124-407: In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among
248-402: A multiplicative identity denoted 1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element x is standardly denoted x − 1 . {\displaystyle x^{-1}.} The following identities , often called exponent rules , hold for all integer exponents, provided that the base
372-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
496-551: A base raised to one power times the same base raised to another power, the powers add. Extending this rule to the power zero gives b 0 × b n = b 0 + n = b n {\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}} , and dividing both sides by b n {\displaystyle b^{n}} gives b 0 = b n / b n = 1 {\displaystyle b^{0}=b^{n}/b^{n}=1} . That is,
620-417: A corresponding inverse function , and an analog among the hyperbolic functions . The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then
744-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
868-637: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
992-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
1116-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of
1240-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it
1364-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
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#17330849716051488-445: A rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities hold for any angle θ and any integer k . The algebraic expressions for the most important angles are as follows: Writing
1612-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
1736-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
1860-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating
1984-595: A superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote sin ( x ) ⋅ sin ( x ) , {\displaystyle \sin(x)\cdot \sin(x),} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from
2108-473: A value and what value to assign may depend on context. For more details, see Zero to the power of zero . Exponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b : Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity ( ∞ {\displaystyle \infty } ). This definition of exponentiation with negative exponents
2232-516: A way similar to that of Chuquet, for example iii 4 for 4 x . The word exponent was coined in 1544 by Michael Stifel. In the 16th century, Robert Recorde used the terms square, cube, zenzizenzic ( fourth power ), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth). Biquadrate has been used to refer to the fourth power as well. In 1636, James Hume used in essence modern notation, when in L'algèbre de Viète he wrote A for A . Early in
2356-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to
2480-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
2604-411: Is flat " and "a field is always a ring ". Exponentiation In mathematics , exponentiation is an operation involving two numbers : the base and the exponent or power . Exponentiation is written as b , where b is the base and n is the power ; often said as " b to the power n ". When n is a positive integer , exponentiation corresponds to repeated multiplication of
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#17330849716052728-668: Is 360° (particularly in elementary mathematics ). However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, or as solutions to differential equations given particular initial values ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in
2852-464: Is a mistranslation of the ancient Greek δύναμις ( dúnamis , here: "amplification" ) used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios . In The Sand Reckoner , Archimedes proved the law of exponents, 10 · 10 = 10 , necessary to manipulate powers of 10 . He then used powers of 10 to estimate the number of grains of sand that can be contained in
2976-429: Is also obtained by the empty product convention, which may be used in every algebraic structure with a multiplication that has an identity . This way the formula also holds for n = 0 {\displaystyle n=0} . The case of 0 is controversial. In contexts where only integer powers are considered, the value 1 is generally assigned to 0 but, otherwise, the choice of whether to assign it
3100-489: Is an arbitrary integer. Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. More precisely, defining one has the following series expansions: The following continued fractions are valid in
3224-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
3348-509: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
3472-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
3596-408: Is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k
3720-469: Is implied if they belong to a structure that is commutative . Otherwise, if a and b are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra , many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes ^^ instead of ^ ) for exponentiation with non-commuting bases, which
3844-487: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
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3968-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
4092-464: Is non-zero: Unlike addition and multiplication, exponentiation is not commutative : for example, 2 3 = 8 {\displaystyle 2^{3}=8} , but reversing the operands gives the different value 3 2 = 9 {\displaystyle 3^{2}=9} . Also unlike addition and multiplication, exponentiation is not associative : for example, (2 ) = 8 = 64 , whereas 2 ) = 2 = 512 . Without parentheses,
4216-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
4340-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
4464-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and 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. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
4588-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
4712-554: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
4836-608: Is the definition of square root: b 1 / 2 = b {\displaystyle b^{1/2}={\sqrt {b}}} . The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define b x {\displaystyle b^{x}} for any positive real base b {\displaystyle b} and any real number exponent x {\displaystyle x} . More involved definitions allow complex base and exponent, as well as certain types of matrices as base or exponent. Exponentiation
4960-728: Is the logarithmic derivative of sin z {\displaystyle \sin z} . From this, it can be deduced also that Euler's formula relates sine and cosine to the exponential function : Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as
5084-470: Is the only one that allows extending the identity b m + n = b m ⋅ b n {\displaystyle b^{m+n}=b^{m}\cdot b^{n}} to negative exponents (consider the case m = − n {\displaystyle m=-n} ). The same definition applies to invertible elements in a multiplicative monoid , that is, an algebraic structure , with an associative multiplication and
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5208-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
5332-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
5456-500: Is the unique solution with y (0) = 0 . The basic trigonometric functions can be defined by the following power series expansions. These series are also known as the Taylor series or Maclaurin series of these trigonometric functions: The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on
5580-403: Is then a definition of the real number π {\displaystyle \pi } which is independent of geometry. Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} , so the tangent function satisfies the ordinary differential equation It
5704-416: Is then called non-commutative exponentiation . For nonnegative integers n and m , the value of n is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation ). Such functions can be represented as m - tuples from an n -element set (or as m -letter words from an n -letter alphabet). Some examples for particular values of m and n are given in
5828-434: Is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after
5952-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
6076-454: Is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as
6200-500: Is used extensively in many fields, including economics , biology , chemistry , physics , and computer science , with applications such as compound interest , population growth , chemical reaction kinetics , wave behavior, and public-key cryptography . The term exponent originates from the Latin exponentem , the present participle of exponere , meaning "to put forth". The term power ( Latin : potentia, potestas, dignitas )
6324-1233: Is usually shown as a superscript to the right of the base as b or in computer code as b^n, and may also be called " b raised to the n th power", " b to the power of n ", "the n th power of b ", or most briefly " b to the n ". The above definition of b n {\displaystyle b^{n}} immediately implies several properties, in particular the multiplication rule: b n × b m = b × ⋯ × b ⏟ n times × b × ⋯ × b ⏟ m times = b × ⋯ × b ⏟ n + m times = b n + m . {\displaystyle {\begin{aligned}b^{n}\times b^{m}&=\underbrace {b\times \dots \times b} _{n{\text{ times}}}\times \underbrace {b\times \dots \times b} _{m{\text{ times}}}\\[1ex]&=\underbrace {b\times \dots \times b} _{n+m{\text{ times}}}\ =\ b^{n+m}.\end{aligned}}} That is, when multiplying
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#17330849716056448-437: The n th term lead to absolutely convergent series: Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which
6572-574: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
6696-459: The Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow
6820-753: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
6944-506: The Pythagorean identity . The other trigonometric functions can be found along the unit circle as By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change
7068-505: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
7192-524: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
7316-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
7440-709: The function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean sin ( x ) + y , {\displaystyle \sin(x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).} A positive integer appearing as
7564-806: The reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x : {\displaystyle \arcsin x\colon } The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case,
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#17330849716057688-408: The (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).} However, the exponent − 1 {\displaystyle {-1}} is commonly used to denote the inverse function , not
7812-638: The 17th century, the first form of our modern exponential notation was introduced by René Descartes in his text titled La Géométrie ; there, the notation is introduced in Book I. I designate ... aa , or a in multiplying a by itself; and a in multiplying it once more again by a , and thus to infinity. Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials , for example, as ax + bxx + cx + d . Samuel Jeake introduced
7936-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
8060-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
8184-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
8308-620: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
8432-408: The angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended,
8556-445: The area of a square with side-length b is b . (It is true that it could also be called " b to the second power", but "the square of b " and " b squared" are more traditional) Similarly, the expression b = b · b · b is called "the cube of b " or " b cubed", because the volume of a cube with side-length b is b . When an exponent is a positive integer , that exponent indicates how many copies of
8680-502: The base are multiplied together. For example, 3 = 3 · 3 · 3 · 3 · 3 = 243 . The base 3 appears 5 times in the multiplication, because the exponent is 5 . Here, 243 is the 5th power of 3 , or 3 raised to the 5th power . The word "raised" is usually omitted, and sometimes "power" as well, so 3 can be simply read "3 to the 5th", or "3 to the 5". The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations . The definition of
8804-463: The base: that is, b is the product of multiplying n bases: b n = b × b × ⋯ × b × b ⏟ n times . {\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.} In particular, b 1 = b {\displaystyle b^{1}=b} . The exponent
8928-574: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
9052-456: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
9176-417: The conventional order of operations for serial exponentiation in superscript notation is top-down (or right -associative), not bottom-up (or left -associative). That is, which, in general, is different from The powers of a sum can normally be computed from the powers of the summands by the binomial formula However, this formula is true only if the summands commute (i.e. that ab = ba ), which
9300-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is
9424-590: The definition for fractional powers: b n / m = b n m . {\displaystyle b^{n/m}={\sqrt[{m}]{b^{n}}}.} For example, b 1 / 2 × b 1 / 2 = b 1 / 2 + 1 / 2 = b 1 = b {\displaystyle b^{1/2}\times b^{1/2}=b^{1/2\,+\,1/2}=b^{1}=b} , meaning ( b 1 / 2 ) 2 = b {\displaystyle (b^{1/2})^{2}=b} , which
9548-454: The definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry. Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include: Sine and cosine can be defined as
9672-458: The degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. The six trigonometric functions can be defined as coordinate values of points on
9796-485: The denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures. When radians (rad) are employed,
9920-553: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
10044-474: The domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed. Conventionally, an abbreviation of each trigonometric function's name
10168-498: The domain of trigonometric functions to be extended to all positive and negative real numbers. Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects
10292-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
10416-406: The exponentiation as an iterated multiplication can be formalized by using induction , and this definition can be used as soon as one has an associative multiplication: The base case is and the recurrence is The associativity of multiplication implies that for any positive integers m and n , and As mentioned earlier, a (nonzero) number raised to the 0 power is 1 : This value
10540-453: The exponents must be constant. As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For example Konrad Zuse introduced floating point arithmetic in his 1938 computer Z1. One register contained representation of leading digits, and a second contained representation of the exponent of 10. Earlier Leonardo Torres Quevedo contributed Essays on Automation (1914) which had suggested
10664-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",
10788-416: The floating-point representation of numbers. The more flexible decimal floating-point representation was introduced in 1946 with a Bell Laboratories computer. Eventually educators and engineers adopted scientific notation of numbers, consistent with common reference to order of magnitude in a ratio scale . The expression b = b · b is called "the square of b " or " b squared", because
10912-670: The following table: In the base ten ( decimal ) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 10 = 1000 and 10 = 0.0001 . Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 299 792 458 m/s (the speed of light in vacuum, in metres per second ) can be written as 2.997 924 58 × 10 m/s and then approximated as 2.998 × 10 m/s . SI prefixes based on powers of 10 are also used to describe small or large quantities. For example,
11036-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
11160-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
11284-421: The letters mīm (m) and kāf (k), respectively, by the 15th century, as seen in the work of Abu'l-Hasan ibn Ali al-Qalasadi . Nicolas Chuquet used a form of exponential notation in the 15th century, for example 12 to represent 12 x . This was later used by Henricus Grammateus and Michael Stifel in the 16th century. In the late 16th century, Jost Bürgi would use Roman numerals for exponents in
11408-731: The line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give
11532-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
11656-819: The multiplication rule implies the definition b 0 = 1. {\displaystyle b^{0}=1.} A similar argument implies the definition for negative integer powers: b − n = 1 / b n . {\displaystyle b^{-n}=1/b^{n}.} That is, extending the multiplication rule gives b − n × b n = b − n + n = b 0 = 1 {\displaystyle b^{-n}\times b^{n}=b^{-n+n}=b^{0}=1} . Dividing both sides by b n {\displaystyle b^{n}} gives b − n = 1 / b n {\displaystyle b^{-n}=1/b^{n}} . This also implies
11780-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
11904-440: The numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. Such simple expressions generally do not exist for other angles which are rational multiples of a right angle. The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees. G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that
12028-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains
12152-514: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
12276-417: The position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities hold for any angle θ and any integer k . The same
12400-405: The prefix kilo means 10 = 1000 , so a kilometre is 1000 m . The first negative powers of 2 have special names: 2 − 1 {\displaystyle 2^{-1}} is a half ; 2 − 2 {\displaystyle 2^{-2}} is a quarter . Powers of 2 appear in set theory , since a set with n members has a power set ,
12524-654: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
12648-434: The right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies
12772-413: The same ordinary differential equation Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 . One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi }
12896-401: The same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table. In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn
13020-422: The set of all of its subsets , which has 2 members. Integer powers of 2 are important in computer science . The positive integer powers 2 give the number of possible values for an n - bit integer binary number ; for example, a byte may take 2 = 256 different values. The binary number system expresses any number as a sum of powers of 2 , and denotes it as a sequence of 0 and 1 , separated by
13144-420: The simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis . The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has
13268-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
13392-561: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
13516-457: The superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use. If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are
13640-461: The term indices in 1696. The term involution was used synonymously with the term indices , but had declined in usage and should not be confused with its more common meaning . In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing: Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions , since in those
13764-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
13888-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
14012-738: The trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. Various mnemonics can be used to remember these definitions. In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent
14136-504: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
14260-684: The unique solution to the initial value problem : Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of
14384-549: The unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and
14508-531: The universe. In the 9th century, the Persian mathematician Al-Khwarizmi used the terms مَال ( māl , "possessions", "property") for a square —the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property" —and كَعْبَة ( Kaʿbah , "cube") for a cube , which later Islamic mathematicians represented in mathematical notation as
14632-418: The values of all trigonometric functions for any arbitrary real value of θ in the following manner. The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is, In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with
14756-415: The whole complex plane . Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation. Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that
14880-452: The whole complex plane: The last one was used in the historically first proof that π is irrational . There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: This identity can be proved with the Herglotz trick. Combining the (– n ) th with
15004-406: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
15128-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
15252-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
15376-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
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