@@6123 In vector calculus , the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field . At every point in the field , the curl of that field is represented by a vector . The attributes of this vector ( length and direction ) characterize the rotation at that point . The direction of the curl is the axis of rotation , as determined by the right-hand rule , and the magnitude of the curl is the magnitude of rotation . If the vector field represents the flow velocity of a moving fluid , then the curl is the circulation density of the fluid . A vector field whose curl is zero is called irrotational . The curl is a form of differentiation for vector fields . The corresponding form of the fundamental theorem of calculus is Stokes ' theorem , which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve . The alternative terminology ' ' rotor ' ' or ' ' rotational ' ' and alternative notations rot F and F are often used ( the former especially in many European countries , the latter , using the del operator and the cross product , is more used in other countries ) for ' ' curl ' ' and curl F . Unlike the gradient and divergence , curl does not generalize as simply to other dimensions ; some generalizations are possible , but only in three dimensions is the geometrically defined curl of a vector field again a vector field . This is a similar phenomenon as in the 3 dimensional cross product , and the connection is reflected in the notation for the curl . The name curl was first suggested by James Clerk Maxwell in 1871. # Definition # The curl of a vector field F , denoted by curl F , or F , or rot F , at a point is defined in terms of its projection onto various lines through the point . If is any unit vector , the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point , divided by the area enclosed . As such , the curl operator maps continuously differentiable functions f : R 3 R 3 to continuous functions g : R 3 R 3 . In fact , it maps ' ' C ' ' k functions in R 3 to ' ' C ' ' k-1 functions in R 3 . Implicitly , curl is defined by : : where is a line integral along the boundary of the area in question , and ' ' A ' ' is the magnitude of the area . If is an outward pointing in-plane normal , whereas is the unit vector perpendicular to the plane ( see caption at right ) , then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R 3 ( right-hand rule ) . The above formula means that the curl of a vector field is defined as the infinitesimal area density of the ' ' circulation ' ' of that field . To this definition fit naturally the Kelvin-Stokes theorem , as a global formula corresponding to the definition , and the following easy to memorize definition of the curl in curvilinear orthogonal coordinates , e.g. in Cartesian coordinates , spherical , cylindrical , or even elliptical or parabolical coordinates : : : , : : , : : . Note that the equation for each component , can be obtained by exchanging each occurrence of a subscript 1 , 2 , 3 in cyclic permutation : 12 , 23 , and 31 ( where the subscripts represent the relevant indices ) . If ( ' ' x ' ' 1 , ' ' x ' ' 2 , ' ' x ' ' 3 ) are the Cartesian coordinates and ( ' ' u ' ' 1 , ' ' u ' ' 2 , ' ' u ' ' 3 ) are the orthogonal coordinates , then : is the length of the coordinate vector corresponding to ' ' u i ' ' . The remaining two components of curl result from cyclic permutation of indices : 3,1,2 1,2,3 2,3,1. # Intuitive interpretation # Suppose the vector field describes the velocity field of a fluid flow ( such as a large tank of liquid or gas ) and a small ball is located within the fluid or gas ( the centre of the ball being fixed at a certain point ) . If the ball has a rough surface , the fluid flowing past it will make it rotate . The rotation axis ( oriented according to the right hand rule ) points in the direction of the curl of the field at the centre of the ball , and the angular speed of the rotation is half the magnitude of the curl at this point . # Usage # In practice , the above definition is rarely used because in virtually all cases , the curl operator can be applied using some set of curvilinear coordinates , for which simpler representations have been derived . The notation F has its origins in the similarities to the 3 dimensional cross product , and it is useful as a mnemonic in Cartesian coordinates if we take as a vector differential operator del . Such notation involving operators is common in physics and algebra . However , in certain coordinate systems , such as polar-toroidal coordinates ( common in plasma physics ) , using the notation F will yield an incorrect result . Expanded in Cartesian coordinates ( see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations ) , F is , for F composed of ' ' F ' ' x , ' ' F ' ' y , ' ' F ' ' z : : where i , j , and k are the unit vectors for the ' ' x ' ' - , ' ' y ' ' - , and ' ' z ' ' -axes , respectively . This expands as follows : : Although expressed in terms of coordinates , the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection . In a general coordinate system , the curl is given by : where denotes the Levi-Civita symbol , the metric tensor is used to lower the index on F , and the Einstein summation convention implies that repeated indices are summed over . Equivalently , : where e ' ' k ' ' are the coordinate vector fields . Equivalently , using the exterior derivative , the curl can be expressed as : : Here and are the musical isomorphisms , and is the Hodge dual . This formula shows how to calculate the curl of F in any coordinate system , and how to extend the curl to any oriented three-dimensional Riemannian manifold . Since this depends on a choice of orientation , curl is a chiral operation . In other words , if the orientation is reversed , then the direction of the curl is also reversed . # Examples # # A simple vector field # Take the vector field , which depends on ' ' x ' ' and ' ' y ' ' linearly : : Its plot looks like this : Simply by visual inspection , we can see that the field is rotating . If we place a paddle wheel anywhere , we see immediately its tendency to rotate clockwise . Using the right-hand rule , we expect the curl to be into the page . If we are to keep a right-handed coordinate system , into the page will be in the negative z direction . The lack of x and y directions is analogous to the cross product operation . If we calculate the curl : : Which is indeed in the negative ' ' z ' ' direction , as expected . In this case , the curl is actually a constant , irrespective of position . The amount of rotation in the above vector field is the same at any point ( ' ' x ' ' , ' ' y ' ' ) . Plotting the curl of ' ' F ' ' is not very interesting : # A more involved example # Suppose we now consider a slightly more complicated vector field : : Its plot : We might not see any rotation initially , but if we closely look at the right , we see a larger field at , say , x=4 than at x=3 . Intuitively , if we placed a small paddle wheel there , the larger current on its right side would cause the paddlewheel to rotate clockwise , which corresponds to a curl in the negative z direction . By contrast , if we look at a point on the left and placed a small paddle wheel there , the larger current on its left side would cause the paddlewheel to rotate counterclockwise , which corresponds to a curl in the positive z direction . Let 's check out our guess by doing the math : : Indeed the curl is in the positive z direction for negative x and in the negative z direction for positive x , as expected . Since this curl is not the same at every point , its plot is a bit more interesting : We note that the plot of this curl has no dependence on y or z ( as it should n't ) and is in the negative z direction for positive x and in the positive z direction for negative x. # Identities # Consider the example ( v F ) . Using Cartesian coordinates , it can be shown that : : In the case where the vector field v and are interchanged : : : which introduces the Feynman subscript notation F , which means the subscripted gradient operates only on the factor F . Another example is ( F ) . Using Cartesian coordinates , it can be shown that : : : which can be construed as a special case of the previous example with the substitution v . ( Note : 2 F represents the vector Laplacian of F ) The curl of the gradient of ' ' any ' ' scalar field is always the zero vector : : : If is a scalar valued function and F is a vector field , then : : # Descriptive examples # In a vector field describing the linear velocities of each part of a rotating disk , the curl has the same value at all points . Of the four Maxwell 's equations , two -- Faraday 's law and Ampre 's law -- can be compactly expressed using curl . Faraday 's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field , while Ampre 's law relates the curl of the magnetic field to the current and rate of change of the electric field . # Generalizations # The vector calculus operations of grad , curl , and div are most easily generalized and understood in the context of differential forms , which involves a number of steps . In a nutshell , they correspond to the derivatives of 0-forms , 1-forms , and 2-forms , respectively . The geometric interpretation of curl as rotation corresponds to identifying bivectors ( 2-vectors ) in 3 dimensions with the special orthogonal Lie algebra so ( 3 ) of infinitesimal rotations ( in coordinates , skew-symmetric 3 3 matrices ) , while representing rotations by vectors corresponds to identifying 1-vectors ( equivalently , 2-vectors ) and so ( 3 ) , these all being 3-dimensional spaces . # Differential forms # In 3 dimensions , a differential 0-form is simply a function ' ' f ' ' ( ' ' x ' ' , ' ' y ' ' , ' ' z ' ' ) ; a differential 1-form is the following expression : a differential 2-form is the formal sum : and a differential 3-form is defined by a single term : ( Here the a-coefficients are real functions ; the wedge products , e.g. can be interpreted as some kind of oriented area elements , , etc . ) The exterior derivative of a ' ' k ' ' -form in R 3 is defined as the ( ' ' k ' ' +1 ) -form from above ( and in R ' ' n ' ' if , e.g. , : in this expression ensures that the total area under the curve ' ' ' ' ( ' ' x ' ' ) is equal to one proof . The in the exponent ensures that the distribution has unit variance ( and therefore also unit standard deviation ) . This function is symmetric around ' ' x ' ' =0 , where it attains its maximum value *43;8204;math scriptstyle 1/sqrt2pi ; and has inflection points at +1 and 1 . # in calculus # is the value of such that the gradient of ' ' a ' ' ' ' x ' ' at 0 equals 1 . This is the blue curve , . Functions ( dotted curve ) and ( dashed curve ) are also shown ; they are not tangent to the line of slope 1 ( red ) . The principal motivation for introducing the number , particularly in calculus , is to perform differential and integral calculus with exponential functions and logarithms . A general exponential function ' ' a ' ' ' ' x ' ' has derivative given as the limit : : The limit on the far right is independent of the variable : it depends only on the base . When the base is , this limit is equal to 1 , and so is symbolically defined by the equation : : Consequently , the exponential function with base is particularly suited to doing calculus . Choosing , as opposed to some other number , as the base of the exponential function makes calculations involving the derivative much simpler . Another motivation comes from considering the base- logarithm . Considering the definition of the derivative of as the limit : : where the substitution ' ' h ' ' / ' ' x ' ' was made in the last step . The last limit appearing in this calculation is again an undetermined limit that depends only on the base , and if that base is , the limit is equal to 1 . So symbolically , : The logarithm in this special base is called the natural logarithm and is represented as ; it behaves well under differentiation since there is no undetermined limit to carry through the calculations . There are thus two ways in which to select a special number ' ' e ' ' . One way is to set the derivative of the exponential function to , and solve for . The other way is to set the derivative of the base logarithm to and solve for . In each case , one arrives at a convenient choice of base for doing calculus . In fact , these two solutions for are actually ' ' the same ' ' , the number . # Alternative characterizations # Other characterizations of are also possible : one is as the limit of a sequence , another is as the sum of an infinite series , and still others rely on integral calculus . So far , the following two ( equivalent ) properties have been introduced : 1 . The number is the unique positive real number such that : 2 . The number is the unique positive real number such that : The following three characterizations can be proven equivalent : 3 . The number is the limit : Similarly : : 4 . The number is the sum of the infinite series : where is the factorial of . 5 . The number is the unique positive real number such that : # Properties # # Calculus # As in the motivation , the exponential function is important in part because it is the unique nontrivial function ( up to multiplication by a constant ) which is its own derivative : and therefore its own antiderivative as well : : # Exponential-like functions # The global maximum for the function : occurs at ' ' e ' ' . Similarly , 1/ ' ' e ' ' is where the global minimum occurs for the function : defined for positive . More generally , ' ' e ' ' 1/ ' ' n ' ' is where the global minimum occurs for the function : for any . The infinite tetration : or converges if and only if ( or approximately between 0.0660 and 1.4447 ) , due to a theorem of Leonhard Euler. # Number theory # The real number is irrational . Euler proved this by showing that its simple continued fraction expansion is infinite . ( See also Fourier 's proof that is irrational . ) Furthermore , by the LindemannWeierstrass theorem , is transcendental , meaning that it is not a solution of any non-constant polynomial equation with rational coefficients . It was the first number to be proved transcendental without having been specifically constructed for this purpose ( compare with Liouville number ) ; the proof was given by Charles Hermite in 1873 . It is conjectured that is normal , meaning that when is expressed in any base the possible digits in that base are uniformly distributed ( occur with equal probability in any sequence of given length ) . # Complex numbers # The exponential function may be written as a Taylor series : Because this series keeps many important properties for even when is complex , it is commonly used to extend the definition of to the complex numbers . This , with the Taylor series for sin and cos , allows one to derive Euler 's formula : : which holds for all . The special case with is Euler 's identity : : from which it follows that , in the principal branch of the logarithm , : Furthermore , using the laws for exponentiation , : which is de Moivre 's formula . The expression : is sometimes referred to as . # Differential equations # The general function : is the solution to the differential equation : : # Representations # The number can be represented as a real number in a variety of ways : as an infinite series , an infinite product , a continued fraction , or a limit of a sequence . The chief among these representations , particularly in introductory calculus courses is the limit : given above , as well as the series : given by evaluating the above power series for at 1 . Less common is the continued fraction . : which written out looks like : This continued fraction for converges three times as quickly : : which written out looks like : Many other series , sequence , continued fraction , and infinite product representations of have been developed . # Stochastic representations # In addition to exact analytical expressions for representation of , there are stochastic techniques for estimating . One such approach begins with an infinite sequence of independent random variables , ... , drawn from the uniform distribution on 0 , 1 . Let be the least number such that the sum of the first samples exceeds 1 : : Then the expected value of is : ' ' e ' ' . # Known digits # The number of known digits of has increased dramatically during the last decades . This is due both to the increased performance of computers and to algorithmic improvements . # In computer culture # In contemporary internet culture , individuals and organizations frequently pay homage to the number . For instance , in the IPO filing for Google in 2004 , rather than a typical round-number amount of money , the company announced its intention to raise $2,718,281,828 , which is billion dollars to the nearest dollar . Google was also responsible for a billboard that appeared in the heart of Silicon Valley , and later in Cambridge , Massachusetts ; Seattle , Washington ; and Austin , Texas . It read first 10-digit prime found in consecutive digits of . com . Solving this problem and visiting the advertised ( now defunct ) web site led to an even more difficult problem to solve , which in turn led to Google Labs where the visitor was invited to submit a resume . The first 10-digit prime in is 7427466391 , which starts at the 99th digit . In another instance , the computer scientist Donald Knuth let the version numbers of his program Metafont approach . The versions are 2 , 2.7 , 2.71 , 2.718 , and so forth . Similarly , the version numbers of his TeX program approach . # Notes # # Further reading # Maor , Eli ; ' ' : The Story of a Number ' ' , ISBN 0-691-05854-7 of the book ' ' Prime Obsession ' ' for another stochastic representation
@@10603 In abstract algebra , a field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element , or equivalently a ring whose nonzero elements form an abelian group under multiplication . As such it is an algebraic structure with notions of addition , subtraction , multiplication , and division satisfying the appropriate abelian group equations and distributive law . The most commonly used fields are the field of real numbers , the field of complex numbers , and the field of rational numbers , but there are also finite fields , fields of functions , algebraic number fields , ' ' p ' ' -adic fields , and so forth . Any field may be used as the scalars for a vector space , which is the standard general context for linear algebra . The theory of field extensions ( including Galois theory ) involves the roots of polynomials with coefficients in a field ; among other results , this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge , as well as a proof of the AbelRuffini theorem on the algebraic insolubility of quintic equations . In modern mathematics , the theory of fields ( or field theory ) plays an essential role in number theory and algebraic geometry . As an algebraic structure , every field is a ring , but not every ring is a field . The most important difference is that fields allow for division ( though not division by zero ) , while a ring need not possess multiplicative inverses ; for example the integers form a ring , but 2 ' ' x ' ' = 1 has no solution in integers . Also , the multiplication operation in a field is required to be commutative . A ring in which division is possible but commutativity is not assumed ( such as the quaternions ) is called a ' ' division ring ' ' or ' ' skew field ' ' . ( Historically , division rings were sometimes referred to as fields , while fields were called ' ' commutative fields ' ' . ) As a ring , a field may be classified as a specific type of integral domain , and can be characterized by the following ( not exhaustive ) chain of class inclusions : : Commutative rings integral domains integrally closed domains unique factorization domains principal ideal domains Euclidean domains fields finite fields . # Definition and illustration # Intuitively , a field is a set ' ' F ' ' that is a commutative group with respect to two compatible operations , addition and multiplication , with compatible being formalized by ' ' distributivity , ' ' and the caveat that the additive identity ( 0 ) has no multiplicative inverse ( one can not divide by 0 ) . The most common way to formalize this is by defining a ' ' field ' ' as a set together with two operations , usually called ' ' addition ' ' and ' ' multiplication ' ' , and denoted by + and , respectively , such that the following axioms hold ; ' ' subtraction ' ' and ' ' division ' ' are defined implicitly in terms of the inverse operations of addition and multiplication : *18;191367;ref That is , the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ) ; this is not taken as given , but is implicitly defined in terms of addition as is the unique ' ' b ' ' such that , implicitly because it is defined in terms of solving an equationand one then defines the binary operation of subtraction , also denoted by , as in terms of addition and additive inverse . In the same way , one defines the binary operation of division in terms of the assumed binary operation of multiplication and the implicitly defined operation of reciprocal ( multiplicative inverse ) . ; ' ' Closure ' ' of ' ' F ' ' under addition and multiplication : For all ' ' a ' ' , ' ' b ' ' in ' ' F ' ' , both ' ' a ' ' + ' ' b ' ' and ' ' a ' ' ' ' b ' ' are in ' ' F ' ' ( or more formally , + and are binary operations on ' ' F ' ' ) . ; ' ' Associativity ' ' of addition and multiplication : For all ' ' a ' ' , ' ' b ' ' , and ' ' c ' ' in ' ' F ' ' , the following equalities hold : ' ' a ' ' + ( ' ' b ' ' + ' ' c ' ' ) = ( ' ' a ' ' + ' ' b ' ' ) + ' ' c ' ' and ' ' a ' ' ( ' ' b ' ' ' ' c ' ' ) = ( ' ' a ' ' ' ' b ' ' ) ' ' c ' ' . ; ' ' Commutativity ' ' of addition and multiplication : For all ' ' a ' ' and ' ' b ' ' in ' ' F ' ' , the following equalities hold : ' ' a ' ' + ' ' b ' ' = ' ' b ' ' + ' ' a ' ' and ' ' a ' ' ' ' b ' ' = ' ' b ' ' ' ' a ' ' . ; Existence of additive and multiplicative ' ' identity elements ' ' : There exists an element of ' ' F ' ' , called the ' ' additive identity ' ' element and denoted by 0 , such that for all ' ' a ' ' in ' ' F ' ' , ' ' a ' ' + 0 = ' ' a ' ' . Likewise , there is an element , called the ' ' multiplicative identity ' ' element and denoted by 1 , such that for all ' ' a ' ' in ' ' F ' ' , ' ' a ' ' 1 = ' ' a ' ' . To exclude the trivial ring , the additive identity and the multiplicative identity are required to be distinct . ; Existence of ' ' additive inverses ' ' and ' ' multiplicative inverses ' ' : For every ' ' a ' ' in ' ' F ' ' , there exists an element ' ' a ' ' in ' ' F ' ' , such that ' ' a ' ' + ( ' ' a ' ' ) = 0 . Similarly , for any ' ' a ' ' in ' ' F ' ' other than 0 , there exists an element ' ' a ' ' 1 in ' ' F ' ' , such that ' ' a ' ' ' ' a ' ' 1 = 1 . ( The elements ' ' a ' ' + ( ' ' b ' ' ) and ' ' a ' ' ' ' b ' ' 1 are also denoted ' ' a ' ' ' ' b ' ' and ' ' a ' ' / ' ' b ' ' , respectively . ) In other words , ' ' subtraction ' ' and ' ' division ' ' operations exist . ; ' ' Distributivity ' ' of multiplication over addition : For all ' ' a ' ' , ' ' b ' ' and ' ' c ' ' in ' ' F ' ' , the following equality holds : ' ' a ' ' ( ' ' b ' ' + ' ' c ' ' ) = ( ' ' a ' ' ' ' b ' ' ) + ( ' ' a ' ' ' ' c ' ' ) . A field is therefore an algebraic structure ' ' F ' ' , + , , , 1 , 0 , 1 ; of type 2 , 2 , 1 , 1 , 0 , 0 , consisting of two abelian groups : ' ' F ' ' under + , , and 0 ; ' ' F ' ' 0 under , 1 , and 1 , with 0 1 , with distributing over + . # First example : rational numbers # A simple example of a field is the field of rational numbers , consisting of numbers which can be written as fractions ' ' a ' ' / ' ' b ' ' , where ' ' a ' ' and ' ' b ' ' are integers , and ' ' b ' ' 0 . The additive inverse of such a fraction is simply ' ' a ' ' / ' ' b ' ' , and the multiplicative inverse ( provided that ' ' a ' ' 0 ) is ' ' b ' ' / ' ' a ' ' . To see the latter , note that : The abstractly required field axioms reduce to standard properties of rational numbers , such as the law of distributivity : : : : : or the law of commutativity and law of associativity . # Second example : a field with four elements # In addition to familiar number systems such as the rationals , there are other , less immediate examples of fields . The following example is a field consisting of four elements called O , I , A and B. The notation is chosen such that O plays the role of the additive identity element ( denoted 0 in the axioms ) , and I is the multiplicative identity ( denoted 1 above ) . One can check that all field axioms are satisfied . For example : : A ( B + A ) = A I = A , which equals A B + A A = I + B = A , as required by the distributivity . The above field is called a finite field with four elements , and can be denoted F 4 . Field theory is concerned with understanding the reasons for the existence of this field , defined in a fairly ad-hoc manner , and describing its inner structure . For example , from a glance at the multiplication table , it can be seen that any non-zero element ( i.e. , I , A , and B ) is a power of A : A = A 1 , B = A 2 = A A , and finally I = A 3 = A A A. This is not a coincidence , but rather one of the starting points of a deeper understanding of ( finite ) fields . # Alternative axiomatizations # As with other algebraic structures , there exist alternative axiomatizations . Because of the relations between the operations , one can alternatively axiomatize a field by explicitly assuming that there are four binary operations ( add , subtract , multiply , divide ) with axioms relating these , or ( by functional decomposition ) in terms of two binary operations ( add and multiply ) and two unary operations ( additive inverse and multiplicative inverse ) , or other variants . The usual axiomatization in terms of the two operations of addition and multiplication is brief and allows the other operations to be defined in terms of these basic ones , but in other contexts , such as topology and category theory , it is important to include all operations as explicitly given , rather than implicitly defined ( compare topological group ) . This is because without further assumptions , the implicitly defined inverses may not be continuous ( in topology ) , or may not be able to be defined ( in category theory ) . Defining an inverse requires that one is working with a set , not a more general object . For a very economical axiomatization of the field of real numbers , whose primitives are merely a set R with , addition , and a binary relation , *3626;191387; 3 + ' ' y ' ' 3 = ' ' z ' ' 3 . In the language of field extensions detailed below , Q ( ) is a field extension of degree 2 . Algebraic number fields are by definition finite field extensions of Q , that is , fields containing Q having finite dimension as a Q -vector space . # Reals , complex numbers , and ' ' p ' ' -adic numbers # Take the real numbers R , under the usual operations of addition and multiplication . When the real numbers are given the usual ordering , they form a ' ' complete ordered field ' ' ; it is this structure which provides the foundation for most formal treatments of calculus . The complex numbers C consist of expressions : ' ' a ' ' + ' ' b ' ' i where i is the imaginary unit , i.e. , a ( non-real ) number satisfying i 2 = 1 . Addition and multiplication of real numbers are defined in such a way that all field axioms hold for C . For example , the distributive law enforces : ( ' ' a ' ' + ' ' b ' ' i ) ( ' ' c ' ' + ' ' d ' ' i ) = ' ' ac ' ' + ' ' bc ' ' i + ' ' ad ' ' i + ' ' bd ' ' i 2 , which equals ' ' ac ' ' ' ' bd ' ' + ( ' ' bc ' ' + ' ' ad ' ' ) i . The real numbers can be constructed by completing the rational numbers , i.e. , filling the gaps : for example *38;195015;span 2 is such a gap . By a formally very similar procedure , another important class of fields , the field of ' ' p ' ' -adic numbers Q ' ' p ' ' is built . It is used in number theory and ' ' p ' ' -adic analysis . Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers . # Constructible numbers # In antiquity , several geometric problems concerned the ( in ) feasibility of constructing certain numbers with compass and straightedge . For example it was unknown to the Greeks that it is in general impossible to trisect a given angle . Using the field notion and field theory allows these problems to be settled . To do so , the field of constructible numbers is considered . It contains , on the plane , the points 0 and 1 , and all complex numbers that can be constructed from these two by a finite number of construction steps using only compass and straightedge . This set , endowed with the usual addition and multiplication of complex numbers does form a field . For example , multiplying two ( real ) numbers ' ' r ' ' 1 and ' ' r ' ' 2 that have already been constructed can be done using construction at the right , based on the intercept theorem . This way , the obtained field ' ' F ' ' contains all rational numbers , but is bigger than Q , because for any ' ' f ' ' ' ' F ' ' , the square root of ' ' f ' ' is also a constructible number . A closely related concept is that of a Euclidean field , namely an ordered field whose positive elements are closed under square root . The real constructible numbers form the least Euclidean field , and the Euclidean fields are precisely the ordered extensions thereof . # Finite fields # ' ' Finite fields ' ' ( also called ' ' Galois fields ' ' ) are fields with finitely many elements . The above introductory example F 4 is a field with four elements . F 2 consists of two elements , 0 and 1 . This is the smallest field , because by definition a field has at least two distinct elements 1 0 . Interpreting the addition and multiplication in this latter field as XOR and AND operations , this field finds applications in computer science , especially in cryptography and coding theory . In a finite field there is necessarily an integer ' ' n ' ' such that ( ' ' n ' ' repeated terms ) equals 0 . It can be shown that the smallest such ' ' n ' ' must be a prime number , called the ' ' characteristic ' ' of the field . If a ( necessarily infinite ) field has the property that is never zero , for any number of summands , such as in Q , for example , the characteristic is said to be zero . A basic class of finite fields are the fields F ' ' p ' ' with ' ' p ' ' elements ( ' ' p ' ' a prime number ) : : F ' ' p ' ' = Z / ' ' p ' ' Z = 0 , 1 , ... , ' ' p ' ' 1 , where the operations are defined by performing the operation in the set of integers Z , dividing by ' ' p ' ' and taking the remainder ; see modular arithmetic . A field ' ' K ' ' of characteristic ' ' p ' ' necessarily contains F ' ' p ' ' , and therefore may be viewed as a vector space over F ' ' p ' ' , of finite dimension if ' ' K ' ' is finite . Thus a finite field ' ' K ' ' has prime power order , i.e. , ' ' K ' ' has ' ' q ' ' = ' ' p ' ' ' ' n ' ' elements ( where ' ' n ' ' 0 is the number of elements in a basis of ' ' K ' ' over F ' ' p ' ' ) . By developing more field theory , in particular the notion of the splitting field of a polynomial ' ' f ' ' over a field ' ' K ' ' , which is the smallest field containing ' ' K ' ' and all roots of ' ' f ' ' , one can show that two finite fields with the same number of elements are isomorphic , i.e. , there is a one-to-one mapping of one field onto the other that preserves multiplication and addition . Thus we may speak of ' ' the ' ' finite field with ' ' q ' ' elements , usually denoted by F ' ' q ' ' or GF ( ' ' q ' ' ) . # Archimedean fields # An Archimedean field is an ordered field such that for each element there exists a finite expression whose value is greater than that element , that is , there are no infinite elements . Equivalently , the field contains no infinitesimals ; or , the field is isomorphic to a subfield of the reals . A necessary condition for an ordered field to be complete is that it be Archimedean , since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational , whence the sequence 1/2 , 1/3 , 1/4 , &hellip ; , every element of which is greater than every infinitesimal , has no limit . ( And since every proper subfield of the reals also contains such gaps , up to isomorphism the reals form the unique complete ordered field. ) # Field of functions # Given a geometric object ' ' X ' ' , one can consider functions on such objects . Adding and multiplying them pointwise , i.e. , ( ' ' f ' ' ' ' g ' ' ) ( ' ' x ' ' ) = ' ' f ' ' ( ' ' x ' ' ) ' ' g ' ' ( ' ' x ' ' ) this leads to a field . However , due to the presence of possible zeros , i.e. , points ' ' x ' ' ' ' X ' ' where ' ' f ' ' ( ' ' x ' ' ) = 0 , one has to take poles into account , i.e. , formally allowing ' ' f ' ' ( ' ' x ' ' ) = . If ' ' X ' ' is an algebraic variety over ' ' F ' ' , then the rational functions ' ' X ' ' ' ' F ' ' , i.e. , functions defined almost everywhere , form a field , the function field of ' ' X ' ' . Likewise , if ' ' X ' ' is a Riemann surface , then the meromorphic functions ' ' S ' ' C form a field . Under certain circumstances , namely when ' ' S ' ' is compact , ' ' S ' ' can be reconstructed from this field . # Local and global fields # Another important distinction in the realm of fields , especially with regard to number theory , are local fields and global fields . Local fields are completions of global fields at a given place . For example , Q is a global field , and the attached local fields are Q ' ' p ' ' and R ( Ostrowski 's theorem ) . Algebraic number fields and function fields over F ' ' q ' ' are further global fields . Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locallythis technique is called local-global principle . # Some first theorems # Every finite subgroup of the multiplicative group ' ' F ' ' is cyclic . This applies in particular to F ' ' q ' ' , it is cyclic of order . In the introductory example , a generator of F 4 is the element A. From the point of view of algebraic geometry , fields are points , because the spectrum ' ' Spec F ' ' has only one point , corresponding to the 0-ideal . This entails that a commutative ring is a field if and only if it has no ideals except 0 and itself . Equivalently , an integral domain is field if and only if its Krull dimension is 0 . Isomorphism extension theorem # Constructing fields # # Closure operations # Assuming the axiom of choice , for every field ' ' F ' ' , there exists a field , called the algebraic closure of ' ' F ' ' , which contains ' ' F ' ' , is algebraic over ' ' F ' ' , which means that any element ' ' x ' ' of satisfies a polynomial equation : ' ' f ' ' ' ' n ' ' ' ' x ' ' ' ' n ' ' + ' ' f ' ' ' ' n ' ' 1 ' ' x ' ' ' ' n ' ' 1 + + ' ' f ' ' 1 ' ' x ' ' + ' ' f ' ' 0 = 0 , with coefficients ' ' f ' ' ' ' n ' ' , ... , ' ' f ' ' 0 ' ' F ' ' , and is algebraically closed , i.e. , any such polynomial does have at least one solution in . The algebraic closure is unique up to isomorphism inducing the identity on ' ' F ' ' . However , in many circumstances in mathematics , it is not appropriate to treat as being uniquely determined by ' ' F ' ' , since the isomorphism above is not itself unique . In these cases , one refers to such a as ' ' an ' ' algebraic closure of ' ' F ' ' . A similar concept is the separable closure , containing all roots of separable polynomials , instead of all polynomials . For example , if ' ' F ' ' = Q , the algebraic closure is also called ' ' field of algebraic numbers ' ' . The field of algebraic numbers is an example of an algebraically closed field of characteristic zero ; as such it satisfies the same first-order sentences as the field of complex numbers C . In general , all algebraic closures of a field are isomorphic . However , there is in general no preferable isomorphism between two closures . Likewise for separable closures . # Subfields and field extensions # A ' ' subfield ' ' is , informally , a small field contained in a bigger one . Formally , a subfield ' ' E ' ' of a field ' ' F ' ' is a subset containing 0 and 1 , closed under the operations + , , and multiplicative inverses and with its own operations defined by restriction . For example , the real numbers contain several interesting subfields : the real algebraic numbers , the computable numbers and the rational numbers are examples . The notion of field extension lies at the heart of field theory , and is crucial to many other algebraic domains . A field extension ' ' F ' ' / ' ' E ' ' is simply a field ' ' F ' ' and a subfield ' ' E ' ' ' ' F ' ' . Constructing such a field extension ' ' F ' ' / ' ' E ' ' can be done by adding new elements or ' ' adjoining elements ' ' to the field ' ' E ' ' . For example , given a field ' ' E ' ' , the set ' ' F ' ' = ' ' E ' ' ( ' ' X ' ' ) of rational functions , i.e. , equivalence classes of expressions of the kind : where ' ' p ' ' ( ' ' X ' ' ) and ' ' q ' ' ( ' ' X ' ' ) are polynomials with coefficients in ' ' E ' ' , and ' ' q ' ' is not the zero polynomial , forms a field . This is the simplest example of a transcendental extension of ' ' E ' ' . It also is an example of a domain ( the ring of polynomials in this case ) being embedded into its field of fractions . The ring of formal power series is also a domain , and again the ( equivalence classes of ) fractions of the form ' ' p ' ' ( ' ' X ' ' ) / ' ' q ' ' ( ' ' X ' ' ) where ' ' p ' ' and ' ' q ' ' are elements of form the field of fractions for . This field is actually the ring of Laurent series over the field ' ' E ' ' , denoted . In the above two cases , the added symbol ' ' X ' ' and its powers did not interact with elements of ' ' E ' ' . It is possible however that the adjoined symbol may interact with ' ' E ' ' . This idea will be illustrated by adjoining an element to the field of real numbers R . As explained above , C is an extension of R . C can be obtained from R by adjoining the imaginary symbol i which satisfies i 2 = 1 . The result is that R i= C . This is different from adjoining the symbol ' ' X ' ' to R , because in that case , the powers of ' ' X ' ' are all distinct objects , but here , i 2 =1 is actually an element of R . Another way to view this last example is to note that i is a zero of the polynomial ' ' p ' ' ( ' ' X ' ' ) = ' ' X ' ' 2 + 1 . The quotient ring can be mapped onto C using the map . Since the ideal ( ' ' X ' ' 2 +1 ) is generated by a polynomial irreducible over R , the ideal is maximal , hence the quotient ring is a field . This nonzero ring map from the quotient to C is necessarily an isomorphism of rings . The above construction generalises to any irreducible polynomial in the polynomial ring ' ' E ' ' ' ' X ' ' , i.e. , a polynomial ' ' p ' ' ( ' ' X ' ' ) that can not be written as a product of non-constant polynomials . The quotient ring ' ' F ' ' = ' ' E ' ' ' ' X ' ' / ( ' ' p ' ' ( ' ' X ' ' ) , is again a field . Alternatively , constructing such field extensions can also be done , if a bigger container is already given . Suppose given a field ' ' E ' ' , and a field ' ' G ' ' containing ' ' E ' ' as a subfield , for example ' ' G ' ' could be the algebraic closure of ' ' E ' ' . Let ' ' x ' ' be an element of ' ' G ' ' not in ' ' E ' ' . Then there is a smallest subfield of ' ' G ' ' containing ' ' E ' ' and ' ' x ' ' , denoted ' ' F ' ' = ' ' E ' ' ( ' ' x ' ' ) and called ' ' field extension F / E generated by x in G ' ' . Such extensions are also called ' ' simple extensions ' ' . Many extensions are of this type ; see the primitive element theorem . For instance , Q ( ' ' i ' ' ) is the subfield of C consisting of all numbers of the form ' ' a ' ' + ' ' bi ' ' where both ' ' a ' ' and ' ' b ' ' are rational numbers . One distinguishes between extensions having various qualities . For example , an extension ' ' K ' ' of a field ' ' k ' ' is called ' ' algebraic ' ' , if every element of ' ' K ' ' is a root of some polynomial with coefficients in ' ' k ' ' . Otherwise , the extension is called ' ' transcendental ' ' . The aim of Galois theory is the study of ' ' algebraic extensions ' ' of a field . # Rings vs fields # Adding multiplicative inverses to an integral domain ' ' R ' ' yields the field of fractions of ' ' R ' ' . For example , the field of fractions of the integers Z is just Q . Also , the field ' ' F ' ' ( ' ' X ' ' ) is the quotient field of the ring of polynomials ' ' F ' ' ' ' X ' ' . Getting back the ring from the field is sometimes possible ; see discrete valuation ring . Another method to obtain a field from a commutative ring ' ' R ' ' is taking the quotient # Ultraproducts # If ' ' I ' ' is an index set , ' ' U ' ' is an ultrafilter on ' ' I ' ' , and ' ' F ' ' ' ' i ' ' is a field for every ' ' i ' ' in ' ' I ' ' , the ultraproduct of the ' ' F ' ' ' ' i ' ' with respect to ' ' U ' ' is a field . For example , a non-principal ultraproduct of finite fields is a pseudo finite field ; i.e. , a PAC field having exactly one extension of any degree . This construction is important to the study of the elementary theory of finite fields . # Galois theory # Galois theory aims to study the algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication . The fundamental theorem of Galois theory shows that there is a strong relation between the structure of the symmetry group and the set of algebraic extensions . In the case where ' ' F ' ' / ' ' E ' ' is a finite ( Galois ) extension , Galois theory studies the algebraic extensions of ' ' E ' ' that are subfields of ' ' F ' ' . Such fields are called intermediate extensions . Specifically , the Galois group of ' ' F ' ' over ' ' E ' ' , denoted Gal ( ' ' F ' ' / ' ' E ' ' ) , is the group of field automorphisms of ' ' F ' ' that are trivial on ' ' E ' ' ( i.e. , the bijections : ' ' F ' ' ' ' F ' ' that preserve addition and multiplication and that send elements of ' ' E ' ' to themselves ) , and the fundamental theorem of Galois theory states that there is a one-to-one correspondence between subgroups of Gal ( ' ' F ' ' / ' ' E ' ' ) and the set of intermediate extensions of the extension ' ' F ' ' / ' ' E ' ' . The theorem , in fact , gives an explicit correspondence and further properties . To study all ( separable ) algebraic extensions of ' ' E ' ' at once , one must consider the absolute Galois group of ' ' E ' ' , defined as the Galois group of the separable closure , ' ' E ' ' sep , of ' ' E ' ' over ' ' E ' ' ( i.e. , Gal ( ' ' E ' ' sep / ' ' E ' ' ) . It is possible that the degree of this extension is infinite ( as in the case of ' ' E ' ' = Q ) . It is thus necessary to have a notion of Galois group for an infinite algebraic extension . The Galois group in this case is obtained as a limit ( specifically an inverse limit ) of the Galois groups of the finite Galois extensions of ' ' E ' ' . In this way , it acquires a topology . *18;195055;ref As an inverse limit of finite discrete groups , it is equipped with the profinite topology , making it a profinite topological group The fundamental theorem of Galois theory can be generalized to the case of infinite Galois extensions by taking into consideration the topology of the Galois group , and in the case of ' ' E ' ' sep / ' ' E ' ' it states that there this a one-to-one correspondence between ' ' closed ' ' subgroups of Gal ( ' ' E ' ' sep / ' ' E ' ' ) and the set of all separable algebraic extensions of ' ' E ' ' ( technically , one only obtains those separable algebraic extensions of ' ' E ' ' that occur as subfields of the ' ' chosen ' ' separable closure ' ' E ' ' sep , but since all separable closures of ' ' E ' ' are isomorphic , choosing a different separable closure would give the same Galois group and thus an equivalent set of algebraic extensions ) . # Generalizations # There are also proper classes with field structure , which are sometimes called Fields , with a capital F : The surreal numbers form a Field containing the reals , and would be a field except for the fact that they are a proper class , not a set . The nimbers form a Field . The set of nimbers with birthday smaller than 2 2 ' ' n ' ' , the nimbers with birthday smaller than any infinite cardinal are all examples of fields . In a different direction , differential fields are fields equipped with a derivation . For example , the field R ( ' ' X ' ' ) , together with the standard derivative of polynomials forms a differential field . These fields are central to differential Galois theory . Exponential fields , meanwhile , are fields equipped with an exponential function that provides a homomorphism between the additive and multiplicative groups within the field . The usual exponential function makes the real and complex numbers exponential fields , denoted R exp and C exp respectively . Generalizing in a more categorical direction yields the field with one element and related objects . # Exponentiation # One does not in general study generalizations of fields with ' ' three ' ' binary operations . The familiar addition/subtraction , multiplication/division , *30;195075;TOOLONG operations from the natural numbers to the reals , each built up in terms of iteration of the last , mean that generalizing exponentiation as a binary operation is tempting , but has generally not proven fruitful ; instead , an exponential field assumes a unary exponential function from the additive group to the multiplicative group , not a partially defined binary function . Note that the exponential operation of is neither associative nor commutative , nor has a unique inverse ( are both square roots of 4 , for instance ) , unlike addition and multiplication , and further is not defined for many pairsfor example , does not define a single number . These all show that even for rational numbers exponentiation is not nearly as well-behaved as addition and multiplication , which is why one does not in general axiomatize exponentiation. # Applications # The concept of a field is of use , for example , in defining vectors and matrices , two structures in linear algebra whose components can be elements of an arbitrary field . Finite fields are used in number theory , Galois theory , coding theory and combinatorics ; and again the notion of algebraic extension is an important tool . Fields of characteristic 2 are useful in computer science .
@@14220 The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and , to a lesser extent , an investigation into the mathematical methods and notation of the past . Before the modern age and the worldwide spread of knowledge , written examples of new mathematical developments have come to light only in a few locales . The most ancient mathematical texts available are ' ' Plimpton 322 ' ' ( Babylonian mathematics c. 1900 BC ) , the ' ' Rhind Mathematical Papyrus ' ' ( Egyptian mathematics c. 2000-1800 BC ) and the ' ' Moscow Mathematical Papyrus ' ' ( Egyptian mathematics c. 1890 BC ) . All of these texts concern the so-called Pythagorean theorem , which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry . The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans , who coined the term mathematics from the ancient Greek ' ' ' ' ( ' ' mathema ' ' ) , meaning subject of instruction . Greek mathematics greatly refined the methods ( especially through the introduction of deductive reasoning and mathematical rigor in proofs ) and expanded the subject matter of mathematics . Chinese mathematics made early contributions , including a place value system . The Hindu-Arabic numeral system and the rules for the use of its operations , in use throughout the world today , likely evolved over the course of the first millennium AD in India and were transmitted to the west via Islamic mathematics through the work of Muammad ibn Ms al-Khwrizm . Islamic mathematics , in turn , developed and expanded the mathematics known to these civilizations . Many Greek and Arabic texts on mathematics were then translated into Latin , which led to further development of mathematics in medieval Europe . From ancient times through the Middle Ages , bursts of mathematical creativity were often followed by centuries of stagnation . Beginning in Renaissance Italy in the 16th century , new mathematical developments , interacting with new scientific discoveries , were made at an increasing pace that continues through the present day . # Prehistoric mathematics # The origins of mathematical thought lie in the concepts of number , magnitude , and form . Modern studies of animal cognition have shown that these concepts are not unique to humans . Such concepts would have been part of everyday life in hunter-gatherer societies . The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one , two , and many , but not of numbers larger than two . The oldest known possibly mathematical object is the Lebombo bone , discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC . It consists of 29 distinct notches cut into a baboon 's fibula . Also prehistoric artifacts discovered in Africa and France , dated between 35,000 and 20,000 years old , suggest early attempts to quantify time . The Ishango bone , found near the headwaters of the Nile river ( northeastern Congo ) , may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone . Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar . In the book ' ' How Mathematics Happened : The First 50,000 Years ' ' , Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division , which he dates to after 10,000 BC , with prime numbers probably not being understood until about 500 BC . He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two , prime numbers between 10 and 20 , and some numbers that are almost multiples of 10 . The Ishango bone , according to scholar Alexander Marshack , may have influenced the later development of mathematics in Egypt as , like some entries on the Ishango bone , Egyptian arithmetic also made use of multiplication by 2 ; this , however , is disputed . Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs . It has been claimed that megalithic monuments in England and Scotland , dating from the 3rd millennium BC , incorporate geometric ideas such as circles , ellipses , and Pythagorean triples in their design . All of the above are disputed however , and the currently oldest undisputed mathematical usage is in Babylonian and dynastic Egyptian sources . # Babylonian mathematics # Babylonian mathematics refers to any mathematics of the people of Mesopotamia ( modern Iraq ) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity . It is named Babylonian mathematics due to the central role of Babylon as a place of study . Later under the Arab Empire , Mesopotamia , especially Baghdad , once again became an important center of study for Islamic mathematics . In contrast to the sparsity of sources in Egyptian mathematics , our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s . Written in Cuneiform script , tablets were inscribed whilst the clay was moist , and baked hard in an oven or by the heat of the sun . Some of these appear to be graded homework . The earliest evidence of written mathematics dates back to the ancient Sumerians , who built the earliest civilization in Mesopotamia . They developed a complex system of metrology from 3000 BC . From around 2500 BC onwards , the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems . The earliest traces of the Babylonian numerals also date back to this period . The majority of recovered clay tablets date from 1800 to 1600 BC , and cover topics which include fractions , algebra , quadratic and cubic equations , and the calculation of regular reciprocal pairs . The tablets also include multiplication tables and methods for solving linear and quadratic equations . The Babylonian tablet YBC 7289 gives an approximation of 2 accurate to five decimal places . Babylonian mathematics were written using a sexagesimal ( base-60 ) numeral system . From this derives the modern day usage of 60 seconds in a minute , 60 minutes in an hour , and 360 ( 60 x 6 ) degrees in a circle , as well as the use of seconds and minutes of arc to denote fractions of a degree . Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors . Also , unlike the Egyptians , Greeks , and Romans , the Babylonians had a true place-value system , where digits written in the left column represented larger values , much as in the decimal system . They lacked , however , an equivalent of the decimal point , and so the place value of a symbol often had to be inferred from the context . On the other hand , this defect is equivalent to the modern-day usage of floating point arithmetic ; moreover , the use of base 60 means that any reciprocal of an integer which is a multiple of divisors of 60 necessarily has a finite expansion to the base 60 . ( In decimal arithmetic , only reciprocals of multiples of 2 and 5 have finite decimal expansions . ) Accordingly , there is a strong argument that arithmetic Old Babylonian style is considerably more sophisticated than that of current usage . The interpretation of Plimpton 322 was the source of controversy for many years after its significance in the context of Pythagorean triangles was realized . In historical context , inheritance problems involving equal-area subdivision of triangular and trapezoidal fields ( with integer length sides ) quickly convert into the need to calculate the square root of 2 , or to solve the Pythagorean equation in integers . Rather than considering a square as the sum of two squares , we can equivalently consider a square as a difference of two squares . Let a , b and c be integers that form a Pythagorean Triple : a2 + b2 = c2 . Then c2 - a2 = b2 , and using the expansion for the difference of two squares we get ( c-a ) ( c+a ) = b2 . Dividing by b2 , it becomes the product of two rational numbers giving 1 : ( c/b - a/b ) ( c/b + a/b ) = 1 . We require two rational numbers which are reciprocals and which differ by 2(a/b) . This is easily solved by consulting a table of reciprocal pairs . E.g. , ( 1/2 ) ( 2 ) = 1 is a pair of reciprocals which differ by 3/2 = 2(a/b) Thus a/b = 3/4 , giving a=3 , b=4 and so c=5 . Solutions of the original equation are thus constructed by choosing a rational number x , from which Pythagorean-triples are 2x , x2-1 , x2+1 . Other triples are made by scaling these by an integer ( the scaling integer being half the difference between the largest and one other side ) . All Pythagorean triples arise in this way , and the examples provided in Plimpton 322 involve some quite large numbers , by modern standards , such as ( 4601 , 4800 , 6649 ) in decimal notation . # Egyptian mathematics # Egyptian mathematics refers to mathematics written in the Egyptian language . From the Hellenistic period , Greek replaced Egyptian as the written language of Egyptian scholars . Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics , when Arabic became the written language of Egyptian scholars . The most extensive Egyptian mathematical text is the Rhind papyrus ( sometimes also called the Ahmes Papyrus after its author ) , dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000-1800 BC . It is an instruction manual for students in arithmetic and geometry . In addition to giving area formulas and methods for multiplication , division and working with unit fractions , it also contains evidence of other mathematical knowledge , including composite and prime numbers ; arithmetic , geometric and harmonic means ; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory ( namely , that of the number 6 ) . It also shows how to solve first order linear equations as well as arithmetic and geometric series . Another significant Egyptian mathematical text is the Moscow papyrus , also from the Middle Kingdom period , dated to c. 1890 BC . It consists of what are today called ' ' word problems ' ' or ' ' story problems ' ' , which were apparently intended as entertainment . One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum : If you are told : A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top . You are to square this 4 , result 16 . You are to double 4 , result 8 . You are to square 2 , result 4 . You are to add the 16 , the 8 , and the 4 , result 28 . You are to take one third of 6 , result 2 . You are to take 28 twice , result 56 . See , it is 56 . You will find it right . Finally , the Berlin Papyrus 6619 ( c. 1800 BC ) shows that ancient Egyptians could solve a second-order algebraic equation . # Greek mathematics # Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus ( 600 BC ) to the closure of the Academy of Athens in 529 AD . Greek mathematicians lived in cities spread over the entire Eastern Mediterranean , from Italy to North Africa , but were united by culture and language . Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics . Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures . All surviving records of pre-Greek mathematics show the use of inductive reasoning , that is , repeated observations used to establish rules of thumb . Greek mathematicians , by contrast , used deductive reasoning . The Greeks used logic to derive conclusions from definitions and axioms , and used mathematical rigor to prove them . Greek mathematics is thought to have begun with Thales of Miletus ( c. 624c.546 BC ) and Pythagoras of Samos ( c. 582c. 507 BC ) . Although the extent of the influence is disputed , they were probably inspired by Egyptian and Babylonian mathematics . According to legend , Pythagoras traveled to Egypt to learn mathematics , geometry , and astronomy from Egyptian priests . Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore . He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales ' Theorem . As a result , he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed . Pythagoras established the Pythagorean School , whose doctrine it was that mathematics ruled the universe and whose motto was All is number . It was the Pythagoreans who coined the term mathematics , and with whom the study of mathematics for its own sake begins . The Pythagoreans are credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history , and with the proof of the existence of irrational numbers . Plato ( 428/427 BC 348/347 BC ) is important in the history of mathematics for inspiring and guiding others . His Platonic Academy , in Athens , became the mathematical center of the world in the 4th century BC , and it was from this school that the leading mathematicians of the day , such as Eudoxus of Cnidus , came . Plato also discussed the foundations of mathematics , clarified some of the definitions ( e.g. that of a line as breadthless length ) , and reorganized the assumptions . The analytic method is ascribed to Plato , while a formula for obtaining Pythagorean triples bears his name . Eudoxus ( 408c.355 BC ) developed the method of exhaustion , a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes . The former allowed the calculations of areas and volumes of curvilinear figures , while the latter enabled subsequent geometers to make significant advances in geometry . Though he made no specific technical mathematical discoveries , Aristotle ( 384c.322 BC ) contributed significantly to the development of mathematics by laying the foundations of logic . In the 3rd century BC , the premier center of mathematical education and research was the Musaeum of Alexandria . It was there that Euclid ( c. 300 BC ) taught , and wrote the ' ' Elements ' ' , widely considered the most successful and influential textbook of all time . The ' ' Elements ' ' introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today , that of definition , axiom , theorem , and proof . Although most of the contents of the ' ' Elements ' ' were already known , Euclid arranged them into a single , coherent logical framework . The ' ' Elements ' ' was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today . In addition to the familiar theorems of Euclidean geometry , the ' ' Elements ' ' was meant as an introductory textbook to all mathematical subjects of the time , such as number theory , algebra and solid geometry , including proofs that the square root of two is irrational and that there are infinitely many prime numbers . Euclid also wrote extensively on other subjects , such as conic sections , optics , spherical geometry , and mechanics , but only half of his writings survive . The first woman mathematician recorded by history was Hypatia of Alexandria ( AD 350 - 415 ) . She succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics . Because of a political dispute , the Christian community in Alexandria punished her , presuming she was involved , by stripping her naked and scraping off her skin with clamshells ( some say roofing tiles ) . Archimedes ( c.287212 BC ) of Syracuse , widely considered the greatest mathematician of antiquity , used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus . He also showed one could use the method of exhaustion to calculate the value of with as much precision as desired , and obtained the most accurate value of then known , 3 *25820;36701; Throughout the 19th century mathematics became increasingly abstract . In the 19th century lived Carl Friedrich Gauss ( 17771855 ) . Leaving aside his many contributions to science , in pure mathematics he did revolutionary work on functions of complex variables , in geometry , and on the convergence of series . He gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law . This century saw the development of the two forms of non-Euclidean geometry , where the parallel postulate of Euclidean geometry no longer holds . The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival , the Hungarian mathematician Jnos Bolyai , independently defined and studied hyperbolic geometry , where uniqueness of parallels no longer holds . In this geometry the sum of angles in a triangle add up to less than 180 . Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann ; here no parallel can be found and the angles in a triangle add up to more than 180 . Riemann also developed Riemannian geometry , which unifies and vastly generalizes the three types of geometry , and he defined the concept of a manifold , which generalizes the ideas of curves and surfaces . The 19th century saw the beginning of a great deal of abstract algebra . Hermann Grassmann in Germany gave a first version of vector spaces , William Rowan Hamilton in Ireland developed noncommutative algebra . The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra , in which the only numbers were 0 and 1 . Boolean algebra is the starting point of mathematical logic and has important applications in computer science . Augustin-Louis Cauchy , Bernhard Riemann , and Karl Weierstrass reformulated the calculus in a more rigorous fashion . Also , for the first time , the limits of mathematics were explored . Niels Henrik Abel , a Norwegian , and variste Galois , a Frenchman , proved that there is no general algebraic method for solving polynomial equations of degree greater than four ( AbelRuffini theorem ) . Other 19th-century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle , to construct the side of a cube twice the volume of a given cube , nor to construct a square equal in area to a given circle . Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks . On the other hand , the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers . Abel and Galois 's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory , and the associated fields of abstract algebra . In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry . In the later 19th century , Georg Cantor established the first foundations of set theory , which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics . Cantor 's set theory , and the rise of mathematical logic in the hands of Peano , L. E. J. Brouwer , David Hilbert , Bertrand Russell , and A.N . Whitehead , initiated a long running debate on the foundations of mathematics . The 19th century saw the founding of a number of national mathematical societies : the London Mathematical Society in 1865 , the Socit Mathmatique de France in 1872 , the Circolo Matematico di Palermo in 1884 , the Edinburgh Mathematical Society in 1883 , and the American Mathematical Society in 1888 . The first international , special-interest society , the Quaternion Society , was formed in 1899 , in the context of a vector controversy . In 1897 , Hensel introduced p-adic numbers . # 20th century # The 20th century saw mathematics become a major profession . Every year , thousands of new Ph.D.s in mathematics were awarded , and jobs were available in both teaching and industry . An effort to catalogue the areas and applications of mathematics was undertaken in Klein 's encyclopedia . In a 1900 speech to the International Congress of Mathematicians , David Hilbert set out a list of 23 unsolved problems in mathematics . These problems , spanning many areas of mathematics , formed a central focus for much of 20th-century mathematics . Today , 10 have been solved , 7 are partially solved , and 2 are still open . The remaining 4 are too loosely formulated to be stated as solved or not . Notable historical conjectures were finally proven . In 1976 , Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem . Andrew Wiles , building on the work of others , proved Fermat 's Last Theorem in 1995 . Paul Cohen and Kurt Gdel proved that the continuum hypothesis is independent of ( could neither be proved nor disproved from ) the standard axioms of set theory . In 1998 Thomas Callister Hales proved the Kepler conjecture . Mathematical collaborations of unprecedented size and scope took place . An example is the classification of finite simple groups ( also called the enormous theorem ) , whose proof between 1955 and 1983 required 500-odd journal articles by about 100 authors , and filling tens of thousands of pages . A group of French mathematicians , including Jean Dieudonn and Andr Weil , publishing under the pseudonym Nicolas Bourbaki , attempted to exposit all of known mathematics as a coherent rigorous whole . The resulting several dozen volumes has had a controversial influence on mathematical education . Differential geometry came into its own when Einstein used it in general relativity . Entire new areas of mathematics such as mathematical logic , topology , and John von Neumann 's game theory changed the kinds of questions that could be answered by mathematical methods . All kinds of structures were abstracted using axioms and given names like metric spaces , topological spaces etc . As mathematicians do , the concept of an abstract structure was itself abstracted and led to category theory . Grothendieck and Serre recast algebraic geometry using sheaf theory . Large advances were made in the qualitative study of dynamical systems that Poincar had begun in the 1890s . Measure theory was developed in the late 19th and early 20th centuries . Applications of measures include the Lebesgue integral , Kolmogorov 's axiomatisation of probability theory , and ergodic theory . Knot theory greatly expanded . Quantum mechanics led to the development of functional analysis . Other new areas include , Laurent Schwartz 's distribution theory , fixed point theory , singularity theory and Ren Thom 's catastrophe theory , model theory , and Mandelbrot 's fractals . Lie theory with its Lie groups and Lie algebras became one of the major areas of study . Non-standard analysis , introduced by Abraham Robinson , rehabillitated the infinitesimal approach to calculus , which had fallen into disrepute in favour of the theory of limits , by extending the field of real numbers to the Hyperreal numbers which include infinitesimal and infinite quantities . An even larger number system , the surreal numbers were discovered by John Horton Conway in connection with combinatorial games . The development and continual improvement of computers , at first mechanical analog machines and then digital electronic machines , allowed industry to deal with larger and larger amounts of data to facilitate mass production and distribution and communication , and new areas of mathematics were developed to deal with this : Alan Turing 's computability theory ; complexity theory ; Derrick Henry Lehmer 's use of ENIAC to further number theory and the Lucas-Lehmer test ; Claude Shannon 's information theory ; signal processing ; data analysis ; optimization and other areas of operations research . In the preceding centuries much mathematical focus was on calculus and continuous functions , but the rise of computing and communication networks led to an increasing importance of discrete concepts and the expansion of combinatorics including graph theory . The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations , leading to areas such as numerical analysis and symbolic computation . Some of the most important methods and algorithms of the 20th century are : the simplex algorithm , the Fast Fourier Transform , error-correcting codes , the Kalman filter from control theory and the RSA algorithm of public-key cryptography . At the same time , deep insights were made about the limitations to mathematics . In 1929 and 1930 , it was proved the truth or falsity of all statements formulated about the natural numbers plus one of addition and multiplication , was decidable , i.e. could be determined by some algorithm . In 1931 , Kurt Gdel found that this was not the case for the natural numbers plus both addition and multiplication ; this system , known as Peano arithmetic , was in fact incompletable . ( Peano arithmetic is adequate for a good deal of number theory , including the notion of prime number . ) A consequence of Gdel 's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic ( including all of analysis and geometry ) , truth necessarily outruns proof , i.e. there are true statements that can not be proved within the system . Hence mathematics can not be reduced to mathematical logic , and David Hilbert 's dream of making all of mathematics complete and consistent needed to be reformulated . One of the more colorful figures in 20th-century mathematics was Srinivasa Aiyangar Ramanujan ( 18871920 ) , an Indian autodidact who conjectured or proved over 3000 theorems , including properties of highly composite numbers , the partition function and its asymptotics , and mock theta functions . He also made major investigations in the areas of gamma functions , modular forms , divergent series , hypergeometric series and prime number theory . Paul Erds published more papers than any other mathematician in history , working with hundreds of collaborators . Mathematicians have a game equivalent to the Kevin Bacon Game , which leads to the Erds number of a mathematician . This describes the collaborative distance between a person and Paul Erds , as measured by joint authorship of mathematical papers . Emmy Noether has been described by many as the most important woman in the history of mathematics , she revolutionized the theories of rings , fields , and algebras . As in most areas of study , the explosion of knowledge in the scientific age has led to specialization : by the end of the century there were hundreds of specialized areas in mathematics and the Mathematics Subject Classification was dozens of pages long . More and more mathematical journals were published and , by the end of the century , the development of the world wide web led to online publishing . # 21st century # In 2000 , the Clay Mathematics Institute announced the seven Millennium Prize Problems , and in 2003 the Poincar conjecture was solved by Grigori Perelman ( who declined to accept an award on this point ) . Most mathematical journals now have online versions as well as print versions , and many online-only journals are launched . There is an increasing drive towards open access publishing , first popularized by the arXiv. # Future of mathematics # There are many observable trends in mathematics , the most notable being that the subject is growing ever larger , computers are ever more important and powerful , the application of mathematics to bioinformatics is rapidly expanding , the volume of data to be analyzed being produced by science and industry , facilitated by computers , is explosively expanding . # See also # History of algebra History of calculus History of combinatorics History of geometry History of logic History of mathematical notation History of number theory History of statistics History of trigonometry History of writing numbers Kenneth O. May Prize List of important publications in mathematics Lists of mathematicians Timeline of mathematics # References # # External articles # Eves , Howard , ' ' An Introduction to the History of Mathematics ' ' , Saunders , 1990 , ISBN 0-03-029558-0 , Burton , David M. ' ' The History of Mathematics : An Introduction ' ' . McGraw Hill : 1997. Katz , Victor J. ' ' A History of Mathematics : An Introduction ' ' , 2nd Edition . Addison-Wesley : 1998. Scimone , Aldo ( 2006 ) . Talete , chi era costui ? Vita e opere dei matematici incontrati a scuola . Palermo : Palumbo Pp. 228 . ; Books on a specific period Cite book . Maier , Annaliese ( 1982 ) , ' ' At the Threshold of Exact Science : Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy ' ' , edited by Steven Sargent , Philadelphia : University of Pennsylvania Press . Cite book . van der Waerden , B. L. , ' ' Geometry and Algebra in Ancient Civilizations ' ' , Springer , 1983 , ISBN 0-387-12159-5 . ; Books on a specific topic Hoffman , Paul , ' ' The Man Who Loved Only Numbers : The Story of Paul Erds and the Search for Mathematical Truth ' ' . New York : Hyperion , 1998 ISBN 0-7868-6362-5. ; Documentaries BBC ( 2008 ) . ' ' The Story of Maths ' ' . ( John J. O'Connor and Edmund F. Robertson ; University of St Andrews , Scotland ) . An award-winning website containing detailed biographies on many historical and contemporary mathematicians , as well as information on notable curves and various topics in the history of mathematics . ( David E. Joyce ; Clark University ) . Articles on various topics in the history of mathematics with an extensive bibliography . ( David R. Wilkins ; Trinity College , Dublin ) . Collections of material on the mathematics between the 17th and 19th century . ( Simon Fraser University ) . ( Jeff Miller ) . Contains information on the earliest known uses of terms used in mathematics . ( Jeff Miller ) . Contains information on the history of mathematical notations . ( John Aldrich , University of Southampton ) Discusses the origins of the modern mathematical word stock . ( Larry Riddle ; Agnes Scott College ) . ( Scott W. Williams ; University at Buffalo ) . ( Steven W. Rockey ; Cornell University Library ) . ; Organizations ; Journals ' ' Historia Mathematica ' ' , the Mathematical Association of America 's online Math History Magazine ; Directories ( The British Society for the History of Mathematics ) Math Archives ( University of Tennessee , Knoxville ) The Math Forum ( Drexel University ) ( Courtright Memorial Library ) . ( David Calvis ; Baldwin-Wallace College ) ( Universidad de La La guna ) ( Universidade de Coimbra ) ( Bruno Kevius ) ( Roberta Tucci ) as : #
@@15287 A series is , informally speaking , the sum of the terms of a sequence . Finite sequences and series have defined first and last terms , whereas infinite sequences and series continue indefinitely . In mathematics , given an infinite sequence of numbers ' ' a ' ' ' ' n ' ' , a series is informally the result of adding all those terms together : ' ' a ' ' 1 + ' ' a ' ' 2 + ' ' a ' ' 3 + . These can be written more compactly using the summation symbol . An example is the famous series from Zeno 's dichotomy and its mathematical representation : : The terms of the series are often produced according to a certain rule , such as by a formula , or by an algorithm . As there are an infinite number of terms , this notion is often called an infinite series . Unlike finite summations , infinite series need tools from mathematical analysis , and specifically the notion of limits , to be fully understood and manipulated . In addition to their ubiquity in mathematics , infinite series are also widely used in other quantitative disciplines such as physics , computer science , and finance . # Basic properties # # Definition # For any sequence of rational numbers , real numbers , complex numbers , functions thereof , etc. , the associated series is defined as the ordered formal sum : . The sequence of partial sums associated to a series is defined for each as the sum of the sequence from to : By definition the series converges to a limit if and only if the associated sequence of partial sums converges to . This definition is usually written as : More generally , if is a function from an index set I to a set G , then the series associated to is the formal sum of the elements over the index elements denoted by the : When the index set is the natural numbers , the function is a sequence denoted by . A series indexed on the natural numbers is an ordered formal sum and so we rewrite as in order to emphasize the ordering induced by the natural numbers . Thus , we obtain the common notation for a series indexed by the natural numbers : When the set is a semigroup , the sequence of partial sums associated to a sequence is defined for each as the sum of the terms : When the semigroup is also a topological space , then the series converges to an element if and only if the associated sequence of partial sums converges to . This definition is usually written as : # Convergent series # A series&thinsp ; ' ' a n ' ' &thinsp ; is said to ' converge ' or to ' be convergent ' when the sequence ' ' S ' ' ' ' N ' ' of partial sums has a finite limit . If the limit of ' ' S ' ' ' ' N ' ' is infinite or does not exist , the series is said to diverge . When the limit of partial sums exists , it is called the sum of the series : An easy way that an infinite series can converge is if all the ' ' a ' ' ' ' n ' ' are zero for ' ' n ' ' sufficiently large . Such a series can be identified with a finite sum , so it is only infinite in a trivial sense . Working out the properties of the series that converge even if infinitely many terms are non-zero is the essence of the study of series . Consider the example : It is possible to visualize its convergence on the real number line : we can imagine a line of length 2 , with successive segments marked off of lengths 1 , , , etc . There is always room to mark the next segment , because the amount of line remaining is always the same as the last segment marked : when we have marked off , we still have a piece of length unmarked , so we can certainly mark the next . This argument does not prove that the sum is ' ' equal ' ' to 2 ( although it is ) , but it does prove that it is ' ' at most ' ' 2 . In other words , the series has an upper bound . Given that the series converges , proving that it is equal to 2 requires only elementary algebra . If the series is denoted ' ' S ' ' , it can be seen that : Therefore , : Mathematicians extend the idiom discussed earlier to other , equivalent notions of series . For instance , when we talk about a recurring decimal , as in : we are talking , in fact , just about the series : But since these series always converge to real numbers ( because of what is called the completeness property of the real numbers ) , to talk about the series in this way is the same as to talk about the numbers for which they stand . In particular , it should offend no sensibilities if we make no distinction between 0.111 and 1 / 9 . Less clear is the argument that , but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations . See 0.999 ... for more . # Examples # A ' ' geometric series ' ' is one where each successive term is produced by multiplying the previous term by a constant number ( called the common ratio in this context ) . Example : : : : In general , the geometric series : : : converges if and only if ' ' z ' ' *194;41802; 3 + 5 over 2 + 7 over 4 + 9 over 8 + 11 over 16 + cdots=sumn=0infty(3+2n) over 2n. The ' ' harmonic series ' ' is the series : : : The harmonic series is divergent . An ' ' alternating series ' ' is a series where terms alternate signs . Example : : : The p-series : : : converges if ' ' r ' ' 1 and diverges for ' ' r ' ' 1 , which can be shown with the integral criterion described below in convergence tests . As a function of ' ' r ' ' , the sum of this series is Riemann 's zeta function . A telescoping series : : : converges if the sequence ' ' b ' ' ' ' n ' ' converges to a limit ' ' L ' ' as ' ' n ' ' goes to infinity . The value of the series is then ' ' b ' ' 1 &minus ; ' ' L ' ' . # Calculus and partial summation as an operation on sequences # Partial summation takes as input a sequence , ' ' a ' ' ' ' n ' ' , and gives as output another sequence , ' ' S ' ' ' ' N ' ' . It is thus a unary operation on sequences . Further , this function is linear , and thus is a linear operator on the vector space of sequences , denoted . The inverse operator is the finite difference operator , . These behave as discrete analogs of integration and differentiation , only for series ( functions of a natural number ) instead of functions of a real variable . For example , the sequence 1 , 1 , 1 , ... has series 1 , 2 , 3 , 4 , ... as its partial summation , which is analogous to the fact that In computer science it is known as prefix sum . # Properties of series # Series are classified not only by whether they converge or diverge , but also by the properties of the terms a n ( absolute or conditional convergence ) ; type of convergence of the series ( pointwise , uniform ) ; the class of the term a n ( whether it is a real number , arithmetic progression , trigonometric function ) ; etc. # Non-negative terms # When ' ' a n ' ' is a non-negative real number for every ' ' n ' ' , the sequence ' ' S N ' ' of partial sums is non-decreasing . It follows that a series ' ' a n ' ' with non-negative terms converges if and only if the sequence ' ' S N ' ' of partial sums is bounded . For example , the series : is convergent , because the inequality : and a telescopic sum argument implies that the partial sums are bounded by 2. # Absolute convergence # A series : is said to converge absolutely if the series of absolute values : converges . This is sufficient to guarantee not only that the original series converges to a limit , but also that any reordering of it converges to the same limit . # Conditional convergence # A series of real or complex numbers is said to be conditionally convergent ( or semi-convergent ) if it is convergent but not absolutely convergent . A famous example is the alternating series : which is convergent ( and its sum is equal to ln 2 ) , but the series formed by taking the absolute value of each term is the divergent harmonic series . The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series , and moreover , if the ' ' a ' ' ' ' n ' ' are real and ' ' S ' ' is any real number , that one can find a reordering so that the reordered series converges with sum equal to ' ' S ' ' . Abel 's test is an important tool for handling semi-convergent series . If a series has the form : where the partial sums ' ' B ' ' ' ' N ' ' = are bounded , ' ' ' ' ' ' n ' ' has bounded variation , and exists : :