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Elementary Algebra

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

Algebra (disambiguation)

Algebra is a branch of mathematics. Algebra may also mean:
- elementary algebra
- abstract algebra
- linear algebra
- universal algebra
- computer algebra
- Boolean algebra, in formal logic
- algebra over a commutative ring (or R-algebra), in ring theory In set theory:
- algebra over a set
- sigma algebra In linear algebra, and the study of vector spaces:
- algebra over a field
- associative algebra In category theory:
- F-algebra
- F-coalgebra

Algebra

:This article is about the branch of mathematics. For other uses of the term see algebra (disambiguation). Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. The study of Algebra is the cause for some debate as the level taught to High School students is rarely applicable in the real world.

History

The origins of algebra can be traced to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.
- Circa 300 BC: Greek mathematician Euclid, who taught and died at Alexandria in Egypt, in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.
- Circa 100 BC: algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.
- Circa 150 AD: Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.
- Circa 200 AD: Greek mathematician Diophantus, often referred to as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
- 476 AD Indian mathematician, Aryabhata obtains whole number solutions to linear equations by a method equivalent to modern one. Bhaskara II (1114 AD), who wrote the text Bijaganita (algebra), was the first to recognize that a positive number has two square roots. The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. They treated indeterminate quadratic equations.
- 820 AD The word algebra is derived from the name of the treatise first written by Persian mathematician Khwarizmi titled: Al-Jabr wa-al-Muqabilah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr means "reunion".
- 1202 AD Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci .

Classification

Algebra may be roughly divided into the following categories:
- elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra);
- abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated;
- linear algebra, in which the specific properties of vector spaces are studied (including matrices);
- universal algebra, in which properties common to all algebraic structures are studied. In advanced studies, axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. The list includes:
- Normed linear spaces
- Banach spaces
- Hilbert spaces
- Banach algebras
- Normed algebras
- Topological algebras
- Topological groups

Algebras

The word algebra is also used for various algebraic structures:
- algebra over a field
- algebra over a set
- Boolean algebra
- sigma-algebra
- F-algebra and F-coalgebra in category theory

References

- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- George Gheverghese Joseph, The Crest of the Peacock : The Non-European Roots of Mathematics (Princeton University Press, 2000).

External links

- [http://www.mathleague.com/help/algebra/algebra.htm Explanation of Basic Topics]
- [http://www.sparknotes.com/math/#algebra1 Sparknotes' Review of Algebra I and II]
- [http://www.jamesbrennan.org/algebra/ Understanding Algebra.] An online algebra text by James W. Brennan. Category:Algebra Category:Arabic words ko:대수학 ms:Algebra ja:代数学 simple:Algebra

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

Number

: This article is about numbers such as counting numbers and measurements. For other uses of the term, see Number (disambiguation). A number originally was a count or a measurement. Mathematicians have extended this concept to include abstractions such as the square root of minus one. In common usage, number symbols are often used as labels (highway numbers) or to indicate order (serial numbers).

Examples

The most familiar numbers are the counting numbers or natural numbers. Some writers include 0, thus: . Others do not: . In the base ten number system, now in almost universal use worldwide, the symbols for natural numbers are written using ten digits, 0 through 9. The symbol for the set of all natural numbers is N. If the negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers Z (from the German word "zahlen"). (Some authors use W for the whole numbers, but other authors use W for the natural numbers, so the W symbol is ambiguous.) Negative numbers are used to indicate an opposite. If a positive number is used to indicate distance to the right of some fixed point, a negative number indicates distance to the left. If a positive number indicates a bank deposit, a negative number indicates a withdrawal. Rational numbers are made up of all numbers that can be expressed as a fraction, with integer numerator and non-zero natural number denominator. The fraction m/n represents the quantity arrived at when a whole is divided into n equal parts, and m of those equal parts are chosen. If m is greater than n, the fraction is greater than one. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face Q (for "quotient"). The real numbers are made up of all numbers that can be expressed as a decimal. These are the measuring numbers, and in the base ten number system are written as a string of digits, with a dot (US) or a comma (Europe) to the right of the ones place. The symbol for the real numbers is R. All measurements are necessarily approximations; the accuracy of the approximation depends on the accuracy of the measuring device. Therefore all measurements are properly represented by decimals that end, the last decimal place indicating the accuracy of the measurement. For example, 1.23 inches indicates a measurement accurate to the nearest hundredth of an inch. However, mathematically, when a rational number is expressed as a decimal, it may never end. Thus 1/3 becomes 0.3333... (unending threes). Mathematicians, therefore, consider both decimals that end and decimals that go on forever. The latter represent an infinite series. Some real numbers can be written as fractions, 0.3333... for example. Others cannot, 0.1010010001... for example. A decimal that can be written as a fraction is called rational, a decimal that cannot be written as a fraction is called irrational. A decimal is rational when it either ends or repeats forever. There is a technical sense in which the real numbers are the ideal set of numbers. They are the only complete ordered field. Moving to a greater level of abstraction, and away from counting and measuring, the real numbers can be extended to the complex numbers C. This set of numbers arose, historically, from consideration of the question of whether or not there was any sense in which negative numbers can have a square root. A new number was invented, the square root of negative one, denoted by i, a symbol assigned to this new number by Leonhard Euler. The complex numbers consist of all numbers of the form a + bi, where a and b are real numbers. If b is zero, then a + bi is real. If a is zero, then a + bi is called imaginary. The complex numbers are an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors with complex coefficients. The above symbols are often written in blackboard bold, thus: :

\mathbb\sub\mathbb\sub\mathbb\sub\mathbb\sub\mathbb

While the natural numbers and the real numbers suffice for most everyday purposes, mathematicians have invented many other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example the roots of polynomials with rational coefficients are called the algebraic numbers. Real numbers that are not algebraic are called transcendental numbers. The Gaussian integers are complex numbers a + bi where a and b are integers. Sets of numbers that are not subsets of the complex numbers include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative.

Further generalizations

Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.

Numerals and numbering

Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral 5 and by the Roman numeral V. Notations used to represent numbers are discussed in the article numeral systems. Numbers are often used to give objects unique names. Examples are telephone numbers, social security numbers, and ISBNs.

Extensions

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While real numbers may have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left, with digits in base p, where p is prime. This leads to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; but they differ in the infinite case.) The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

External links

- [http://freepages.history.rootsweb.com/~catshaman/13comp/0numer.htm Mesopotamian and Germanic numbers]

References

- Erich Friedman, [http://www.stetson.edu/~efriedma/numbers.html What's special about this number?]
- [http://www.cut-the-knot.org/do_you_know/numbers.shtml What's a Number?] at cut-the-knot Category:Group theory Category:Numbers __NOTOC__ ko:수 (수학) ja:数 simple:Number th:จำนวน

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

Equation

:This article is about equations in mathematics. For the chemistry term, see chemical equation. An equation is a mathematical statement, in symbols, that two things are the same. Equations are written with the equals sign, as in :2 + 3 = 5. Equations are often used to state the equality of two expressions containing one or more variables. For example, given any value of x, it is always true that :x − x = 0. The two equations above are examples of identities: equations that are true regardless of the values of any variables that appear within them. The following equation is not an identity: :x + 1 = 2. The above equation is false for almost all conceivable values of x. Therefore, if the equation is known to be true, it carries information about the value of x. In this example, one concludes that x = 1. In general, the values of the variables for which the equation is true are called solutions. To solve an equation means to find its solutions. Many authors reserve the term equation for an equality which is not an identity. The distinction between the two concepts can be subtle; for example, :(x + 1)² = x² + 2x + 1 is an identity, while :(x + 1)² = 2x² + x + 1 is an equation, whose solutions for x are 0 and 1. Whether a statement is meant to be an identity or an equation carrying information about its variables can usually be determined from its context. Letters from the beginning of the alphabet like a, b, c, ... are often considered constants in the context of the discussion at hand, while letters from end of the alphabet, like x, y, z, are usually considered variables.

Properties

If an equation in algebra is known to be true, it can be manipulated to produce another true equation in a variety of ways: # Any quantity can be added to both sides. # Any quantity can be subtracted from both sides. # Any quantity can be multiplied to both sides. # Any nonzero quantity can divide both sides. # Generally, any function can be applied to both sides. If a function that is not injective is applied to both sides of a true equation, the resulting equation will still be true, but it may be less useful. Formally, one has an implication, not an equivalence, so the solution set may get larger. The functions implied in properties (1), (2), and (4) are always injective, as is (3) if we do not multiply by zero. Some generalized products, such as a dot product, are never injective.

External links

- Free Online Equation Interpreter and Plotter: [http://www.wessa.net/math.wasp Mathematical Equation Plotter]. Plots 2D mathematical equations, computes integrals, and finds solutions.
- Solve 2D equations graphically and numerically: [http://deadline.3x.ro DeadLine]. Free Windows software.
- [http://eqworld.ipmnet.ru/en/solutions/ae.htm Algebraic Equations and Systems of Algebraic Equations] at EqWorld: The World of Mathematical Equations. Category:Elementary algebra Category:Equations ko:방정식 ja:方程式 simple:Equation

Expression (mathematics)

An expression combines numbers, operators, and/or free variables and bound variables: bound variables are defined in the expression (they are for internal use), free variables are taken from the context. For a given combination of values for the free variables, an expression may be evaluated to a value, and is said to have that value, although for some combinations of values of the free variables, the expression may be undefined. Thus an expression represents a function of the values for the free variables. The evaluation of an expression is dependent on the definition of the mathematical operators and system of values that forms the context of an expression. Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same value, i.e., they represent the same function. Example: The expression :

\sum_^ 2xy

has free variable x, bound variable y, constants 1, 2, and 3, two occurrences of an implicit multiplication operator, and a summation operator. The expression is equivalent with the simpler expression 12x. The value for x=3 is 36. Expressions and their evaluation were formalised by Alonzo Church and Stephen Kleene in the 1930s in their lambda calculus. The lambda calculus has been a major influence in the development of modern mathematics and computer programming languages. One of the more interesting results of the lambda calculus is that the equivalence of two expressions in the lambda calculus is in some cases undecidable. This is also true of any expression in any system that has power equivalent to the lambda calculus.

External links

- [http://www.mathematics21.org/theory-of-formulas-index.html Axiomatic Theory of Formulas] - theory of expressions on high abstraction level.
- [http://www.algebra.com/services/rendering/ Plot mathematical expressions] this system plots math equations, graphs, diagrams, and even animated cartoons of transformation of math expressions and arithmetic operations. Knowledge of TeX not required. Category:Abstract algebra Category:Algebra ja:数式

Identity (mathematics)

:For other senses of this word, see identity (disambiguation). In mathematics, identity can refer to an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. Alternatively, in algebra, an identity or identity element of a set S with a binary operation is an element e which combined with any element s of S produces s. Yet a third meaning is that an identity is a function f from a set S to itself, such that f(x) = x for all x in S. A common example of the first meaning is the trigonometric identity when

\theta

is considered over the set of real numbers (since that is the domain of sin and cos) :

( \sin \theta)^2 + ( \cos \theta)^2 = 1,\,

which is true for all values of

\theta

, as opposed to :

\cos \theta = 1,\,

which is true only for a subset of the domain. A common example of the second meaning is addition in the real numbers, where 0 is the identity. This means that for all

a\in\Bbb

, :

0 + a = a,\,

a + 0 = a,\,

and :

0 + 0 = 0.\,

A common example of the third meaning is the identity permutation, which sends each element of the set to itself. See also list of mathematical identities. Category:Elementary algebra Category:Identities ja:恒等式

Order of operations

In arithmetic and algebra, certain rules are used for the order in which the operations in expressions are to be evaluated. These precedence rules (which are mere notational conventions, not mathematical facts) are also used in many programming languages and by most modern calculators. In computing, the standard algebraic notation is known as infix notation. This article assumes the reader is familiar with addition, subtraction, multiplication, division, exponents, and roots (such as square roots, cube roots, and so on).

The standard order of operations

The order of operations is expressed in the following chart. :::exponents and roots ::: :::multiplication and division ::: :::addition and subtraction In the absence of parentheses, do all the exponents and roots first. Stacked exponents must be done from right to left. Root symbols have a bar over the radicand which acts as a symbol of grouping. Then do all the multiplication and division, from left to right. Finally, do all of the addition and subtraction, from left to right. If there are parentheses, in arithmetic do the expression inside the innermost parentheses first, and work outward. In algebra, the distributive law can sometimes be used to remove parentheses. The chart which gives the order of operations can help in remembering that roots and exponents distribute over multiplciation and division, while multiplication and division distribute over addition and subtraction.

Examples

:1. Evaluate subexpressions contained within parentheses, starting with the innermost expressions. (Brackets [ ] are used here to indicate what is evaluated next.) ::

(4+10/2)/9=(4+[10/2])/9=[4+5]/9=1 \,

:2. Evaluate exponential powers; for iterated powers, start from the right: ::

2^=2^=[2^9]=512 \,

:3. Evaluate multiplications and divisions, starting from the left: ::

8/2\times3=[8/2]\times3=[4\times3]=12 \,

:4. Evaluate additions and subtractions, starting from the left: ::

7-2-4+1=[7-2]-4+1=[5-4]+1=[1+1]=2 \,

The expression: 2 + 3 × 4 is evaluated to 14, and not 20, because multiplication precedes addition. If the intention is to perform the addition first, parentheses must be used: (2 + 3) × 4 = 20. In Australia and Canada, an acronym BEDMAS is often used as a mnemonic for Brackets, Exponents, Division, Multiplication, Addition, and Subtraction. In the UK and New Zealand, the acronym BODMAS is used for Brackets, Orders, Division, Multiplication, Addition, Subtraction. This is sometimes written as BOMDAS, BIDMAS or BIMDAS where I stands for Indices. In the US, the acronym PEMDAS (for Parentheses, Exponentiation, Multiplication/Division, Addition/Subtraction) is used instead, sometimes expressed as the mnemonic "Please Excuse My Dear Aunt Sally". Warning: In some grade school textbooks in use in the United States, students are instructed incorrectly to add (Aunt) before subtracting (Sally). These textbooks give the answer to ::

10 - 3 + 2 \,

as 5 rather than the correct answer, 9. Parents should check their children's textbooks to see if they contain this error.

More examples

- Given: ::

3-(5-(7+1))^2\times(-5)+2 \,

- Evaluate the innermost subexpression (7 + 1): ::

3-(5-8)^2\times(-5)+2 \,

- Evaluate the subexpression within the remaining parentheses (5 − 8): ::

3-(-3)^2\times(-5)+2 \,

- Evaluate the power of (−3)²: ::

3-9\times(-5)+2 \,

- Evaluate the multiplication 9 × (−5): ::

3-(-45)+2 \,

- Evaluate the subtraction 3 − (−45): ::

48+2 \,

- Evaluate the addition 48 + 2: ::

48+2=50 \,

Proper use of parentheses and other grouping symbols

When you are restricted to using a straight text editor, parentheses (or more generally "grouping symbols") must be used generously to make up for the lack of graphics, like square root symbols. Here are some rules for doing so: 1) Whenever there is a fraction formed with a slash, put the numerator (the numbers on top of the fraction) in one set of parentheses, and the denominator (the numbers on the bottom of the fraction) in another set of parentheses. This is not required for fractions formed with underlines: :y = (x+1)/(x+2) 2) Whenever there is an exponent using the caret (^) symbol, put the base in one set of parentheses, and the exponent in another set of parentheses: :y = (x+1)^(x+2) 3) Whenever there is a trig function, you put the argument of the function, typically shown in bold and/or italics, in parentheses: :y = sin(x+1) 4) The rule for trig functions also applies to any other function, such as "sqrt". That is, the argument of the function should be contained in parentheses: :y = sqrt(x+1) 5) An exception to the rules requiring parentheses applies when only one character is present. While correct either way, it is more readable if parentheses around a single character are omitted: :y = (3)/(x) or y = 3/x :y = (3)/(2x) or y = 3/(2x) :y = (x)^(5) or y = x^5 :y = (2x)^(5) or y = (2x)^5 :y = (x)^(5z) or y = x^(5z) Calculators generally require parentheses around the argument of any function. Printed or handwritten expressions sometimes omit the parentheses, provided the argument is a single character. Thus, a calculator or computer program requires: :y = sqrt(2) :y = tan(x) A printed text may have: :y = sqrt 2 :y = tan x 6) Whenever anything can be interpreted multiple ways, put the part to be done first in parentheses, to make it clear. 7) You may alternate use of the different grouping symbols (parentheses, brackets, and braces) to make it more readable. For example: :y = is more readable than: :y = ( 2 / ( 3 / ( 4 / 5 ) ) ) Note that certain applications, like computer programming, will restrict you to certain grouping symbols.

Special cases

In the case that a factorial is in an expression, it is evaluated before exponents and roots, unless parentheses or other grouping symbols dictate otherwise. When new operations are defined they are generally presumed to take precedence over other operations unless defined otherwise. In the case where repeated operators of the same type are used, such as in ::

a/b/c

the expression is evaluated from left to right, as ::

((a/b)/c)

. (Recall that exponents are an exception, always evaluated from right to left.)

External links

-
- For a rationale behind the use of the order of operations, see [http://www.mathandtext.blogspot.com/2005/05/order-of-operations.html MathandText]. Category:Abstract algebra Category:Algebra

Commutative operation

: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.

Elementary algebra

Laws of elementary algebra

Examples

Linear equations

Quadratic equations

System of linear equations

See also

External links

Algebra (disambiguation)

Algebra

History

Classification

Algebras

References

See also

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

Number

Examples

Further generalizations

Numerals and numbering

Extensions

See also

External links

References

Real number

History

Definition

Construction from the rational numbers

Axiomatic approach

Properties

Completeness

"The complete ordered field"

Advanced properties

Generalizations and extensions

Equation

Properties

See also

External links

Expression (mathematics)

See also

External links

Identity (mathematics)

Order of operations

The standard order of operations

Examples

More examples

Proper use of parentheses and other grouping symbols

Special cases

See also

External links

Commutative operation

Mathematical meaning

Neurophysiological meaning

See also

Subt