integers

A number originally was a count or a measurement. Mathematicians have extended this concept to include abstractions such as the square root of minus one. In common usage, number symbols are often used as labels (highway numbers) or to indicate order (serial numbers). Naturals {0,1,2,3..} Primes { 2,3,5,7,11,.. } Integers {..-1,0,1,..} Rationals Constructibles Irrational numbers Real numbers () Imaginary numbers Complex (), Algebraic numbers Transcendentals Transfinite numbers Computable numbers
Including:

• Number - Further generalizations
• Number - Numerals and numbering
• Number - Extensions

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Addition is the most basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series. Repeated addition of the number one is the most basic form of counting. Addition can also be defined for mathematical objects other than numbers — for example, matrices or ...

Including:

• Addition - Notation and terminology
• Addition - Interpretations
• Addition - Extending a measure
• Addition - Combining translations
• Addition - Properties
• Addition - Commutativity
• Addition - Associativity
• Addition - Zero and one
• Addition - Units
• Addition - Definitions and proofs for the real numbers
• Addition - Naturals
• Addition - Integers
• Addition - Rationals
• Addition - Reals
• Addition - Generalizations
• Addition - In algebra
• Addition - Addition of sets
• Addition - Related operations
• Addition - Notes

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Until its proof in 1995, the most famous of all conjectures was the mis-named Fermat's last theorem - this conjecture became a true theorem only after its proof. In the process, a special case of the Taniyama-Shimura conjecture, itself a longstanding open problem, was proven; this conjecture has since been completely proven. Other famous conjectures include: There are no odd perfect numbers Goldbach's conjecture The twin prime conjecture The Collatz conjecture The Riemann hypothesis P ≠ NP The Poinca ...

See also:

Conjecture, Conjecture - Famous conjectures, Conjecture - Counterexamples, Conjecture - Use of conjectures in conditional proofs, Conjecture - Undecidable conjectures, Conjecture - Usage outside of mathematics

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In mathematics, the axiom of determinacy (abbreviated as AD) is an axiom in set theory. It states the following: Consider infinite two-person games with perfect information. Then, every game of length ω where both players choose integers is determined, i.e., one of the two players has a winning strategy. The axiom of determinacy is inconsistent with the axiom of choice (AC); however, it has been shown that it implies that all sets of re ...

Including:

• Axiom of determinacy - Types of game that are determined
• Axiom of determinacy - Why the axiom of choice contradicts the axiom of determinacy
• Axiom of determinacy - Infinite logic and the axiom of determinacy

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In computer programming, an array, also known as a vector or list, is one of the simplest data structures. Arrays hold a series of data elements, usually of the same size and data type. Individual elements are accessed by index using a consecutive range of integers, as opposed to an associative array. Some arrays are multi-dimensional, meaning they are indexed by a fixed number of integers, for example by a tuple of four integers. Generally, one- and ...

Including:

• Array - Advantages and disadvantages
• Array - Uses
• Array - Indices into arrays
• Array - Multi-dimensional arrays

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A character encoding consists of a code that pairs a set of characters (representations of graphemes or grapheme-like units, such as might appear in an alphabet or syllabary for the communication of a natural language) with a set of something else, such as numbers or electrical pulses, in order to facilitate the storage of text in computers and the transmission of text through telecommunication networks. Common examples include Morse code, which encodes letters of the Latin alphabet as series of long and short depressions of a telegraph key; and ASCII, which encodes letters, numerals, and other symbols, both as ...

Including:

• Character encoding - Character repertoire
• Character encoding - Encoding forms and encoding schemes
• Character encoding - Popular character encodings

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Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business. Arithmetic - Arithmetic operations.

Including:

• Arithmetic - Arithmetic operations
• Arithmetic - Number theory
• Arithmetic - Pronunciation

Read more here: » Arithmetic: Encyclopedia - Arithmetic

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

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In mathematics, a base (or radix) is the number of single digits, including zero, denoting different values in a positional numeral system. For example, the decimal system, the most common system in use today, uses base ten, hence the maximum number a single digit will ever reach is 9, after this it is necessary to add another digit to achieve a higher number. Sometimes, a subscript notation is used where the base number is written in subscript after the number represented. For example, 238 indicates that the ...

Including:

• Base mathematics - System
• Base mathematics - Conversion between bases
• Base mathematics - Applications
• Base mathematics - Historical systems
• Base mathematics - Computing

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The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of this encyclopedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — m ...

Including:

• Areas of mathematics - Foundations / general
• Areas of mathematics - Algebra
• Areas of mathematics - Analysis
• Areas of mathematics - Geometry
• Areas of mathematics - Applied mathematics
• Areas of mathematics - Probability and statistics
• Areas of mathematics - Computational sciences
• Areas of mathematics - Physical sciences
• Areas of mathematics - Non-physical sciences

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Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. Sometimes known as "the prince of mathematicians", Gauss had a remarkable influence in many fields of mathematics and science and is ranked beside Euler, Newton ...

Including:

• Carl Friedrich Gauss - Biography
• Carl Friedrich Gauss - Early years
• Carl Friedrich Gauss - Middle years
• Carl Friedrich Gauss - Later years death and afterwards
• Carl Friedrich Gauss - Family
• Carl Friedrich Gauss - Personality
• Carl Friedrich Gauss - Commemorations

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On a computer, arbitrary-precision arithmetic, also called bignum arithmetic, is a technique that allows computer programs to perform calculations on integers and rational numbers with an arbitrary number of digits of precision, limited only by the available memory of the host system. It typically works by storing a number as a variable-length array of digits in some base, in contrast to most computer arithmetic which uses a fixed number of bits given by the size of the processor registers. Rational numbers can be stored as a p ...

Including:

• Arbitrary-precision arithmetic - Applications
• Arbitrary-precision arithmetic - Algorithms
• Arbitrary-precision arithmetic - Arbitrary-precision software

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Sir Andrew John Wiles (born April 11, 1953) is a British mathematician living in the United States. He was educated at The Leys School Cambridge and in 1974 he graduated from the University of Oxford. He then completed his Ph.D. at the University of Cambridge in 1979 and is currently a Professor and the chair of the department of mathematics at Princeton University. In one of the great success stories in the history of mathematics, Wiles (with help from Richard Taylor) proved Fermat's Last Theorem in 1994. Andrew Wiles - ...

Including:

• Andrew Wiles - Early life and work
• Andrew Wiles - Interest in the Fermat problem
• Andrew Wiles - Announcement and aftermath
• Andrew Wiles - Awards

Read more here: » Andrew Wiles: Encyclopedia - Andrew Wiles

The mathematical constant π is a real number which may be defined as the ratio of a circle's circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. The name of the Greek letter π is pi (pronounced pie in English), and this spelling can be used in typographical contexts where the Greek letter is not available. π is also known as Archimedes' constant (not to be confused with Archime ...

Including:

• Pi - Properties
• Pi - Formulae involving π
• Pi - Geometry
• Pi - Analysis
• Pi - Continued fractions
• Pi - Number theory
• Pi - Dynamical systems and ergodic theory
• Pi - Physics
• Pi - Probability and statistics
• Pi - History of π
• Pi - Numerical approximations of π
• Pi - Miscellaneous formulae
• Pi - Less accurate approximations
• Pi - Open questions
• Pi - The nature of π
• Pi - Fictional references
• Pi - π culture

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Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attrib ...

Including:

• Infinity - History
• Infinity - Ancient view of infinity
• Infinity - Views from the Renaissance to modern times
• Infinity - Modern philosophical views
• Infinity - Infinity symbol
• Infinity - Mathematical infinity
• Infinity - Infinity in real analysis
• Infinity - Infinity in complex analysis
• Infinity - Arithmetic properties of infinity
• Infinity - Infinity in set theory
• Infinity - Mathematics without infinity
• Infinity - Use of infinity in common speech
• Infinity - Physical infinity
• Infinity - Infinity in cosmology
• Infinity - Three types of infinities
• Infinity - Infinity in science fiction
• Infinity - Note

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In mathematics, a set can be thought of as any collection of distinct things considered as a whole. Though a simple idea, it is nevertheless one of the most important and fundamental concepts in modern mathematics, and the study of the structure of possible sets, set theory, is quite rich. Set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as primary school. It is the language in which modern mathematics is described. Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from wh ...

Including:

• Set - Definition
• Set - Describing sets
• Set - Descriptions using words or lists
• Set - Descriptions using mathematical notation
• Set - Set membership
• Set - Cardinality of a set
• Set - Subsets
• Set - Special sets
• Set - Unions
• Set - Intersections
• Set - Complements

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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FALSE is an esoteric programming language designed by Wouter van Oortmerssen in 1993, named after his favourite boolean value. It is a small Forth-like stack-oriented language, with syntax designed to make the code inherently obfuscated, confusing, and unreadable. It is also noteworthy for having a compiler of only 1024 bytes (written in 68000 assembly). According to van Oortmerssen, FALSE provided the inspiration for various well known esoteric ...

Including:

• FALSE - The stack
• FALSE - Data types
• FALSE - Basic operators
• FALSE - Variables
• FALSE - Subroutines
• FALSE - Control flow
• FALSE - If
• FALSE - While
• FALSE - Strings
• FALSE - Input / Output
• FALSE - Comments
• FALSE - Code examples

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In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets). A class that is not a set is called a proper class. A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fr ...

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In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form: anxn + an−1xn−1 + ··· + a1x + a0 = 0 wh ...

Including:

• Algebraic number - The field of algebraic numbers
• Algebraic number - Numbers defined by radicals
• Algebraic number - Algebraic integers
• Algebraic number - Special classes of algebraic number

Read more here: » Algebraic number: Encyclopedia - Algebraic number