In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and , where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or or . It is the Dedekind–MacNeille completion of the real numbers.
When the meaning is clear from context, the symbol is often written simply as .
It is often useful to describe the behavior of a function , as either the argument or the function value gets "infinitely large" in some sense. For example, consider the function
The graph of this function has a horizontal asymptote at . Geometrically, when moving increasingly farther to the right along the -axis, the value of approaches 0. This limiting behavior is similar to the limit of a function in which the real number approaches , except that there is no real number to which approaches.
By adjoining the elements and to , it enables a formulation of a "limit at infinity", with topological properties similar to those for .
To make things completely formal, the Cauchy sequences definition of allows defining as the set of all sequences of rational numbers, such that every is associated with a corresponding for which for all . The definition of can be constructed similarly.
In measure theory, it is often useful to allow sets that have infinite measure and integrals whose value may be infinite.
Such measures arise naturally out of calculus. For example, in assigning a measure to that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering improper 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 would not make sense.
The affinely extended real number system can be turned into a totally ordered set, by defining for all . With this order topology, has the desirable property of compactness: every subset of has a supremum and an infimum (the infimum of the empty set is and its supremum is ). Moreover, with this topology, is homeomorphic to the unit interval . Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric that is an extension of the ordinary metric on .
In this topology, a set is a neighborhood of , if and only if it contains a set for some real number . The notion of the neighborhood of can be defined similarly. Using this characterization of extended-real neighborhoods, the specially defined limits for tending to and , and the specially defined concepts of limits equal to and reduce to the general topological definition of limits.
The arithmetic operations of can be partially extended to as follows:
For exponentiation, see Exponentiation § Limits of powers. Here, "" means both "" and "", while "" means both "" and "".
The expressions and (called indeterminate forms) are usually left undefined. These rules are modeled on the laws for infinite limits. However, in the context of probability or measure theory, is often defined as .
When dealing with both positive and negative extended real numbers, the expression is usually left undefined, because, although it is true that for every real nonzero sequence that converges to , the reciprocal sequence is eventually contained in every neighborhood of , it is not true that the sequence must itself converge to either or Said another way, if a continuous function achieves a zero at a certain value then it need not be the case that tends to either or in the limit as tends to . This is the case for the limits of the identity function when tends to 0, and of (for the latter function, neither nor is a limit of , even if only positive values of are considered).
However, in contexts where only non-negative values are considered, it is often convenient to define . For example, when working with power series, the radius of convergence of a power series with coefficients is often defined as the reciprocal of the limit-supremum of the sequence . Thus, if one allows to take the value , then one can use this formula regardless of whether the limit-supremum is or not.
With these definitions, is not even a semigroup, let alone a group, a ring or a field as in the case of . However, it has several convenient properties:
In general, all laws of arithmetic are valid in —as long as all occurring expressions are defined.
Several functions can be continuously extended to by taking limits. For instance, one may define the extremal points of the following functions as:
Some singularities may additionally be removed. For example, the function can be continuously extended to (under some definitions of continuity), by setting the value to for , and for and . On the other hand, the function can not be continuously extended, because the function approaches as approaches from below, and as approaches from above.
A similar but different real-line system, the projectively extended real line, does not distinguish between and (i.e. infinity is unsigned). As a result, a function may have limit on the projectively extended real 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 at . On the other hand, and correspond on the projectively extended real 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, the functions and cannot be made continuous at on the projectively extended real line.
In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
In probability theory, the expected value of a random variable , denoted or , is a generalization of the weighted average, and is intuitively the arithmetic mean of a large number of independent realizations of . The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment. Expected value is a key concept in economics, finance, and many other subjects.
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.
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.
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
In mathematics, a norm is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.
In real analysis, the projectively extended real line, is the extension of the set of the real numbers, by a point denoted ∞. It is thus the set with the standard arithmetic operations extended where possible, and is sometimes denoted by The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis.
In mathematics, a real coordinate space of dimension n, written Rn or , is a coordinate space over the real numbers. This means that it is the set of the n-tuples of real numbers. With component-wise addition and scalar multiplication, it is a real vector space.
In number theory, natural density is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval [1, n] as n grows large.
In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|z| for some real-valued constant C as |z| → ∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ(z) as opposed to ez.
In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name blancmange comes from its resemblance to a pudding of the same name. It is a special case of the more general de Rham curve; see also fractal curve.
In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.
In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value. The term can take on several different meanings depending on the context. For example:
In mathematics, a square-integrable function, also called a quadratically integrable function or function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows.