Return the greatest common divisor of the specified integer arguments.If any of the arguments is nonzero, then the returned value is the largestpositive integer that is a divisor of all arguments. If all argumentsare zero, then the returned value is 0. gcd() without argumentsreturns 0.
The IEEE 754 special values of NaN, inf, and -inf will behandled according to IEEE rules. Specifically, NaN is not consideredclose to any other value, including NaN. inf and -inf are onlyconsidered close to themselves.
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Return the least common multiple of the specified integer arguments.If all arguments are nonzero, then the returned value is the smallestpositive integer that is a multiple of all arguments. If any of the argumentsis zero, then the returned value is 0. lcm() without argumentsreturns 1.
Return the IEEE 754-style remainder of x with respect to y. Forfinite x and finite nonzero y, this is the difference x - n*y,where n is the closest integer to the exact value of the quotient x /y. If x / y is exactly halfway between two consecutive integers, thenearest even integer is used for n. The remainder r = remainder(x,y) thus always satisfies abs(r)
Special cases follow IEEE 754: in particular, remainder(x, math.inf) isx for any finite x, and remainder(x, 0) andremainder(math.inf, x) raise ValueError for any non-NaN x.If the result of the remainder operation is zero, that zero will havethe same sign as x.
Return x with the fractional partremoved, leaving the integer part. This rounds toward 0: trunc() isequivalent to floor() for positive x, and equivalent to ceil()for negative x. If x is not a float, delegates to x.__trunc__, which should return an Integral value.
Return e raised to the power x, minus 1. Here e is the base of naturallogarithms. For small floats x, the subtraction in exp(x) - 1can result in a significant loss of precision; the expm1()function provides a way to compute this quantity to full precision:
Return x raised to the power y. Exceptional cases followthe IEEE 754 standard as far as possible. In particular,pow(1.0, x) and pow(x, 0.0) always return 1.0, evenwhen x is a zero or a NaN. If both x and y are finite,x is negative, and y is not an integer then pow(x, y)is undefined, and raises ValueError.
Return the complementary error function at x. The complementary errorfunction is defined as1.0 - erf(x). It is used for large values of x where a subtractionfrom one would cause a loss of significance.
CPython implementation detail: The math module consists mostly of thin wrappers around the platform Cmath library functions. Behavior in exceptional cases follows Annex F ofthe C99 standard where appropriate. The current implementation will raiseValueError for invalid operations like sqrt(-1.0) or log(0.0)(where C99 Annex F recommends signaling invalid operation or divide-by-zero),and OverflowError for results that overflow (for example,exp(1000.0)). A NaN will not be returned from any of the functionsabove unless one or more of the input arguments was a NaN; in that case,most functions will return a NaN, but (again following C99 Annex F) thereare some exceptions to this rule, for example pow(float('nan'), 0.0) orhypot(float('nan'), float('inf')).
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,[1] algebra,[2] geometry,[1] and analysis,[3] respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
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 from 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.[5][6]
Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[7] 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[a] and infinitesimal calculus were introduced as new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both.[8] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method,[9] 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.
The word mathematics comes from Ancient Greek mthma (tag_hash_110), meaning "that which is learnt",[10] "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in Classical times.[b] Its adjective is mathmatiks (), meaning "related to learning" or "studious", which likewise further came to mean "mathematical".[14] In particular, mathmatik tkhn ( ; Latin: ars mathematica) meant "the mathematical art".[10]
In Latin, and in 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", is sometimes mistranslated as a condemnation of mathematicians.[16]
Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes.[20] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.[21]
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.[25][9] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.[26] Some of these areas correspond to the older division, as 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.[27]
Number theory began with 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 is mostly used for numerical calculations.[28] Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria.[29] 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.[30]
Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example 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.[31] 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.[32]
Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).[27]
Geometry 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.[33]
A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it 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, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.[34][35]
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.[c][33] 152ee80cbc
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