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135 relations: Absolute value, Absolute value (algebra), Accuracy and precision, Additive inverse, Adele ring, Adelic algebraic group, Algebraic closure, Algebraic extension, Algebraic geometry, Algebraic number field, Algebraically closed field, Almost all, American Mathematical Monthly, Andrew Wiles, Arithmetic, Axiom of choice, Balanced ternary, Base (topology), Bijective numeration, Binary number, Bit numbering, Brightness, C-minimal theory, Calculus, Cambridge University Press, Cantor set, Cardinality, Cardinality of the continuum, Carry (arithmetic), Cauchy sequence, Characteristic (algebra), Chromaticity, Colorfulness, Commutative ring, Compact space, Compactification (mathematics), Complete metric space, Complex number, Composite number, Constructive proof, Coprime integers, Countable set, Counterexamples in Topology, Cyclotomic field, Decimal representation, Dedekind domain, Derivative, Divisor, Dyadic rational, E (mathematical constant), ..., Encyclopedia of Mathematics, Endomorphism ring, Equivalence class, Eric Hehner, Ernst Kummer, Euclidean distance, Euler's theorem, Exponentiation, Factorial, Fermat's Last Theorem, Field (mathematics), Field of fractions, Fractional ideal, Fundamental theorem of arithmetic, Galois extension, Galois group, Global field, Graduate Texts in Mathematics, Hasse principle, Helmut Hasse, Hensel's lemma, Hue, If and only if, Integer, Integral domain, Interaction, Inverse limit, John Stillwell, Kurt Hensel, Limit of a sequence, Local field, Localization of a ring, Locally compact space, Locally constant function, Magnitude (mathematics), Mahler's theorem, Mathematics, Metric (mathematics), Metric space, Modular arithmetic, Natural number, Nigel Horspool, Number, Number theory, Numerical digit, Order theory, Ordered field, Ostrowski's theorem, P-adic analysis, P-adic order, P-adic quantum mechanics, Power series, Prüfer group, Prime ideal, Prime number, Prime power, Profinite integer, Prosolvable group, Quadratic form, Quote notation, Radix, Rational number, Real number, Residue field, Ring (mathematics), Series (mathematics), SIAM Journal on Computing, Solvable group, Subring, Subspace topology, Tate module, Ternary relation, Topological space, Torsion (algebra), Two's complement, Ultrametric space, Uncountable set, Valuation (algebra), Variable (mathematics), Volkenborn integral, Wiles's proof of Fermat's Last Theorem, Zero divisor, 0.999..., 1 + 2 + 4 + 8 + ⋯, 2. Expand index (85 more) »
Absolute value
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
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Absolute value (algebra)
In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain.
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Accuracy and precision
Precision is a description of random errors, a measure of statistical variability.
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Additive inverse
In mathematics, the additive inverse of a number is the number that, when added to, yields zero.
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Adele ring
In mathematics, the adele ring (also adelic ring or ring of adeles) is defined in class field theory, a branch of algebraic number theory.
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Adelic algebraic group
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A.
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Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
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Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic number field
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
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Algebraically closed field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.
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Almost all
In mathematics, the term "almost all" means "all but a negligible amount".
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American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.
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Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory.
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Arithmetic
Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.
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Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
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Balanced ternary
Balanced ternary is a non-standard positional numeral system (a balanced form), used in some early computers and useful in the solution of balance puzzles.
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Base (topology)
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.
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Bijective numeration
Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits.
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Binary number
In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).
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Bit numbering
In computing, bit numbering (or sometimes bit endianness) is the convention used to identify the bit positions in a binary number or a container for such a value.
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Brightness
Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light.
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C-minimal theory
In model theory, a branch of mathematical logic, a C-minimal theory is a theory that is "minimal" with respect to a ternary relation C with certain properties.
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Calculus
Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.
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Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
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Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.
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Carry (arithmetic)
In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits.
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Cauchy sequence
In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Chromaticity
Chromaticity is an objective specification of the quality of a color regardless of its luminance.
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Colorfulness
Colorfulness, chroma and saturation are attributes of perceived color relating to chromatic intensity.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Compactification (mathematics)
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.
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Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Composite number
A composite number is a positive integer that can be formed by multiplying together two smaller positive integers.
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Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object.
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Coprime integers
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
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Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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Counterexamples in Topology
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.
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Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to, the field of rational numbers.
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Decimal representation
A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum where a0 is a nonnegative integer, and a1, a2,...
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Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
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Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
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Dyadic rational
In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator, when the ratio is in minimal (coprime) terms, is a power of two, i.e., a number of the form \frac where a is an integer and b is a natural number; for example, 1/2 or 3/8, but not 1/3.
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E (mathematical constant)
The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.
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Encyclopedia of Mathematics
The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.
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Endomorphism ring
In abstract algebra, the endomorphism ring of an abelian group X, denoted by End(X), is the set of all endomorphisms of X (i.e., the set of all homomorphisms of X into itself) endowed with an addition operation defined by pointwise addition of functions and a multiplication operation defined by function composition.
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Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
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Eric Hehner
Eric C. R. Hehner, called Rick, is a Canadian computer scientist.
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Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.
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Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.
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Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if n and a are coprime positive integers, then where \varphi(n) is Euler's totient function.
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Exponentiation
Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.
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Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.
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Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.
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Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains.
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Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
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Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
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Galois group
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
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Global field
In mathematics, a global field is a field that is either.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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Hasse principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number.
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Helmut Hasse
Helmut Hasse (25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local class field theory and diophantine geometry (Hasse principle), and to local zeta functions.
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Hensel's lemma
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number, then this root corresponds to a unique root of the same equation modulo any higher power of, which can be found by iteratively "lifting" the solution modulo successive powers of.
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Hue
Hue is one of the main properties (called color appearance parameters) of a color, defined technically (in the CIECAM02 model), as "the degree to which a stimulus can be described as similar to or different from stimuli that are described as red, green, blue, and yellow", (which in certain theories of color vision are called unique hues).
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Integral domain
In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
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Interaction
Interaction is a kind of action that occur as two or more objects have an effect upon one another.
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Inverse limit
In mathematics, the inverse limit (also called the projective limit or limit) is a construction that allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects.
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John Stillwell
John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University.
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Kurt Hensel
Kurt Wilhelm Sebastian Hensel (29 December 1861 – 1 June 1941) was a German mathematician born in Königsberg.
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Limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.
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Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.
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Localization of a ring
In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.
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Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
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Locally constant function
In mathematics, a function f from a topological space A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant.
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Magnitude (mathematics)
In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind.
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Mahler's theorem
In mathematics, Mahler's theorem, introduced by, expresses continuous ''p''-adic functions in terms of polynomials.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Metric (mathematics)
In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.
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Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
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Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).
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Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
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Nigel Horspool
R.
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Number
A number is a mathematical object used to count, measure and also label.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Numerical digit
A numerical digit is a single symbol (such as "2" or "5") used alone, or in combinations (such as "25"), to represent numbers (such as the number 25) according to some positional numeral systems.
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Order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.
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Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.
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Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.
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P-adic analysis
In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers.
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P-adic order
In number theory, for a given prime number, the -adic order or -adic valuation of a non-zero integer is the highest exponent such that divides.
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P-adic quantum mechanics
p-adic quantum mechanics is a collection of related research efforts in quantum physics that replace real numbers with ''p''-adic numbers.
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Power series
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
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Prüfer group
In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique ''p''-group in which every element has p different p-th roots.
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Prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Prime power
In mathematics, a prime power is a positive integer power of a single prime number.
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Profinite integer
In mathematics, a profinite integer is an element of the ring where p runs over all prime numbers, \mathbb_p is the ring of ''p''-adic integers and \widehat.
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Prosolvable group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups.
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Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.
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Quote notation
Quote notation is a numeral system for representing rational numbers which was designed to be attractive for use in computer architecture.
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Radix
In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Residue field
In mathematics, the residue field is a basic construction in commutative algebra.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
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SIAM Journal on Computing
The SIAM Journal on Computing is a scientific journal focusing on the mathematical and formal aspects of computer science.
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Solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.
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Subring
In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).
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Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).
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Tate module
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A.
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Ternary relation
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Torsion (algebra)
In abstract algebra, the term torsion refers to elements of finite order in groups and to elements of modules annihilated by regular elements of a ring.
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Two's complement
Two's complement is a mathematical operation on binary numbers, best known for its role in computing as a method of signed number representation.
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Ultrametric space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\.
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Uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
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Valuation (algebra)
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field.
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Variable (mathematics)
In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.
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Volkenborn integral
In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions.
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Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves.
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Zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.
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0.999...
In mathematics, 0.999... (also written 0., among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).
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1 + 2 + 4 + 8 + ⋯
In mathematics, is the infinite series whose terms are the successive powers of two.
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2
2 (two) is a number, numeral, and glyph.
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References
[1] https://en.wikipedia.org/wiki/P-adic_number
