are usually left undefined. These rules are modeled on the laws for infinite limits. However, in the context of probability or measure theory, 0 × (±∞) is often defined as 0.

The expression 1/0 is not defined either as +∞ or −∞, because although it is true that whenever f(x) → 0 for a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

f(x) it must be the case that 1/f(x) is eventually contained in every neighborhood of the set {−∞, +∞}, it is not true that 1/f(x) must tend to one of these points. An example is f(x) = 1/(sin(1/x)). (Its modulus
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

1/| f(x) |, nevertheless, does approach +∞.)

## Algebraic properties

With these definitions is not a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

and not even a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

. However, it still has several convenient properties:
• 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 ab and if both a + c and b + c are defined, then a + cb + c.
• if ab and c > 0 and both a × c and b × c are defined, then a × cb × c.

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

## Miscellaneous

Several functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

can be continuously extended to by taking limits. For instance, one defines exp
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

(−∞) = 0, exp(+∞) = +∞, ln
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

(0) = −∞, ln(+∞) = +∞ etc.

Some discontinuities may additionally be removed. For example, the function 1/x2 can be made continuous (under some definitions of continuity) by setting the value to +∞ for x = 0, and 0 for x = +∞ and x = −∞. The function 1/x can not be made continuous because the function approaches −∞ as x approaches 0 from below, and +∞ as x approaches 0 from above.

Compare the real projective line, which does not distinguish between +∞ and −∞. As a result, on one hand a function may have limit ∞ on the real projective line, while in the affinely extended real number system only the absolute value of the function has a limit, e.g. in the case of the function 1/x at x = 0. On the other hand and
correspond on the real projective line to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus ex and arctan(x) cannot be made continuous at x = ∞ on the real projective line.