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Coprime

Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3. The number 1 is coprime to every integer; 0 is coprime only to 1 and −1. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. Euler's totient function (or Euler's phi function) of a positive integer n is the number of integers coprime to integers from 1 to n - 1.

Properties

There are a number of conditions which are equivalent to a and b being coprime:
- There exist integers x and y such that ax + by = 1 (see Bézout's identity).
- The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In other words, b is a unit in the ring Z/aZ of integers modulo a. As a consequence, if a and b are coprime and br ≡ bs (mod a), then r ≡ s (mod a) (because we may "divide by b" when working modulo a). Furthermore, if a and b₁ are coprime, and a and b₂ are coprime, then a and b₁b₂ are also coprime (because the product of units is a unit). If a and b are coprime and a divides a product bc, then a divides c. This can be viewed as a generalisation of Euclid's lemma, which states that if p is prime, and p divides a product bc, then either p divides b or p divides c. The two integers a and b are coprime if and only if the point with coordinates (a, b) in an Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a, b). The probability that two randomly chosen integers are coprime is 6/π² (see pi), which is about 60%. Two natural numbers a and b are coprime if and only if the numbers 2^a − 1 and 2^b − 1 are coprime.

Cross notation, group

If n≥1 is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation; it is written as (Z/nZ)^× or Z_n^-.

Generalizations

Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime. If the ideals A and B of R are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese Remainder Theorem is an important statement about coprime ideals. The concept of being relatively prime can also be extended any finite set of integers S = to mean that the greatest common divisor of the elements of the set is 1. If every pair of integers in the set is relatively prime, then the set is called pairwise relatively prime. Every pairwise relatively prime set is relatively prime; however, the converse is not true: is relatively prime, but not pairwise relative prime. (In fact, each pair of integers in the set has a non-trivial common factor.)

Mathematics

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. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Integer

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set. The set of all integers is usually denoted in mathematics by a boldface Z (or blackboard bold,

\mathbb

), which stands for Zahlen (German for "numbers"). The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as the Gaussian integers.

Algebraic properties

Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. The following table lists some of the basic properties of addition and multiplication for any integers a, b and c. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... 1 or (−1) + (−1) + ... + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, note that not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. All the properties from the above table taken together say that Z together with addition and multiplication is a commutative ring with unity. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the field of rational numbers. This process can be mimicked to form the quotient field of any integral domain, where an integral domain is a commutative ring with unity such that whenever ab = 0, either a = 0 or b = 0. Although ordinary division is not defined on Z, it does possess an important property called the division algorithm: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The integer q is called the quotient and r is called the remainder, resulting from division of a by b. This is the basis for the Euclidean algorithm for computing greatest common divisors. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.

Order-theoretic properties

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by : ... < −2 < −1 < 0 < 1 < 2 < ... An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: # if a < b and c < d, then a + c < b + d # if a < b and 0 < c, then ac < bc. (From this fact, one can show that if c < 0, then ac > bc.)

Integers in computing

An integer (sometimes known as an "int", from the name of a datatype in the C programming language) is often a primitive datatype in computer languages. However, integer datatypes can only represent a subset of all integers, since practical computers are of finite capacity. Variable-length representations of integers, such as bignums, can store any integer that fits in the computers memory. Other integer datatypes are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10). In contrast, theoretical models of digital computers, such as Turing machines, typically do have infinite (but only countable) capacity.

Quotations

God invented the integers, all else is the work of man. Kronecker

External links

- [http://www.positiveintegers.org The Positive Integers - divisor tables and numeral representation tools] Category:Elementary mathematics Category:Group theory Category:Integers Category:Elementary number theory Category:Set theory ko:정수 ja:整数 th:จำนวนเต็ม

Divisor

:For divisors in algebraic geometry, see divisor (algebraic geometry). In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder.

Explanation

For example, 7 is a divisor of 42 because 42/7 = 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 or 7 divides 42 and we usually write 7 | 42. For example, the positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. In general, we say m|n (read: m divides n) for any integers m and n iff there exists an integer k such that n = km. Thus, divisors can be negative as well as positive. 1 and -1 are divisors of every integer, every integer is a divisor of itself, and every integer is a divisor of 0, while 0 is a divisor only of 0 (see also division by zero). Numbers divisible by 2 are called even and those that are not are called odd. A divisor of n that is not 1, -1, n or -n is known as a non-trivial divisor; numbers with non-trivial divisors are known as composite numbers, while prime numbers have no non-trivial divisors. The name comes from the arithmetic operation of division: if a/b=c then a is the dividend, b the divisor, and c the quotient. There are some rules which allow to recognize small divisors of a number from the number's decimal digits.

Further notions and facts

Some elementary rules:
- If a | b and a | c, then a | (b + c), in fact, a | (mb + nc) for all integers m, n.
- If a | b and b | c, then a | c. (transitive relation)
- If a | b and b | a, then a = b or a = -b. The following property is important:
- If a | bc, and gcd(a,b) = 1, then a | c. (Euclid's lemma) A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. (A number which does not evenly divide n, but leaves a remainder, is called an aliquant part of n.) An integer n > 1 whose only proper divisor is 1 is called a prime number. Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of the Fundamental theorem of arithmetic. If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than that sum are said to be deficient, while numbers greater than that sum are said to be abundant. The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42) = 8 = 2×2×2 = d(2)×d(3)×d(7)). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42) = 96 = 3×4×8 = σ(2)×σ(3)×σ(7)). If the prime factorization of n is given by :

n = p_1^ \, p_2^ \, ... \, p_n^

then the number of positive divisors of n is :

d(n) = (\nu_1 + 1) (\nu_2 + 1) ... (\nu_n + 1),

and each of the divisors has the form :

p_1^ \, p_2^ \, ... \, p_n^

where :

\forall i : 0 \le \mu_i \le \nu_i.

Divisibility of numbers

The relation | of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest one is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z. If an integer n is written in base b, and d is an integer with b ≡ 1 (mod d), then n is divisible by d if and only if the sum of its digits is divisible by d. The rules for d=3 and d=9 given above are special cases of this result (b=10). We can generalize this method even further to find how to check divisibility of any integer in any base by any other (smaller integer). Let us say that we want to determine if d | a in base b. Then we first find a pair of integers (n, k) that solves the congruence bⁿ ≡ k (mod d). Now rather than summing the digits, we take a (which has m digits) and multiply the first m-n digits by k and add the product to the last (or more precisely, smallest) k digits and repeat if necessary. If the result is a multiple of d then the original number is divisible by d. A few examples will help demonstrate this. Since 10³ ≡ 1 (mod 37) then the number 1523836638 gives 1+523+836+638 = 1998 which gives 999 which we know is divisible by 37 due to the above congruence. Again, 10² ≡ 2 (mod 7) so 43106 gives 431×2 + 06 = 868 which gives 8×2+68 = 84 which is easily noted as being a multiple of 7. Note that there is no unique triple (n, k, d) since for example 10 ≡ 3 (mod 7) so we could also have done 4310×3 + 6 = 12936 and 1293×3 + 6 = 3885 and 388×3 + 5 = 1169 and 116×3 + 9 = 357 and 35×3 + 7 = 112 and 11×3 + 2 = 35 and 3×3 + 5 = 14 and 1×3 + 4 = 7. Clearly this is not always efficient but note that each number in this series, 43106, 12936, 3885, 1169, 357, 112, 35, 14, 7 is a multiple of 7 and many series could contain trivially identifiable multiples. This method is not necessarily useful for some numbers (for example 10⁴ ≡ 4 (mod 17) is the first n where k < 10) but lends itself to fast calculations in other cases where n and k are relatively small.

Generalization

One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting.

External links

- [http://www.farfarfar.com/math/calculators/factoring/ Factoring Calculator] -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.
- [http://users.adelphia.net/~j.mccranie/ webpage that has program for factoring up to 18 digit numbers] Category:Elementary number theory Category:Elementary arithmetic ko:약수 ja:約数

6 (number)

6 (six) is the natural number following 5 and preceding 7. The SI prefix for 1000⁶ is exa (E), and for its reciprocal atto (a).

In mathematics

Six is the second smallest composite number, its proper divisors being 1, 2 and 3. Since six equals the sum of these proper divisors, six is a perfect number. As a perfect number, 6 is related to the Mersenne prime 3, since 2¹(2² - 1) = 6. The next perfect number is 28. Six is also a unitary perfect number, a harmonic divisor number and a highly composite number. The next highly composite number is 12. The smallest non-abelian group is the symmetric group S₃ which has 3! = 6 elements. S₆, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n isomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in 1-1 correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n=6. Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others. In binary code, six is 110; in ternary code six is 20; in quaternary numeral system code six is 12; in quinary six is 11; in senary six is 10; in septenary code and all codes above (such as octal, decimal and hexadecimal) six is 6. Since it is divisible by the sum of its digits in all these bases, 6 is one of the four all-Harshad numbers. A six-sided polygon is a hexagon. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Six is also an octahedral number. It is a triangular number and so is its square (36). In base 10, 6 is a 1-automorphic number.

The Arabic glyph

Image:Evo6glyph.png The evolution of our modern glyph for 6 appears rather simple when compared with that for the other numerals. Our modern 6 can be traced back to the Brahmin Indians, who wrote it in one stroke like a cursive lowercase e rotated 45 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Ghubar Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G. On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on. For calculators that can display results in hexadecimal, a 6 that looks like a B is not practical. In fonts with text figures, 6 usually has an ascender, for example, Image:TextFigs036.png.

In science

- The atomic number of carbon.
- The number of carbon atoms and carbon-carbon bonds in benzene. In astronomy, : Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster. : The New General Catalogue [http://www.ngcic.org/ object] NGC 6, a spiral galaxy in the constellation Andromeda :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the solar eclipse series which began on -2691 March 16 and ended on -1393 May 3. The duration of Saros series 6 was 1298.1 years, and it contained 73 solar eclipses. :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -2642 July 25 and ended on -1091 February 10. The duration of Saros series 6 was 1550.6 years, and it contained 87 lunar eclipses.

In music

- The number of strings on a standard guitar.
- The number of basic holes or keys on most woodwind instruments (e.g., pennywhistle, clarinet, saxophone, bassoon). These holes or keys are usually not given numbers or letters in the fingering charts.
- "Six Geese a Laying" were given as a present on the sixth day in the popular Christmas carol The Twelve Days of Christmas.
- The number of completed symphonies by Peter Tchaikovsky
- The number of Brandenburg Concertos of Johann Sebastian Bach
- The concerti grossi Opus 3, organ concertos Opus 4 and Opus 7 (each) by Georg Friederich Handel
- The number of string quartets of Béla Bartók

In sports

- In American Football, the number of points received for a touchdown.
- In rugby union, the position of blindside flanker.
- In baseball, six represents the shortstop's position.
- In Australian Rules football, six points are received for a goal.
- In cricket, a "six" or "sixer" is a shot in which the ball clears the boundary without bouncing, scoring six runs. See cricket terminology.
- Retired number of former baseball players Stan Musial and Al Kaline

In technology

- On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry it is the key for J and K.

In television

- The audio component of any broadcast channel 6 is located at 87.7 MHz. This means that people can listen to TV stations such as WSYX using an FM radio when a television is not nearby. See also
- Number Six was the main character in the television series The Prisoner.
- Number Six is a character in the television mini-series and series Battlestar Galactica (2003).
- Six is the name of a character on Blossom.

In other fields

Six is:
- In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June is the sixth month and Sextilis was renamed August.
- Whether Thursday or Friday or Saturday is the sixth day of the week varies across cultures.
- Sextidi was the sixth day of the decade in the French Revolutionary calendar.
- The length in years of a standard term of a United States senator.
- The name of the smallest group of Cub Scouts, traditionally consisting of six people and is led by a 'sixer'. Logically speaking, this isn't always the case, particularly in packs with less than 6 Cub Scouts in it.
- The number of cans in a six-pack.
- The number of feet below ground level a coffin is traditionally buried; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead.
- The number of inhabited continents, and the total number of continents if Eurasia (Europe plus Asia) is considered a single continent.
- The number of large geyser fields in the world.
- The number of points on a Star of David.
- In Astrology, Virgo is the 6th astrological sign of the Zodiac.
- The first number where counting on your fingers requires two hands.
- The number of sides on a cube, hence the highest number on a standard die.
- The highest number on one end of a standard domino.
- Les Six were a group of Lost Generation writers.
- One of A. A. Milne's poetry books was entitled Now We Are Six.
- Extra-sensory perception is sometimes called the "sixth sense".
- Bands with the number six in their name include Eve 6, Sixpence None the Richer, Los Xey (sei is Basque for "six")and Electric Six.
- Six Cardinal Directions : north, south, east, west, heaven, and land.
- The Six dynasties form part of Chinese history.
- The Bionic Six are the heroes of an animated series.
- The Birmingham Six were held in prison for 16 years.
- Six Degrees of Kevin Bacon is a game that can be played with the names of actors and actresses.
- John L. Holland's RIASEC is a system that places occupations into six categories.
- 666 is considered the Number of the Beast.
- The year "6" can refer to: A.D. 6, 6 B.C., or 1906. Hexa is Greek for "six". Thus:
- A hexahedron is a polyhedron with six faces, with a cube being a special case.
- An hexagon is a regular polygon with six sides.
- L' Hexagone is a French nickname for the continental part of France.
- A hexapod is an animal with six legs; this includes all insects.
- Hexameter is a poetic form consisting of six feet per line.
- "Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16. The prefix "hexa-" also occurs in the systematic name of many chemical compounds, such as "hexamethyl". Sex- is a Latin prefix meaning "six". Thus:
- A group of six musicians is called a sextet.
- Six babies delivered in one birth are sextuplets. The first set of sextuplets of whom all six survived are the Dilley sextuplets.
- People with sexdactyly have six fingers on each hand.
- The measuring instrument called a sextant got its name because its shape forms one sixth of a whole circle.
- The ordinal adjective senary.

References

- The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66--68
- A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0521423856
- [http://math.ucr.edu/home/baez/six.html Some Thoughts on the Number Six], John Baez 06 ko:6 ja:6 th:6 (จำนวน)

27 (number)

27 (twenty-seven) is the natural number following 26 and preceding 28. Twenty-seven is the smallest positive integer requiring four syllables to name in English, though it can be unambiguously defined in just two: "three cubed". It is the first composite number not divisible by any of its digits.


Cardinal	twenty-seven
Ordinal	twenty-seventh
Factorization	$3^3 \,\!$
Roman numeral	XXVII
Binary	11011
Hexadecimal	1B

In mathematics

Twenty-seven is a perfect cube, being 3³ = 3 × 3 × 3. 27 is 3↑↑2 (using Knuth's up-arrow notation). It is also a decagonal number. It is the magic result of a prime reciprocal magic square of the multiples of 1/7. It is a Smith number and a Harshad number. There are exactly 27 straight lines on a cubic surface. It is the twenty-eighth (and twenty-ninth) digit in π. (3.141592653589793238462643383279). A 10,000-day-old person is 27 years old.

In astronomy

- The Messier object M27, a magnitude 7.5 planetary nebula in the constellation Vulpecula, also known as the Dumbbell Nebula.
- The New General Catalogue object NGC 27, a spiral galaxy in the constellation Andromeda
- The Saros number of the solar eclipse series which began on -1993 March 9 and ended on -713 April 16. The duration of Saros series 27 was 1280.1 years, and it contained 72 solar eclipses. Further, the Saros number of the lunar eclipse series which began on -1944 July 17 and ended on -411 January 23. The duration of Saros series 27 was 1532.5 years, and it contained 86 lunar eclipses.
- The 27th moon of Jupiter is Sinope.

In other fields

Twenty-seven is also:
- The atomic number of cobalt.
- The total number of books in the New Testament of the Holy Bible.
- The number of letters in the Spanish alphabet
- The current number of Amendments to the United States Constitution.
- The age in years at which musicians Robert Johnson, Jim Morrison, Jimi Hendrix, Brian Jones, Janis Joplin and Kurt Cobain died. Occasionally, this is termed the ideal age to die, and in fact the concept is so famous that it has inspired a brand of clothing, Dead.At.27.
- The code for international direct-dialed phone calls to South Africa.
- "Weird Al" Yankovic hides the number 27 somewhere in many of his songs and videos.
- With the number 37, a key number in comedian Charles Fleischer's concept of [http://www.monkeydog.com/moleeds/ moleeds].
- I-27 is the designation for a US interstate highway in Texas.
- US 27 is the designation of a highway from Fort Wayne, Indiana to Miami, Florida.
- The number of completed, numbered piano concertos of Wolfgang Amadeus Mozart
- Historical years: 27 A.D., 27 B.C., or 1927

External link

- The homepage of Jonathan J. Harris [http://www.number27.org/ number27.org]
- The Mystery of the number 27 [http://www.chrismore.com/projectx/27/ chrismore.com/projectx/27]
- 27 Conspiracy Sightings from [http://www.black-triangle-ufo-roma.com/27Sightings/StrangeParanormalNumber.html black triangle ufo roma]
- Is the name of Billy Martins(good charlottes guitarist)clothing line,Level 27 2 7 ko:27 ja:27 nb:27 (tall)

1 (number)

: This article discusses the number one. For the year AD 1, see 1. For other uses of 1, see 1 (disambiguation) 1 (one) is a number, numeral, and the name of the glyph representing that number. It is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity, and unit is sometimes used as an adjective in this sense. (For example, a line segment of "unit length" is a line segment of length 1.)

History

Some Ancient Greeks did not consider one as a number: they considered it to be the unit, two being the first proper number as it represented a multiplicity.

In mathematics

For any number x: :x·1 = 1·x = x (This expresses the fact that 1 is the multiplicative identity.) As a consequence of this, 1 is a 1-automorphic number in any place-based numbering system. :x/1 = x (see division) :x¹ = x, 1^x = 1, and for nonzero x, x⁰ = 1 (see exponentiation) :x↑↑1 = x and 1↑↑x = 1 (see tetration). Using ordinary addition, we have 1 + 1 = 2; depending on the interpretation of the symbol "+" and the numeral system used, the expression can have many different meanings, listed at one plus one. One cannot be used as the base of a positional numeral system in the ordinary way. Sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this doesn't work in the same way as other positional numeral systems. Related to this, one cannot take logarithms with base 1, since the "exponential function" with base 1 is the constant function 1. In the Von Neumann representation of natural numbers, 1 is defined as the set . This set has cardinality 1 and hereditary rank 1. Sets like this with a single element are called singletons. In a multiplicative group or monoid, the identity element is sometimes denoted "1", but "e" (from the German Einheit, unity) is more traditional. However, "1" is especially common for the multiplicative identity of a ring. (Note that this multiplicative identity is also often called "unity".) One is its own factorial, and its own square and cube (and so on, as 1 × 1 × ... × 1 = 1). As a consequence of its being its own square, one is also a Kaprekar number. One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few. It is also the first and second numbers in the Fibonacci sequence, and is the first number in a lot of mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that didn't already have it, and considered these initial 1's in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases. One is the empty product. One is a harmonic divisor number. One is most often used for representing 'true' as a Boolean datatype in computer science. One is currently considered neither a prime number, nor a composite number - although it used to be considered prime. Defining a prime as a number that is only divisible by one and itself, one is a prime. However, for purposes of factorization and especially the fundamental theorem of arithmetic, it is more convenient to not think of one as a prime factor, or to think of it as an implicit factor that's always there but need not be written down. To exclude the number one from the list of prime numbers, primality is defined as a number having exactly two distinct divisors, one and itself. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899, although Carl Sagan included one in a list of prime numbers in his book Contact in 1985. One is one of three possible return values of the Möbius function. Passed an integer that is square-free with an even number of distinct prime factors, the Möbius function returns one. One is the only odd number that is in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2. One is the only 1-perfect number (see multiply perfect number). One is equal to the sum of its digits in any place-based numbering system, making it an all-Harshad number. One is the number of n × n magic squares for n = 1, 3. One is the number of n-queens problem solutions for n = 1. One is a meandric number, a semi-meandric number, and an open meandric number. By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix. One is the value of the sine and cosine at π/2 and 0 radians, respectively. One is the most common leading digit in many sets of data, a consequence of Benford's law. See also -1.

The Arabic glyph

Image:Evolution1glyph.png The glyph used today in the Western world to represent the number 1, a vertical line, often with a little serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmin Indians, who wrote 1 as a horizontal line (in Chinese today this is the way it is written). The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right, but kept the circle small. This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. In fonts with text figures, 1 is typically the same height as a lowercase X, for example, Image:TextFigs148.png.

In science

One is:

- set equal to celerity (c), the speed of light, in Heaviside notation to simplify calculations.
- the factor in ratios for unit conversions.
- the total density ratio for a flat universe.
- The atomic number of hydrogen In astronomy, : Messier object M1, a magnitude 7.0 supernova remnant in the constellation Taurus, also known as the Crab Nebula. : The New General Catalogue [http://www.ngcic.org/ object] NGC 1, a 13th magnitude 13 spiral galaxy in the constellation Pegasus :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the solar eclipse series which began on -2872 June 4 and ended on -1592 July 11 . The duration of Saros series 1 was 1280.1 years, and it contained 73 solar eclipses. :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -2588 March 2 and ended on -1272 April 30. The duration of Saros series 1 was 1316.2 years, and it contained 74 lunar eclipses.

In human society

Many human cultures have given the concept of one-ness symbolic meanings. Many religions consider God to be a perfect example of one-ness. See monad for a detailed discussion of other types of one-ness. One represents unity, togetherness, and absence of separation or discrimination, e.g. "We are all one" or "everyone". Something is unique if it is the only one of its kind. More loosely and exaggeratingly (especially in advertising) the term is used for something very special. One is also an (archaic) expression of the first person singular ("one is not amused") and of the second person singular ("does one take sugar?"). In Western culture, it is believed by many that the maximum number of girlfriends or boyfriends one may have at one time is 1. Also, it is strongly believed that you can be married to only 1 person at any time - this is called monogamy. Being married to more than one person at any time is called bigamy or polygamy. This is illegal in many Western societies. Among children, or when otherwise calling for subtlety, the phrase "number 1" can refer to the act of urination. This can derive from a traditional U.S. elementary school practice of holding up one or two fingers to indicate the approximate time of a requested absence. "Number 1" can also refer to oneself, or that something is first in its class, the latter being used often as a cheer in sports games. On a clock, 1:00 signals that one full hour has passed since the last change of the "AM" or "PM" meridian.

In music

In harmonic analysis of tonal music, the tonic chord is referred to as I. "One is the loneliest number that you'll ever do", according to the first line of "One" by Three Dog Night. The number appears in the title of songs by Metallica, U2, Creed, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, Harry Nilsson, Three Dog Night, and the Bee Gees. Also the title of a best-selling compilation album of all Beatles songs that reached number 1 in the UK or US charts (see List of Number 1 Hits (USA)). The song Green grow the rushes, O has the line "One and all alone and evermore shall be so."

In religion

There is one god according to monotheism (see also: tawhid). There is one surat al-Fatiha in the Qur'an.

In sports

In some sports, one is the number of a specific position: in rugby union, the number of the loosehead prop; in baseball, the number representing the pitcher's position; in football, the number of the goalkeeper. In 2004, fans of the Philadelphia Eagles NFL team used the phrase "One" to show support for the team as they inched closer to the Super Bowl. The full text of the phrase was "One Team. One City. One Dream."

In technology

One is the DVD region of the United States and Canada. In the DOS Shell and many Windows programs, the function key F1 calls up online help. On most standard phones, the 1 key is not associated with any letters the way other number keys are, but on the BlackBerry, 1 is also the key for the letters E and R. Some cellular phones associate the "1" key with various symbols (i.e. the pound sign, the ampersand, etc.) when users engage in text messaging. In the Rich Text Format specification, 1 is the language code for the Arabic language. All codes for dialects of Arabic are congruent to 1 mod 256. 1 is a punctuation mark indicating exclamation, or the letter "l" in "leetspeak". 1, in binary, stands for 'yes'.

In other fields

One is:
- the denomination of U.S. dollar bill with George Washington's portrait, and the denomination of coin with Sacagawea's portrait. It is also the denomination of the older Eisenhower and Susan B. Anthony tender coins and the American Silver Eagle bullion coin.
- the denomination of Canadian dollar coin with a swimming loon on the reverse, hence its universally employed nickname, the "loonie"
- in cents of a U.S. dollar, the denomination of coin with Abraham Lincoln's portrait, commonly known as a penny.
- in cents of a Canadian dollar, the denomination of coin with two maple leaves on the reverse, also known as a penny.
- the code for international direct dial phone calls to countries participating in the North American Numbering Plan, such as the United States and Canada.
- the designation of many roads, listed at Route 1.
- as Air Force One is the callsign of any United States Air Force aircraft carrying the President of the United States.
- the house number of Number One Observatory Circle, the US Vice-President's residence.
- the address of Apsley House, known simply as Number 1, London.
- a subway service in New York City. See 1 (New York City Subway service).
- the name of a train operating company in East Anglia, England. See: 'one'
- an enneagram personality type.
- part of a nickname for the U. S. Army's First Infantry Division, "The Big Red One".
- as a word in all capitals, ONE, a trademark of Nestlé Purina PetCare for a line of pet food products (acronym for the "optimum nutritional effectiveness" they are claimed to possess).
- as ONE, the self-referential acronym for ONE North East, the regional development agency in North East England.
- used as a pronoun in English grammar to refer to an unspecified person: see generic you.
- in Astrology, Aries is the 1st astrological sign of the Zodiac.
- the number of syllables in a monosyllabic word.
- the number of humps on a dromedary.
- the number of wheels on a unicycle.
- the number of players in solitaire, Tetris, and many video games.
- the year A.D. 1 or the immediately previous year 1 B.C., or the year 1 in a century.
- January 1, the first day of the year in the Gregorian calendar, is New Year's Day.
- April 1 is commonly known as April Fool's Day.
- To Roman Catholics, November 1 is All Saint's Day. Category:Famous numbers 01 ko:1 ja:1 simple:One th:1 (จำนวน)

0 (number)

:This page is about the number and numeral 0. For other uses of 0 or "zero", see 0 (disambiguation) 0 (zero), alternatively called naught or nought, is both a number and a numeral. It was the last numeral to be created in most numerical systems, as it is not a counting number (which is to say, one begins counting at the number 1) and was in many eras and places represented only by a gap or mark very different from the other numerals.

0 as a number

0 is the integer that precedes the positive 1, and all positive integers, and follows -1, and all negative integers. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted. Zero is a number which means nothing, null, void or an absence of value. For example, if the number of one's brothers is zero, then that person has no brothers. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero may be used to describe any event considered so significant that it virtually starts a new time reckoning.

0 as a numeral

Year Zero The modern numeral 0 is normally written as a circle or (rounded) rectangle. On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments (at right), though on some historical calculator models it was written with four line segments. This variant glyph has not caught on. It is important to distinguish the number zero (as in the "zero brothers" example above) from the numeral or digit zero, used in numeral systems where the position of a digit signifies its value. Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. The Babylonian numeral system used two narrow slanting wedges, similar to \\, for the equivalent of a positional zero numeral starting in about 400BC. A zero digit is not always necessary in a positional number system: decimal without a zero provides a possible counterexample. In fonts with text figures, 0 is usually the same height as a lowercase X, for example, Image:TextFigs036.png.

History

Etymology

The word zero comes ultimately from the Arabic sifr (صفر) meaning empty or vacant, a literal translation of the Sanskrit meaning void or empty. Through transliteration this became zephyr or zephyrus in Latin. The word zephyrus already meant "west wind" in Latin; the proper noun

Coprime

Properties

Cross notation, group

Generalizations

See also

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

Common misconceptions

See also

Bibliography

External links

Integer

Algebraic properties

Order-theoretic properties

Integers in computing

Quotations

External links

Divisor

Explanation

Further notions and facts

Divisibility of numbers

Generalization

See also

External links

6 (number)

In mathematics

The Arabic glyph

In science

In music

In sports

In technology

In television

In other fields

References

27 (number)

In mathematics

In astronomy

In other fields

External link

1 (number)

History

In mathematics

The Arabic glyph

In science

In human society

In music

In religion

In sports

In technology

In other fields

0 (number)

0 as a number

0 as a numeral

History

Etymology