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In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced "plus infinity" and "minus infinity"). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted R or [−∞, +∞]. The affinely extended real number system should be distinguished from the projectively extended real numbers by having two infinities, rather than one. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, the real numbers may be described informally in several different ways. ...
The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ...
In mathematics, the real numbers may be described informally in several different ways. ...
In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...
Calculus is an important branch of mathematics. ...
Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...
In mathematics, a measure is a function that assigns a number, e. ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
When the meaning is clear from context, the symbol +∞ is often written simply as ∞.
Motivation
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 of y = 0. Geometrically, as we move farther and farther to the right down the x-axis, the value of gets closer and closer to 0. This limiting behavior is similar to the limit of a function at a real number, except that there is no real number which x is approaching. An asymptote is a straight line or curve which a curve approaches as one moves along the curve. ...
In mathematics, the real numbers may be described informally in several different ways. ...
By adjoining the element +∞ to R, we allow ourselves to formulate a definition of such a "limit at infinity" which is topologically identical to the usual definition at a real number. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
Measure and integration In measure theory, it is often useful to allow sets which have infinite measure and integrals whose value may be infinite. In mathematics, a measure is a function that assigns a number, e. ...
Such measures arise naturally out of calculus. For example, if we are to assign a measure 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 In mathematics, a measure is a function that assigns a number, e. ...
the value "infinity" arises. Finally, we often wish 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 would not make sense. Monotone convergence theorem, in mathematics, may refer to several theorems, all of which are concerned with a monotonic function in one way or another: Monotonic function refers to the convergence of an infinite series that is monotonic Dominated convergence theorem refers to Lebesgues monotone convergence theorem Categories: | | ...
In mathematics, Lebesgues dominated convergence theorem states that if a sequence { fn : n = 1, 2, 3, ... } of real-valued measurable functions on a measure space S converges almost everywhere, and is dominated (explained below) by some nonnegative function g in , then To say that the sequence is dominated by...
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 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 that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to 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). ...
A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...
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. ...
Arithmetic operations The arithmetic operations of R can be partially extended to R as follows: - a + ∞ = +∞ + a = +∞ if a ≠ −∞
- a − ∞ = −∞ + a = −∞ if a ≠ +∞
- a × ±∞ = ±∞ × a = ±∞ if a > 0
- a × ±∞ = ±∞ × a = ∓∞ if a < 0
- a / ±∞ = 0 if −∞ < a < +∞
- ±∞ / a = ±∞ if 0 < a < +∞
- ±∞ / a = ∓∞ if −∞ < a < 0
Here, "a + ∞" means both "a + (+∞)" and "a - (−∞)", and "a − ∞" means both "a − (+∞)" and "a + (−∞)".
The expressions ∞ − ∞, 0 × ±∞ and ±∞ / ±∞ 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 usually defined as 0. In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
Note that 1 / 0 is not defined as either +∞ or −∞, because although it is true that whenever f(x) → 0 for a continuous function f(x), we must have that 1/f(x) is eventually in every neighborhood of the set {−∞, +∞}, it is not true that 1/f(x) must converge to one of these points. An example is f(x) = 1/(sin(1/x)). In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
Algebraic properties 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 properties listed below. ...
- 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.
Miscellaneous Several functions can be continuously extended to R by taking limits. For instance, one defines exp(−∞) = 0, exp(+∞) = +∞, ln(0) = −∞, ln(+∞) = +∞ etc. Partial plot of a function f. ...
In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...
The exponential function is one of the most important functions in mathematics. ...
The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...
Compare the real projective line, which does not distinguish between +∞ and −∞. In mathematics, the projective line is a fundamental example of an algebraic curve. ...
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