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Distributivity

Distributivity

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: : 4 · (2 + 3) = (4 · 2) + (4 · 3) In the left-hand side of the above equation, the 4 multiplies the sum of 2 and 3; on the right-hand side, it multiplies the 2 and the 3 individually, with the results added afterwards. Because these give the same final answer (20), we say that multiplication by 4 distributes over addition of 2 and 3. Since we could have put any real numbers in place of 4, 2, and 3 above, and still gotten a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Definition

Given a set S and two binary operations
- and + on S, we say that
-
- is left-distributive over + if, given any elements x, y, and z of S, ::x
- (y + z) = (x
- y) + (x
- z);
-
- is right-distributive over + if, given any elements x, y, and z of S: ::(y + z)
- x = (y
- x) + (z
- x);
-
- is distributive over + if it is both left- and right-distributive. Notice that when
- is commutative, then the three above conditions are logically equivalent.

Examples

# Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers. # Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive. # Matrix multiplication is distributive over matrix addition, even though it's not commutative. # The union of sets is distributive over intersection, and intersection is distributive over union. Also, intersection is distributive over the symmetric difference. # Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction. Also, conjunction is distributive over exclusive disjunction ("xor"). # For real numbers (or for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: max(a,min(b,c)) = min(max(a,b),max(a,c)) and min(a,max(b,c)) = max(min(a,b),min(a,c)). # For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd(a,lcm(b,c)) = lcm(gcd(a,b),gcd(a,c)) and lcm(a,gcd(b,c)) = gcd(lcm(a,b),lcm(a,c)). # For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) = min(a+b,a+c). Distributivity is most commonly found in rings and distributive lattices. A ring has two binary operations (commonly called "+" and "
- "), and one of the requirements of a ring is that
- must distribute over +. Most kinds of numbers (example 1) and matrices (example 3) form rings. A lattice is another kind of algebraic structure with two binary operations, ^ and v. If either of these operations (say ^) distributes over the other (v), then v must also distribute over ^, and the lattice is called distributive. See also the article on distributivity (order theory). Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras. Rings and distributive lattices are both special kinds of rigs, certain generalisations of rings. Those numbers in example 1 that don't form rings at least form rigs. Near-rigs are a further generalisation of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.

Generalizations of distributivity

In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, others being defined in the presence of only one binary operation. Details of the according definitions and their relations are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice. In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either ≤ or ≥. Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on intervals.

External links

- [http://www.algebra.com/algebra/homework/Distributive-property/proof-of-distributive-property.lesson Proof for children of distributive property with integer as multiplier, with an animation]
- [http://www.algebra.com/algebra/homework/Distributive-property/example-distributive-property-addition.solver Solver showing an animation for user supplied examples of distributive property] Category:Abstract algebra Category:Elementary algebra ja:分配法則

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

Binary operation

In mathematics, a binary operation is a calculation involving two input quantities. Binary operations can be accomplished using either a binary function or binary operator. Binary operations are sometimes called dyadic operations in order to avoid confusion with the binary numeral system. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division. More precisely, a binary operation on a set S is a binary function from S and S to S, in other words a function f from the Cartesian product S × S to S. Sometimes, especially in computer science, the term is used for any binary function. That f takes values in the same set S that provides its arguments is the property of closure. Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more. Most generally, a magma is a set together with any binary operation defined on it. Many binary operations of interest in both algebra and formal logic are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (
- ) of numbers and matrices as well as composition of functions on a single set. Examples of operations that are not commutative are subtraction (-), division (/), exponentiation(^), and super-exponentiation(@). Binary operations are often written using infix notation such as a
- b, a + b, or a · b rather than by functional notation of the form f(a,b). Sometimes they are even written just by juxtaposition: ab. They can also be expressed using prefix or postfix notations. A prefix notation, Polish notation, dispenses with parentheses; it is probably more often encountered now in its postfix form, reverse Polish notation.

External binary operations

An external binary operation is a binary function from K and S to S. This differs from a binary operation in the strict sense in that K need not be S; its elements come from outside. An example of an external binary operation is scalar multiplication in linear algebra. Here K is a field and S is a vector space over that field. An external binary operation may alternatively be viewed as an action; K is acting on S. Category:Algebra Category:Abstract algebra ja:二項演算

Elementary algebra

:This article is about basic algebra in mathematics. For other uses of the term "algebra" see algebra (disambiguation). Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as a, x, y) to denote numbers. This is useful because:
- It allows the general formulation of arithmetical laws (such as

a + b = b + a

for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
- It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance "find a number x such that

3x + 1 = 10

).
- It allows the formulation of functional relationships (such as "if you sell x tickets, then your profit will be

3x - 10

dollars"). These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a much more advanced topic generally taught to college seniors. In algebra, an "expression" may contain numbers, variables and arithmetical operations; a few examples are: :

x + 3\,

y^ - 3\,

z^ + a(b + x^) + 42/y - \pi.\,

An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as

a + (b + c) = (a + b) + c

); these are also known as "identities". Other equations contain symbols for unknown values and we are then interested in finding those values for which the equation becomes true:

x^ - 1 = 4.

These are the "solutions" of the equation.

Laws of elementary algebra

- The order of operations in a mathematical expression are as follows:
- groupings -> exponents -> multiplication -> addition
- Addition is a commutative operation.
- Subtraction is the reverse of addition.
- To subtract is the same as to add a negative number: :::

a - b = a + (-b). \

:: Example: if

5 + x = 3

then

x = -2.

- Multiplication is a commutative operation.
- Division is the reverse of multiplication.
- To divide is the same as to multiply by a reciprocal: :::

= a \left(  \right).

- Exponentiation is not a commutative operation.
- Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with fractional exponents (e.g. square roots).
- Examples: if

3^x = 10

then

x = \log_3 10 .

x^ = 10

then

x = 10^.

- The square roots of negative numbers do not exist in the real number system. (See: complex number system)
- Associative property of addition:

(a + b) + c = a + (b + c).

- Associative property of multiplication:

(ab)c = a(bc).

- Distributive property of multiplication with respect to addition:

c(a + b) = ca + cb.

- Distributive property of exponentiation with respect to multiplication:

(a b)^c = a^c b^c .

- How to combine exponents:

a^b a^c = a^ .

- Power to a power property of exponents:

(a^b)^c = a^ .

- If

a = b

and

b = c

, then

a = c

(transitivity of equality).
-

a = a

(reflexivity of equality).
- If

a = b

then

b = a

(symmetry of equality).
- If

a = b

and

c = d

then

a + c = b + d.

- If

a = b

then

a + c = b + c

for any c (addition property of equality).
- If

a = b

and

c = d

then

ac

bd.

- If

a = b

then

ac = bc

for any c (multiplication property of equality).
- If two symbols are equal, then one can be substituted for the other at will (substitution principle).
- If

a > b

and

b > c

then

a > c

(transitivity of inequality).
- If

a > b

then

a + c > b + c

for any c.
- If

a > b

and

c > 0

then

ac > bc.

- If

a > b

and

c < 0

then

ac < bc.

Examples

Linear equations

The simplest equations to solve are linear equations. They contain only constant numbers and a single variable without an exponent. For example: :

2x + 4 = 12. \,

The central technique is add, subtract, multiply, or divide both sides of the equation by the same thing in such a way to eventually arrive at the value of the unknown variable. If we subtract 4 from both sides in the equation above we get: :

2x = 8 \,

and if we then divide both sides by 2, we get our solution :

x = \frac = 4.

Quadratic equations

Quadratic equations contain variables raised to the first and second (square) power, and can be solved using factorization or the quadratic formula. As an example of factoring: :

x^ + 3x = 0. \,

This is the same thing as :

x(x + 3) = 0. \,

Setting x to 0 or -3 will make this true. All quadratic equations will either have one or two solutions.

System of linear equations

If we have a system of linear equations, for example, two equations in two variables, it is often possible to find two answers that satisfy both. :

4x + 2y = 14 \,

2x - y = 1. \,

Now, multiply the second equation by 2 on both sides, and you have the following equations: :

4x + 2y = 14 \,

4x - 2y = 2. \,

Now we add the two equations together to get: :

8x = 16 \,

x = 2. \,

You can see that since we multiplied the second equation by 2, we can combine the equations and cancel out y, and then we can solve for x. Note that you can multiply by any numbers (positive or negative, but not zero) to both sides of any to get to a point where a variable cancels out when you combine them. To find y, choose either one of the equations from the beginning. :

4x + 2y = 14. \,

Substitute in 2 for x. :

4(2) + 2y = 14. \,

Simplify using the rules of algebra. :

8 + 2y = 14 \,

2y = 6 \,

y = 3. \,

The full solution to this problem is then :

\begin x = 2 \\ y = 3. \end\,

External links

Charles Smith, [http://mathbooks.library.cornell.edu:8085/Dienst/UIMATH/1.0/Display/cul.math/Smit025 A Treatise on Algebra], in [http://historical.library.cornell.edu/math Cornell University Library Historical Math Monographs]. Other example problems can be found at [http://www.exampleproblems.com www.exampleproblems.com].
- Category:School subjects

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 (x_n) 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 |x_n − x_m| 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 x_n eventually come and remain arbitrarily close to each other. A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − 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 2^ω, 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

Addition

Addition is the most basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends, into a single number, the sum. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series. Repeated addition of the number one is the most basic form of counting. Addition can also be defined for mathematical objects other than numbers — for example, matrices or polynomials. Regardless of the nature and number of objects being added, the individual constituents of a sum typically are called summands or terms. (This is to be distinguished from factors, which are multiplied.)

Notation

multiplied Addition is written using the plus sign "+" between the terms. For example, :1 + 1 = 2 :2 + 2 = 4 :5 + 4 + 2 = 11 (see "associativity" below) :3 + 3 + 3 + 3 = 12 (see "multiplication" below) There are also situations where addition is "understood" even though no symbol appears:
- A column of numbers, with the last number in the column underlined, usually (but not always) indicates that the numbers in the column are to be added, with the sum written below the underlined number.
- A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number. For example, ::3¹⁄₂ = 3 + ¹⁄₂ = 3.5. :This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead.

Interpretations

Addition is used to model countless physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations.

Combining sets

Possibly the most fundamental interpretation of addition lies in combining sets:
- When two or more collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. This interpretation is well-suited to quick proofs of the properties of natural number addition, and it is easy to visualize, with little danger of ambiguity. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. See [http://arxiv.org/abs/math.QA/0004133 this article] for an example of the sophistication involved in adding with sets of "fractional cardinality". One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than just combining collections of segments, rods can be joined end-to-end. :This section is under construction.

Extending a measure

- When an original measure is extended by a given amount, the final measure is the sum of the original measure and the measure of the extension. Under this interpretation, the parts of a sum a + b play asymmetric roles; instead of calling both a and b addends, it is more appropriate to call a the augend, since a plays a passive role. In geometry, a might be a point and b a vector; their sum is then another point, the translation of a by b. In analytic geometry, a and b might both be represented by ordered pairs of numbers, but they remain conceptually different. Here, the addition operation is not so much a binary operation as a family of unary operations; the function (+b) is acting on a. The unary and binary views are formally equivalent, in that for any sets A and B there is a natural identification of sets of functions :

A^\cong \left(A^A\right)^B.

(This law of exponentiation may be more familiar for numbers.) The unary view is useful, for example, when discussing subtraction. Addition and subtraction are not inverses as binary operations, but they are inverses as families of unary operations. :This section is under construction.

Combining translations

- When two motions are performed in succession, the measure of the resulting motion is the sum of the measures of the original motions. :This section is under construction.

Basic properties

Commutivity

subtraction Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same. Symbolically, if a and b are any two numbers, then :a + b = b + a. The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".

Associativity

binary operation A somewhat subtler property of addition is associativity, which comes up when one tries to define repeated addition. Should the expression :"a + b + c" be defined to mean (a + b) + c or a + (b + c)? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers a, b, and c, it is true that :(a + b) + c = a + (b + c). Not all operations are associative, so in expressions with operations other than addition, it is important to specify the order of operations.

Zero and one

order of operations If one adds zero to any number, the quantity won't change; zero is the identity element for addition. In symbols, for any a, :a + 0 = 0 + a = a. The sum of any number and its additive inverse (in contexts where such a thing exists) is zero. In the context of integers, addition of one plays a special role: for any integer a, the integer (a + 1) is the least integer greater than a, also known as the successor of a.

Units

In order to numerically add certain types of numbers, such as vulgar fractions and physical quantities with units, they must first be expressed with a common denominator. For example, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is another name for 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in dimensional analysis.

Generalizations

:There are many things that can be added: numbers, vectors, matrices, spaces, shapes, sets, functions, equations, strings, chains... —[http://www.cut-the-knot.org/do_you_know/addition.shtml Alexander Bogomolny] Addition is first defined on the natural numbers. In set theory, addition is then extended to larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers. (In mathematics education, positive fractions are added before negative numbers are even considered; this is also the historical route.) In turn, real addition extends to addition operations on even larger sets, such as the set of complex numbers or a many-dimensional vector space in linear algebra.

In algebra

There are many more sets that support an operation called addition. There are already infinitely many natural numbers, and the set of real numbers is even larger. It is also useful to study addition on smaller sets, even finite ones. In modular arithmetic, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as "exclusive or". The ideas of extending and compacting sets can be combined. In geometry, the sum of two angles is often taken to be their sum as two real numbers modulo 2π. This amounts to an addition operation on the circle, which in turn generalizes to addition operations on many-dimensional tori. A general form of addition occurs in abstract algebra, where addition may be almost any well-defined binary operation on a set. For an operation to be called "addition" in abstract algebra, it is required to be associative and commutative.

Addition of sets

One extraordinary generalization of the addition of natural numbers is the addition of ordinal numbers. Unlike most addition operations, ordinal addition is not commutative. However, passing to the "smaller" class of cardinal numbers, we recover a commutative operation. Cardinal addition is closely related to the disjoint union of two sets. In category theory, the disjoint union is a kind of coproduct, so coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts are named to evoke their connection with addition; see Direct sum and Wedge sum.

Related operations

- Incrementation, also known as the successor operation, is the addition of 1 to a number. In formal treatments of addition, such as the Peano axioms, the successor is an elementary operation, and addition is defined from successors through recursion.
- Summation describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is 0. An infinite summation is known as a series.
- Counting is an intuitive procedure that can be formalized as the summation of 1 over some finite domain. In everyday counting, the domain is typically a small set of physical objects; in mathematics it may be large and abstract, as it is for the prime counting function.
- Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation.
- Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding x and subtracting x are inverse functions.
- Multiplication can be thought of as repeated addition. If a single term x appears in a sum n times, then the sum is the product of n and x. If n is not a natural number, the product may still make sense; for example, multiplication by −1 yields the additive inverse of a number. In many contexts, multiplication can be transformed into addition, and vice versa, through exponentials and logarithms. In general, multiplication operations always distribute over addition.
- Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, usually a real or complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.
- Convolution is used to add two independent random variables defined by distribution functions. Its usual definition combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.

Notes

# Begle (p.57) and Johnson (p.119) prefer "addends" and "sum". Calling both inputs "addends" emphasizes the symmetry of addition; see the section on #Extending a measure for a context in which "augend" is more appropriate. # Devine et al p.263 # Adding it up (p.73) compares adding measuring rods to adding sets of cats: "For example, inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide them into parts, and it seriously changes their nature." # Stewart makes the distinction by writing angle brackets for vectors and parentheses for points, although this notation is not widely used. See the chapter Vectors. # Weaver (p.62) argues for the importance of contrasting the two views, going so far as to term the version of commutivity satisfied by unary addition "pseudocommutivity". # Enderton (p.142, Theorem 6I) discusses this relationship in the context of cardinal arithmetic identities. # Enderton chapters 4 and 5, for example, follow this development. # California standards; see grades [http://www.cde.ca.gov/be/st/ss/mthgrade2.asp 2], [http://www.cde.ca.gov/be/st/ss/mthgrade3.asp 3], and [http://www.cde.ca.gov/be/st/ss/mthgrade4.asp 4]. # Baez (p.37) explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"

References

- Preprint available [http://arxiv.org/abs/math.QA/0004133 here] on arXiv.
-
- [http://www.cde.ca.gov/be/st/ss/mthmain.asp California State Board of Education mathematics content standards] Adopted December 1997, accessed December 2005.
-
-
-
- Available [http://www.nap.edu/books/0309069955/html/index.html here] from the publisher.
-
-
-

External links

;General
- [http://www.cut-the-knot.org/do_you_know/addition.shtml Addition on cut-the-knot.org] An exploration of various kinds of addition. ;Methods and practice
- [http://www.mathsisfun.com/worksheets/addition.php Addition Worksheets or Online Practice]
- [http://www.apples4theteacher.com/flash-cards.html Addition Flash Cards]
- [http://webhome.idirect.com/~totton/abacus/pages.htm#Addition1 Addition on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead] Category:Arithmetic ja:総和 ko:덧셈 simple:Addition th:การบวก

Given any

Category:Logic In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. The resulting statement is a universally quantified statement, and we have universally quantified over the predicate. In symbolic logic, the universal quantifier (typically "∀") is the symbol used to denote universal quantification. Quantification in general is covered in the article quantification, while this article discusses universal quantification specifically.

Basics

Suppose you wish to say : 2·0 = 0 + 0, and 2·1 = 1 + 1, and 2·2 = 2 + 2, etc. This would seem to be a logical conjunction because of the repeated use of "and". But the "etc" can't be interpreted as a conjunction in formal logic. Instead, rephrase the statement as : For any natural number n, 2·n = n + n. This is a single statement using universal quantification. Notice that this statement is really more precise than the original one. It may seem obvious that the phrase "etc" is meant to include all natural numbers, and nothing more, but this wasn't explicitly stated, which is essentially the reason that the phrase couldn't be interpreted formally. In the universal quantification, on the other hand, the natural numbers are mentioned explicitly. This particular example is true, because you could put any natural number in for n and the statement "2·n = n + n" would be true. In contrast, "For any natural number n, 2·n > 2 + n" is false, because you replace n with, say, 1 and get the false statement "2·1 > 2 + 1". It doesn't matter that "2·n > 2 + n" is true for most natural numbers n; even the existence of a single counterexample is enough to prove the universal quantification false. On the other hand, "For any composite number n, 2·n > 2 + n" is true, because none of the counterexamples are composite numbers. This indicates the importance of the domain of discourse, which specifies which values n is allowed to take. Further information on using domains of discourse with quantified statements can be found in the Quantification article. But in particular, note that if you wish to restrict the domain of discourse to consist only of those objects that satisfy a certain predicate, then for universal quantification, you do this with a logical conditional. For example, "For any composite number n, 2·n > 2 + n" is logically equivalent to "For any natural number n, if n is composite, then 2·n > 2 + n". Here the "if ... then" construction indicates the logical conditional. In symbolic logic, we use the universal quantifier "∀" (an upside-down letter "A" in a sans-serif font) to indicate universal quantification. Thus if P(n) is the predicate "2·n > 2 + n" and N is the set of natural numbers, then :

\forall\mathbf\, P(n)

is the (false) statement : For any natural number n, 2·n > 2 + n. Similarly, if Q(n) is the predicate "n is composite", then :

\forall\mathbf\, Q(n)\;\!\;\! \;\!\;\! P(n)

is the (true) statement : For any composite number n, 2·n > 2 + n. Several variations in the notation for quantification (which apply to all forms) can be found in the quantification article. But there is a special notation used only for universal quantification, which we also give here: :

(n\mathbf)\, P(n)

The parentheses indicate universal quantification by default.

Properties

We need a list of algebraic properties of universal quantification, such as distributivity over conjunction, and so on. Also rules of inference.

Negation

Note that a quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The notation mathematicians and logicians utilize to denote negation is:

\lnot\

. For example, let P(x) be the propositional function "x is married"; then, for a Universe of Discourse X of all living human beings, consider the universal quantification "Given any living person x, that person is married":

\forall\mathbf\, P(x)

. A few second's thought demonstrates this as irrevocably false; then, truthfully, we may say, "It is not the case that, given any living person x, that person is married", or, symbolically:

\lnot\ \forall\mathbf\, P(x)

. Take a moment and consider what, exactly, negating the universal quantifier means: if the statement is not true for every element of the Universe of Discourse, then there must be at least one element for which the statement is false. That is, the negation of P(x) is logically equivalent to "There exists a living person x such that he is not married", or:

\exists\mathbf\, \lnot\ P(x)

Generally, then, the negation of a propositional function's universal quantification is an existential quantification of that propositional function's negation; symbolically,

\lnot\ \forall\mathbf\, P(x) \equiv\ \exists\mathbf\, \lnot\ P(x)

Rules of Inference

A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the universal quantifier. Universal instantiation concludes that, if the propositional function is known to be universally true, then it must be true for any arbitrary element of the Universe of Discourse. Symbolically, this is represented as

\forall\mathbf\, P(x) \to\ P(c)

, where c is a completely arbitrary element of the Universe of Discourse. Universal generalization concludes the propositional function must be universally true if it is true for any arbitrary element of the Universe of Discourse. Symbolically, for an arbitrary c,

P(c) \to\ \forall\mathbf\, P(x)

It is especially important to note c must be completely arbitrary; else, the logic does not follow: if c is not arbitrary, and is instead a specific element of the Universe of Discourse, then P(c) only implies an existential quantification of the propositional function. Discuss universally quantified types in type theory.

Commutative

:For other meanings of commutation, see commutation (disambiguation).

Mathematical meaning

In mathematics, especially abstract algebra, a binary operation

\times

on a set S is commutative if :

x\times y = y\times x

for all x and y in S. Otherwise, the operation is noncommutative. Additionally, if :

x\times y = y\times x

for a particular pair of elements x and y, then x and y are said to commute. Every element commutes with itself and, in a group, every element commutes with the identity, with its own inverse, and with its powers. The most well-known examples of commutative binary operations are addition and multiplication of real numbers; for example:
- 4 + 5 = 5 + 4 (since both expressions evaluate to 9)
- 2 × 3 = 3 × 2 (since both expressions evaluate to 6) Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. Among the noncommutative binary operations are subtraction (a − b), division (a/b), exponentiation (a^b), function composition (f o g), tetration (a↑↑b), matrix multiplication, and quaternion multiplication. An abelian group is a group whose group operation is commutative. A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) In a field both addition and multiplication are commutative.

Neurophysiological meaning

In neurophysiology, commutative has much the same meaning as in algebra. Physiologist Douglas A. Tweed and coworkers consider whether certain neural circuits in the brain exhibit noncommutativity and state: :In non-commutative algebra, order makes a difference to multiplication, so that

a\times b\neq b\times a

. This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuits that steer the eyes, head and limbs, and sensory circuits that handle spatial information. This idea is controversial: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain, but none that clearly rules out all commutative models. (Douglas A. Tweed and others, Nature 399, 261 - 263; 20 May 1999). Tweed goes on to demonstrate non-commutative computation in the vestibulo-ocular reflex by showing that subjects rotated in darkness can hold their gaze points stable in space---correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.

Number

: This article is about numbers such as counting numbers and measurements. For other uses of the term, see Number (disambiguation). A nu

Distributivity

Definition

Examples

Generalizations of distributivity

External links

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

Binary operation

External binary operations

Elementary algebra

Laws of elementary algebra

Examples

Linear equations

Quadratic equations

System of linear equations

See also

External links

Real number

History

Definition

Construction from the rational numbers

Axiomatic approach

Properties

Completeness

"The complete ordered field"

Advanced properties

Generalizations and extensions

Addition

Notation

Interpretations

Combining sets

Extending a measure

Combining translations

Basic properties

Commutivity

Associativity

Zero and one

Units

Generalizations

In algebra

Addition of sets

Related operations

See also

Notes

References

External links

Given any

Basics

Properties

Negation

Rules of Inference

Commutative

Mathematical meaning

Neurophysiological meaning

See also

Number