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Commensurability (mathematics)

Commensurability (mathematics)

:This article is about the meaning of 'commensurable' and derived words in mathematics. For other senses, see commensurability.

Commensurability in general

Generally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable. A time measured in weeks and a time measured in minutes are commensurable because a week is a constant number of minutes (10080), so that one can convert between the two units by multiplying or dividing by 10080.

Commensurability in mathematics

In mathematics, two nonzero real numbers a and b are said to be commensurable if a/b is a rational number. The usage comes to us from translations of Euclid's Elements, in which two line segments a and b are called commensurable precisely if there is some third segment c that can be laid end-to-end a whole number of times to produce a segment congruent to a, and also, with a different whole number, a segment congruent to b. Euclid did not use any concept of real number, but he used a notion of congruence of line segments, and of one such segment being longer or shorter than another. That a/b is rational is a necessary and sufficient condition for the existence of some real number c, and integers m and n, such that :a = mc and b = nc. Assuming for simplicity that a and b are positive, one can say that a ruler, marked off in units of length c, could be used to measure out both a line segment of length a, and one of length b. That is, there is a common unit of length in terms of which a and b can both be measured; this is the origin of the term. Otherwise the pair a and b are incommensurable. In group theory, a generalisation to pairs of subgroups is obtained, by noticing that in the case given, the subgroups of the real line as additive group, generated respectively by a and by b, intersect in the subgroup generated by dc, where d is the LCM of m and n. This is of finite index, therefore in each of them. This gives rise to a general notion of commensurable subgroups: two subgroups A and B of a group are commensurable when their intersection has finite index in each of them. Sometimes in fact this relation is called commensurate, and to be commensurable requires only to be conjugate to a commensurate subgroup. A relationship can similarly be defined on subspaces of a vector space, in terms of projections that have finite-dimensional kernel and cokernel. In contrast, two subspaces \mathrm and \mathrm that are given by some moduli space stacks over a Lie algebra \mathcal, are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completions of \mathcal-type modules corresponding to \mathfrak and \mathfrak are not well-defined, then \mathfrak and \mathfrak are also not commensurable. Category:Group theory

Commensurability

Generally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable. A time measured in weeks and a time measured in minutes are commensurable because a week is a constant number of minutes (10080), so that one can convert between the two units by multiplying or dividing by 10080.
- For the commensurability of scientific theories, see commensurability (philosophy of science).
- For the concept of commensurability in mathematics, see commensurability (mathematics). simple:Incommensurability

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.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

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

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

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.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

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

Real number

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247… The three dots indicate that there would still be more digits to come, no matter how many more might be added at the end. Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. Mathematicians use the symbol R (or alternatively, \Bbb , the letter "R" in blackboard bold) to represent the set of all real numbers. The notation Rn refers to an n-dimensional space of real numbers; for example, a value from R3 consists of three real numbers and specifies a location in 3-dimensional space. In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.

History

Vulgar fractions had been used by the Egyptians around 1000 BC; around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers were invented by Indian mathematicians around 600 AD, and then possibly reinvented in China shortly after. They were not used in Europe until the 1600s, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic. The development of calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.

Definition

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.

Axiomatic approach

Let R denote the set of all real numbers. Then:
- The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
- The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
  - if x ≥ y then x + z ≥ y + z;
  - if x ≥ 0 and y ≥ 0 then xy ≥ 0.
- The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object.

Properties

Completeness

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − xm| is less than ε provided that n and m are both greater than N. In other words, a sequence is a Cauchy sequence if its elements xn eventually come and remain arbitrarily close to each other. A sequence (xn) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x. It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true: :Every Cauchy sequence of real numbers is convergent. That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example, the standard series of the exponential function : \mathrm^x = \sum_^ \frac converges to a real number because for every x the sums : \sum_^ \frac can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we do not know ahead of time what the limit is.

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore the field structure. However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way. But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

Advanced properties

The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2ω, i.e., the cardinality of the set of subsets of the natural numbers. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. The real numbers form a metric space: the distance between x and y is defined to be the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected and simply connected), separable metric space of dimension 1, and are everywhere dense. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, continuous, and separable order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra. The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1. The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim-Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of hyperreal numbers is much bigger than R but also satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.

Generalizations and extensions

The real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean. Occasionally, the two formal elements +∞ and −∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers. Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers. Category:Elementary mathematics Category:Real numbers Category:Set theory ko:실수 ja:実数 th:จำนวนจริง

Rational number

In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. Each rational number can be written in infinitely many forms, for example 3/6 = 2/4 = 1/2. The simplest form is when a and b have no common divisors, and every non-zero rational number has exactly one simplest form of this type with positive denominator. The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above 1. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A real number that is not rational is called an irrational number. In mathematics, the term "rational something" means that the underlying field considered is the field \mathbb of rational numbers. For example, rational polynomials or rational prime ideals. The set of all rational numbers is denoted by Q, or in blackboard bold \mathbb. Using the set-builder notation \mathbb is defined as such: :\mathbb = \left\

Arithmetic

:\frac + \frac = \frac   :\frac \cdot \frac = \frac   Two rational numbers \frac and \frac are equal if and only if ad = bc Additive and multiplicative inverses exist in the rational numbers. :- \left( \frac \right) = \frac   :\left(\frac\right)^ = \frac \mbox a \neq 0

History

Egyptian fractions

Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers. For instance, \frac = \frac + \frac + \frac For any positive rational number, there are infinitely many different such representations. These representations are called Egyptian fractions, because the ancient Egyptians used them. The Egyptians also had a different notation for dyadic fractions. See also Egyptian numerals.

Formal construction

Mathematically we may define them as an ordered pair of integers \left(a, b\right), with b not equal to zero. We can define addition and multiplication of these pairs with the following rules: : \left(a, b\right) + \left(c, d\right) = \left(ad + bc, bd\right) : \left(a, b\right) \times \left(c, d\right) = \left(ac, bd\right) To conform to our expectation that 2/4 = 1/2, we define an equivalence relation \sim upon these pairs with the following rule: : \left(a, b\right) \sim \left(c, d\right) \mbox ad = bc This equivalence relation is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. (This construction can be carried out in any integral domain, see quotient field.) We can also define a total order on Q by writing : \left(a, b\right) \le \left(c, d\right) \mbox (bd>0\mbox ad \le bc)\mbox(bd<0\mbox ad \ge bc)

Properties

The set \mathbb, together with the addition and multiplication operations shown above, forms a field, the quotient field of the integers \mathbb. The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of \mathbb. The algebraic closure of \mathbb, i.e. the field of roots of rational polynomials, is the algebraic numbers. The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set. The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. As a totally ordered set, the rationals are uniquely characterized by being countable, dense (in the above sense), and having no least or greatest element.

Real numbers

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expressions of continued fraction. By virtue of their order, the rationals carry an order topology. The rational numbers also carry a subspace topology. The rational numbers form a metric space by using the metric d\left(x, y\right) = |x - y|, and this yields a third topology on \mathbb. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metric space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of \mathbb.

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn \mathbb into a topological field: let p be a prime number and for any non-zero integer a let |a|_p = p^, where p^n is the highest power of p dividing a; in addition write |0|_p = 0. For any rational number \frac, we set \left|\frac\right|_p = \frac. Then d_p\left(x, y\right) = |x - y|_p defines a metric on \mathbb. The metric space \left(\mathbb, d_p\right) is not complete, and its completion is the p-adic number field \mathbb_p. Category:Elementary mathematics Category:Field theory Category:Fractions Category:Real numbers Category:Set theory ko:유리수 ja:有理数 simple:Rational number th:จำนวนตรรกยะ



Euclid's Elements

Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory. The Elements is one of the oldest extant axiomatic deductive treatments of geometry, and has proved instrumental in the development of logic and modern science. It is considered the most successful textbook ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (well over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today. quadrivium

First principles

Euclid based his work in Book I on 23 definitions, such as point, line and surface, five postulates and five "common notions" (both of which are today called axioms). Postulates in Book I: # A straight line segment can be drawn by joining any two points. # A straight line segment can be extended indefinitely in a straight line. # Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center. # All right angles are congruent. # If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. Common notions in Book I: # Things which equal the same thing are equal to one another. # If equals are added to equals, then the sums are equal. # If equals are subtracted from equals, then the remainders are equal. # Things which coincide with one another are equal to one another. # The whole is greater than the part. These basic principles reflect the constructive geometry Euclid, along with his contemporary Greeks, was interested in. The first three postulates basically describe the constructions one can carry out with a compass and an unmarked straightedge or ruler.

Success

The success of Elements is due primarily to its logical presentation of much of the mathematical knowledge available to Euclid. Most of the material is not original to him, although a few of the proofs are his. Its systematic development from a small set of axioms to deep results encouraged its use as a textbook for hundreds of years, and still influences modern geometry books. Throughout history there have been controversies surrounding many of Euclid's axioms and proofs. Nevertheless, the Elements has withstood the test of time and is still considered a masterpiece in the application of logic to mathematics, and, historically, it has been enormously influential in many areas of science. European scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei and especially Sir Isaac Newton were all influenced by the Elements, and applied their knowledge of it to their work. Mathematicians (Bertrand Russell, Alfred North Whitehead) and philosophers (Baruch Spinoza) have also attempted to provide their own Elements; that is, axiomatized deductive structures of their own respective disciplines. Of the five postulates Euclid used, the last, so-called "parallel postulate" seemed less obvious than the others. Many geometers suspected that it may be provable from the other postulates but all attempts to do this failed. By the mid-19th century, it was shown that no such proof exists, because one can construct non-Euclidean geometries where the parallel postulate is false, while the other postulates remain true. Mathematicians say that the parallel postulate is independent of the other postulates. Two alternatives are possible: either an infinite number of parallel lines can be drawn through a point not on a straight line (hyperbolic geometry, also called Lobachevskian geometry), or none can (elliptic geometry, also called Riemannian geometry). That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy. Indeed, Albert Einstein's theory of general relativity shows that the "real" space in which we live can be non-Euclidean (for example, around black holes and neutron stars). That Euclid recognized the independence of the parallel postulate long before other mathematicians accepted it is a testament to Euclid's dedication to a logical development from as few assumptions as possible.

History

Elements was written in approximately 300 BC by Euclid, an ancient Greek mathematician who probably studied under the pupils of Plato. Although most of the theorems had been developed earlier, Elements was so impressive and comprehensive that the Greeks had no use for the older books, and little is known about earlier geometers today. It was translated later into Arabic after being gifted to the Arabs by Byzantium and from those secondary translations into Latin. The first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Copies of the Greek text also exist, e.g. in the Vatican Library and the Bodlean library in Oxford. However, the manuscripts available are of very variable quality and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been drawn about the contents of the original text (copies of which are no longer available). Texts which refer to the Elements itself and mathematical theories which were current at the time it was written are also important in this process. Such analyses are conducted by J.L. Heiberg and Sir Thomas L. Heath in their editions of the text. Also of importance are the scholia, or footnotes to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or elucidation. Some of these are useful and add to the text, but many are not.

Later axiomizations

Surprisingly, mathematicians in the nineteenth century discovered that Euclid's proofs require additional assumptions, ones not stated among either his postulates or his common notions. For example:
- One of his theorems is to the effect that any line segment is part of a triangle, which he constructs in the usual way, by drawing circles around both endpoints and taking their intersection and them as three corners. However, his axioms do not guarantee that the circles actually do intersect.
- It is proven by superposition that two triangles are congruent if two of their sides and the angle between them are respectively equal. The method relies on three assumptions: that the straight line making a given angle with another on one side is unique, that the straight line connecting two points is unique, and that the properties of a figure are independent of where it is situated. David Hilbert gave a revised list containing no fewer than 23 separate axioms

Contents

Although Elements is a geometric work, it also includes results that today would be classified as number theory. The contents of the work are as follows: Books 1 through 4 deal with plane geometry:
- Book 1 contains the basic properties of geometry: the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).
- Book 2 is commonly called the "book of geometrical algebra," because the material it contains may easily be interpreted as algebra.
- Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point.
- Book 4 is concerned with inscribing and circumscribing triangles and regular polygons. Books 5 through 10 introduce ratios and proportions:
- Book 5 is a treatise on proportions of magnitudes.
- Book 6 applies proportions to geometry: Thales' theorem, similar figures.
- Book 7 deals strictly with number theory: divisibility, prime numbers, greatest common divisor, least common multiple.
- Book 8 deals with proportions in number theory and geometric sequences.
- Book 9 applies the results of the preceding two books: the infinitude of prime numbers, the sum of a geometric series, perfect numbers.
- Book 10 attempts to classify incommensurable (in modern language, irrational) magnitudes by using the method of exhaustion, a precursor to integration. Books 11 through 13 deal with spatial geometry:
- Book 11 generalizes the results of Books 1–6 to space: perpendicularity, parallelism, volumes of parallelepipeds.
- Book 12 calculates areas and volumes by using the method of exhaustion: cones, pyramids, cylinders, and the sphere.
- Book 13 generalizes Book 4 to space: golden section, the five regular (or Platonic) solids inscribed in a sphere.

External links


- [http://aleph0.clarku.edu/~djoyce/java/elements/elements.html Euclid's Elements] adapted to the web by David E. Joyce. Includes java applets.
- [http://www.headmap.org/unlearn/euclid/before/princ-trans.htm On the principal editions and translations of the text]
- [http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/byrne.html Oliver Byrne's edition of the Elements of Euclid], published in 1847. These scanned images are presumably now available in the public domain, although it would be polite to contact the site operator first.
- [http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.01.0086 Euclid's Elements] translated with introduction and commentary by sir Thomas L. Heath from the Perseus Digital Library. Complete and fragmentary manuscripts of versions of Euclid's elements:
- Sir Thomas More's [http://www.columbia.edu/acis/textarchive/rare/24.html manuscript]
- [http://www.columbia.edu/acis/textarchive/rare/6.html Latin translation] by Aethelhard of Bath
- Category:Mathematics books Category:Roman era books ja:ユークリッド原論

Necessary and sufficient condition

:This article discusses only the logical relationships implied by necessary and sufficient. For the causal meanings see causation. In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. A necessary and sufficient condition, then, is one which by itself, must happen, and is all that needs to happen, for something else to be the case.
- A necessary condition is one that must be satisfied, for the result to happen. Drinking is necessary to stay alive, but not sufficient; if you drank but did nothing else you would die.
- A sufficient condition is one that, if it is satisfied, the result is certain to happen. Jumping is sufficient to leave the ground, but not necessary; there are many other ways one can leave the ground instead without doing that.
- By contrast, having an ID card is a necessary and sufficient condition for being allowed into some places (you must have an ID card, and also you don't need anything more than an ID card; having an ID card is enough by itself) "Necessary and sufficient" is another way of saying the logical statement "if and only if" (sometimes abbreviated to "iff").

Necessary conditions

To say that A is necessary for B is to say that B cannot be true unless A is true, or that whenever (wherever, etc.) B is true, so is A. We might say that being at least sixteen years old is necessary for having a driver's license. In the sense in which we are using the word "necessary" here, we might also say "smoke is necessary for fire." This is confusing, since smoke comes after fire; but all we are saying is that wherever B is, there is A - ie, fire (B) cannot occur without there being smoke (A). We are trying not to say anything about the direction of time at all. Ordinary language would say "smoke is a necessary outcome of fire." [1] In either case, the important thing is to note that one thing is assumed (a license, fire), and a second thing is derived as "necessarily following." Being at least sixteen is the necessary condition in the first case; smoke is the necessary condition in the second (though, again, we ordinarily would not call it a "condition"). Importantly, it is quite possible for a necessary condition to occur on its own. So, you can be sixteen without having a driver's license, and there are ways to generate smoke without fire. If A is a necessary condition for B, then the logical relation between them is expressed as "If B then A" or "B only if A" or "B → A" (B implies A).

Sufficient conditions

To say that A is sufficient for B is to say precisely the converse: that A cannot occur without B, or whenever A occurs, B occurs. That there is a fire is sufficient for there being smoke. Necessary and sufficient conditions are therefore related. A is a necessary condition for B just in case B is a sufficient condition for A. In the sense in which we are using the word "sufficient", we might also say "Having a license is sufficient for being at least sixteen." This is confusing, since having a license doesn't cause you to be at least sixteen; still, the ordinary sense of it is that if you have a license, you must be at least sixteen (we consider licenses proof of age because we consider them sufficient for age in something like this sense). Try to ignore the causal relationship and the direction of time: we are looking at it just as a logical relationship. In either case, note that one thing is assumed (fire, a license), and this same thing we are identifying as the sufficient condition for another thing (smoke, age) - sufficient in the sense of "enough for the other to be the case". Importantly, a sufficient condition, by definition, is what cannot occur without the thing it is a condition for. So, you cannot have a license without being at least sixteen. If A is a sufficient condition for B, then the logical relation between them is expressed as "If A then B" or "A only if B" or "A → B".

Necessary and sufficient conditions

To say that A is necessary and sufficient for B is to say two things: # A is necessary for B # A is sufficient for B. For example, if Alice always eats steak on Monday, but never on any other day, we might say "Being Monday is a necessary and sufficient condition for Alice eating steak." The converse is also true: "Alice eating steak is a necessary and sufficient condition for it being Monday". Thus, whenever A is necessary and sufficient for B, B is necessary and sufficient for A. Once again, this is confusing, since Alice's act of eating steak doesn't cause it to be Monday. Since the phrase "necessary and sufficient" can express a relation between sentences or between states of affairs, objects, or events, it should not be too quickly conflated with logical equivalence. Alice eating steak is not logically equivalent to it being Monday. However, "A is necessary and sufficient for B" does express the same thing as "A if and only if B".

Footnotes

# For the purposes of this example, we're ignoring the possibility of fire that doesn't create smoke.

See also


- If and only if
- Causation

External links


- Stanford Encyclopedia of Philosophy: [http://plato.stanford.edu/entries/necessary-sufficient/ Necessary and Sufficient Conditions] Category:Logic Category:Technical terminology Category:Mathematical terminology

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

Ruler

:This article is about the drawing instrument. Ruler can also refer to a statesman in charge or ceremonial head of state of a country or minor politically significant principality; for this meaning see Monarch or Lists of incumbents. Lists of incumbents A ruler is an instrument used in geometry and technical drawing to measure short distances and/or to rule straight lines. Strictly speaking, the ruler is the instrument used to rule and calibrated stick for measurement is called a measure. However, common usage is that a ruler is calibrated so that it can measure, creating ambiguity in what a ruler is allowed to do in ruler-and-compass constructions. For instance, a ruler with measurement capability (e.g. its own length) can be used for angle trisection. This is resolved by referring to an instrument that can only rule as a straightedge. Practical rulers have distance markings along their edges. How these distance markings are applied and calibrated should be described here, including a history of old methods.
- A ruler makes a popular tool for use as a pervertible, i.e. for corporal punishment, usually on the hands (either outstretched palms or knuckles) or thighs. Generally, solid, heavy wooden or metal rulers are used for this purpose, but lighter, often flexible plastic ones can also be used punitively, especially in spanking.
- The ruler (calibrated, though numbers are not shown) appears as a charge in heraldry in the arms of Odouze.

See also


- Slide rule
- Architect's scale
- Engineer's scale

External links


- [http://www.markus-bader.de/MB-Ruler/ Virtual Screen Ruler] Category:Dimensional instruments Category:Metalworking measuring instruments Category:Spanking implements ja:定規 th:ไม้บรรทัด

Length

:This article is about the concept and measurement of distance. For usage in cricket, see line and length. In general English usage, length (symbols: l, L) is but one particular instance of distance – an object's length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with "distance". Height is vertical distance; width (or breadth) is a lateral distance; an object's width is less than its length. No one speaks of "the length from here to Alpha Centauri", but rather of "the distance from here to Alpha Centauri," but when one speaks of distance more abstractly, one says "A kilometre or a mile, is a unit of length" or "...of distance", and the two statements are synonymous. Likewise, a mountain might be a mile in height. Length is the metric of one dimension of space. The metric of space itself is volume, or (length)3. Length is commonly considered to be one of the fundamental units, meaning that it cannot be defined in terms of other dimensions. However, a set of units can be constructed where units of length can be derived from fundamental physical constants - see Planck units and Planck length. Colloquially length sometimes refers to duration, especially when used in context of music.

Units of length(SI)

The SI unit of Length is the metre (U.S. spelling: meter), from which can be derived:from the regular basis of the foundation of the whole world
- centimetre
- kilometre

Other units of length

The Imperial and US customary units of length


- inch
- foot
- yard
- mile

Units are used in astronomy


- Astronomical unit
- Light year
- Parsec

See also


- Curve
- Metric space
- Orders of magnitude
- Distance
- Planck length
- International standard ISO 31-1: Quantities and units – Space and time

External links


- [http://www.unitconversion.org/unit_converter/length.html Length Converter: convert between units of length, such as meter, yard, mile, and so on]
- [http://www.unitconversion.org/unit_converter/length-v.html Length Conversion table: convert selected unit to all other units of length]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- Category:Norm ko:길이 ja:長さ





Real line

In mathematics, the real line is simply the set R of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space. The real line has been studied at least since the days of the ancient Greeks, but it was not rigorously defined until 1872. Before and since that date, it has been a prolific example that has played a significant role in many branches of mathematics. The real line carries a standard topology which can be introduced in two different, equivalent ways. First, since the real numbers are totally ordered, they carry an order topology. Second, the real numbers can be turned into a metric space by using the metric given by the absolute value: d(x,y) := |y - x|. This metric induces a topology on R equal to the order topology. As a topological space, the real line is a topological manifold of dimension . It is paracompact and second-countable as well as contractible and locally compact. It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.) Indeed, R was historically the first example to be studied of each of these mathematical structures, so that it serves as the inspiration for these branches of modern mathematics. (Indeed, many of the terms above can't even be defined until R is already in place.) As a vector space, the real line is a vector space over the field R of real numbers (that is, over itself) of dimension . It has a standard inner product, making it an Euclidean space. (The inner product is simply ordinary multiplication of real numbers.) As a vector space, it is not very interesting, and thus it was in fact 2-dimensional Euclidean space that was first studied as a vector space. However, we can still say that R inspired the field of linear algebra, since vector spaces were first studied over R. R is also a premier example of a ring, even a field. It is in fact a real complete field, and was the first such field to be studied, so that it inspired that branch of abstract algebra as well. However, in such purely algebraic contexts, R is rarely called a "line". For more information on R in all of its guises, see real number. Category:Real numbers Category:topological spaces

Lcm

In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. If there is no such positive integer, e.g., if a = 0 or