In mathematics, an exponential function is a function of the form where b is a positive real number, in which the argument x occurs as an exponent. For real numbers c and d, a function of the form f = a b c x + d is an exponential function, as it can be rewritten as a b c x + d = x; as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b: For b = 1 the real exponential function is a constant and the derivative is zero because log e b = 0, for positive a and b > 1 the real exponential functions are monotonically increasing, because the derivative is greater than zero for all arguments, for b < 1 they are monotonically decreasing, because the derivative is less than zero for all arguments. The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself: Since changing the base of the exponential function results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or "the exponential function" and denoted by While both notations are common, the former notation is used for simpler exponents, while the latter tends to be used when the exponent is a complicated expression.
The exponential function satisfies the fundamental multiplicative identity This identity extends to complex-valued exponents. It can be shown that every continuous, nonzero solution of the functional equation f = f f is an exponential function, f: R → R, x ↦ b x, with b > 0. The fundamental multiplicative identity, along with the definition of the number e as e1, shows that e n = e × ⋯ × e ⏟ n terms for positive integers n and relates the exponential function to the elementary notion of exponentiation; the argument of the exponential function can be any real or complex number or an different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change in the dependent variable; this occurs in the natural and social sciences.
The graph of y = e x is upward-sloping, increases faster as x increases. The graph always lies above the x-axis but can be arbitrarily close to it for negative x; the slope of the tangent to the graph at each point is equal to its y-coordinate at that point, as implied by its derivative function. Its inverse function is the natural logarithm, denoted log, ln, or log e; the real exponential function exp: R → R can be characterized in a variety of equivalent ways. Most it is defined by the following power series: exp := ∑ k = 0 ∞ x k k! = 1 + x + x 2 2 + x 3 6 + x 4 24 + ⋯ Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers z ∈ C (see below for the extension
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group, generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, it contains an element g such that every other element of the group may be obtained by applying the group operation to g or its inverse; each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group; every infinite cyclic group is isomorphic to the additive group of the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n; every cyclic group is an abelian group, every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order; the cyclic groups of prime order are thus among the building blocks from which all groups can be built.
For any element g in any group G, one can form the subgroup of all integer powers ⟨g⟩ =, called the cyclic subgroup of g. The order of g is the number of elements in ⟨g⟩. A cyclic group is a group, equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group with order |G| = n, this means G =, where e is the identity element and gj = gk whenever j ≡ k modulo n. An abstract group defined by this multiplication is denoted Cn, we say that G is isomorphic to the standard cyclic group Cn; such a group is isomorphic to Z/nZ, the group of integers modulo n with the addition operation, the standard cyclic group in additive notation. Under the isomorphism χ defined by χ = i the identity element e corresponds to 0, products correspond to sums, powers correspond to multiples. For example, the set of complex 6th roots of unity G = forms a group under multiplication, it is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6: that is, G = ⟨z⟩ = with z6 = 1.
Under a change of letters, this is isomorphic to the standard cyclic group of order 6, defined as C6 = ⟨g⟩ = with multiplication gj · gk = gj+k, so that g6 = g0 = e. These groups are isomorphic to Z/6Z = with the operation of addition modulo 6, with zk and gk corresponding to k. For example, 1 + 2 ≡ 3 corresponds to z1 · z2 = z3, 2 + 5 ≡ 1 corresponds to z2 · z5 = z7 = z1, so on. Any element generates its own cyclic subgroup, such as ⟨z2⟩ = of order 3, isomorphic to C3 and Z/3Z. Instead of the quotient notations Z/nZ, Z/, or Z/n, some authors denote a finite cyclic group as Zn, but this conflicts with the notation of number theory, where Zp denotes a p-adic number ring, or localization at a prime ideal. On the other hand, in an infinite cyclic group G = ⟨g⟩, the powers gk give distinct elements for all integers k, so that G =, G is isomorphic to the standard group C = C∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, the name "cyclic" may be misleading.
To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group". The set of integers Z,with the operation of addition, forms a group, it is an infinite cyclic group, because all integers can be written by adding or subtracting the single number 1. In this group, 1 and −1 are the only generators; every infinite cyclic group is isomorphic to Z. For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, denoted Z/nZ. A modular integer i is a generator of this group if i is prime to n, because these elements can generate all other elements of the group through integer addition; every finite cyclic group G is isomorphic to Z/nZ. The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings denoted Z and Z/nZ or Z/.
If p is a prime Z/pZ is a finite field, is denoted Fp or GF. For every positive integer n, the set of the integers modulo n that are prime to n is written as ×; this group is not always cyclic, bu
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences cited as Sloane's, is an online database of integer sequences. It was maintained by Neil Sloane while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009. Sloane is president of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, is cited; as of September 2018 it contains over 300,000 sequences. Each entry contains the leading terms of the sequence, mathematical motivations, literature links, more, including the option to generate a graph or play a musical representation of the sequence; the database is searchable by subsequence. Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics; the database was at first stored on punched cards.
He published selections from the database in book form twice: A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and assigned M-numbers from M0000 to M5487; the Encyclopedia includes the references to the corresponding sequences in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not. These books were well received and after the second publication, mathematicians supplied Sloane with a steady flow of new sequences; the collection became unmanageable in book form, when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, soon after as a web site. As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998; the database continues to grow at a rate of some 10,000 entries a year.
Sloane has managed'his' sequences for 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000, was added to the database in November 2011. Besides integer sequences, the OEIS catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences: the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence, 1 5, 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4, 4 5, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5.
Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, it still uses a linear form of conventional mathematical notation. Greek letters are represented by their full names, e.g. mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits always referred to with leading zeros, e.g. A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by periods, or spaces. In comments, etc. A represents the nth term of the sequence. Zero is used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists." The value of a is 2. But there is no such 2×2 magic square, so a is 0; this special usage has a solid mathematical basis in certain counting functions.
For example, the totient valence function. There are 4 solutions for 4, but no solutions for 14, hence a of A014197 is 0—there are no solutions. −1 is used for this purpose instead, as in A094076. The OEIS ma
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; the geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. We take n ≥ 3 here; the associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. For example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°; the order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, it states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I; the axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and