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| [[File:quadratic root.svg|thumb|right|Algebraic operations in the solution to the [[quadratic equation]]. The radical sign, √ denoting a [[square root]], is equivalent to [[exponentiation]] to the power of ½. The [[Plus-minus sign|± sign]] represents the equation written with either a + and with a - sign.]]
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| In [[mathematic]]s, an '''algebraic operation''' is any one of the [[Operation (mathematics)|operation]]s [[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]], raising to an integer [[exponentiation|power]], and taking [[nth root|root]]s (fractional power). Algebraic operations are performed on an algebraic variable, term or [[Algebraic expression|expression]],<ref>William Smyth, ''Elementary algebra: for schools and academies'', Publisher Bailey and Noyes, 1864, "[http://books.google.co.uk/books?id=BqQZAAAAYAAJ&lpg=PA55&ots=ex07zH_ljg&dq=%22Algebraic%20operations%22&pg=PA55#v=onepage&q=%22Algebraic%20operations%22&f=false Algebraic Operations]"</ref> and work in the same way as arithmetic operations.<ref>Horatio Nelson Robinson, ''New elementary algebra: containing the rudiments of science for schools and academies'', Ivison, Phinney, Blakeman, & Co., 1866, [http://books.google.co.uk/books?id=dKZXAAAAYAAJ&dq=Elementary%20algebra%20notation&pg=PA7#v=onepage&q=Elementary%20algebra%20notation&f=false page 7]</ref>
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| ==Notation==
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| Multiplication symbols are usually omitted, and implied when there is no operator between two variables or terms, or when a [[coefficient]] is used. For example, 3 × ''x''<sup>2</sup> is written as 3''x''<sup>2</sup>, and 2 × ''x'' × ''y'' is written as 2''xy''.<ref>Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in ''Mathematics Matters Secondary 1 Express Textbook'', Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, [http://books.google.co.uk/books?id=nL5ObMmDvPEC&lpg=PR9-IA8&ots=T_h6l40AE5&dq=%22Algebraic%20notation%22%20multiplication%20omitted&pg=PR9-IA8#v=onepage&q=%22Algebraic%20notation%22%20multiplication%20omitted&f=false page 68]</ref> Sometimes multiplication symbols are replaced with either a dot, or center-dot, so that ''x'' × ''y'' is written as either ''x'' . ''y'' or ''x'' · ''y''. [[Plain text]], [[programming languages]], and [[calculators]] also use a single asterisk to represent the multiplication symbol,<ref>William P. Berlinghoff, Fernando Q. Gouvêa, ''Math through the Ages: A Gentle History for Teachers and Others'', Publisher MAA, 2004, ISBN 0883857367, 9780883857366, [http://books.google.co.uk/books?id=JAXNVaPt7uQC&lpg=PA75&ots=-P78Lrz792&dq=calculator%20asterisk%20multiplication&pg=PA75#v=onepage&q=calculator%20asterisk%20multiplication&f=false page 75]</ref> and it must be explicitly used, for example, 3''x'' is written as 3 * ''x''.
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| Rather than using the [[obelus]] symbol, ÷, division is usual represented with a [[Vinculum (symbol)|vinculum]], a horizontal line, e.g. {{sfrac|3|''x'' + 1}}. In plain text and programming languages a slash (also called a [[Slash (punctuation)|solidus]]) is used, e.g. 3 / (''x'' + 1).
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| Exponents are usually formatted using superscripts, e.g. ''x''<sup>2</sup>. In [[plain text]], and in the [[TeX]] mark-up language, the [[caret]] symbol, ^, represents exponents, so ''x''<sup>2</sup> is written as ''x'' ^ 2.<ref>Ramesh Bangia, ''Dictionary of Information Technology'', Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153, 9789380298153, [http://books.google.co.uk/books?id=zQa5I2sHPKEC&lpg=PA212&ots=s6pWav1Z_D&dq=%22plain%20text%22%20math%20caret%20exponent&pg=PA212#v=onepage&q=exponentiation%20caret&f=false page 212]</ref><ref>George Grätzer, ''First Steps in LaTeX'', Publisher Springer, 1999, ISBN 0817641327, 9780817641320, [http://books.google.co.uk/books?id=mLdg5ZdDKToC&lpg=PP1&ots=V9DFIaAAh0&dq=tex%20math&pg=PA17#v=onepage&q=subscripts%20and%20superscripts%20caret&f=false page 17]</ref> In programming languages such as [[Ada (programming language)|Ada]],<ref>S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, ''Ada 2005 Reference Manual'', Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352, [http://books.google.co.uk/books?id=694P3YtXh-0C&lpg=PA718&ots=O_EgQ75FeB&dq=ada%20%20asterisk&pg=PA12#v=onepage&q=double%20star%20exponentiate&f=false page 13]</ref> [[Fortran]],<ref>C. Xavier, ''Fortran 77 And Numerical Methods'', Publisher New Age International, 1994, ISBN 812240670X, 9788122406702, [http://books.google.co.uk/books?id=WYMgF9WFty0C&lpg=PA20&ots=BTtzs9F-NB&dq=fortran%20asterisk%20exponentiation&pg=PA20#v=onepage&q=fortran%20asterisk%20exponentiation&f=false page 20]</ref> [[Perl]],<ref>Randal Schwartz, brian foy, Tom Phoenix, ''Learning Perl'', Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140, 9781449313142, [http://books.google.co.uk/books?id=l2IwEuRjeNwC&lpg=PA24&ots=5nsYOLHxlD&dq=perl%20asterisk%20exponentiation&pg=PA24#v=onepage&q=double%20asterisk%20exponentiation&f=false page 24]</ref> [[Python (programming language)|Python]]<ref>Matthew A. Telles, ''Python Power!: The Comprehensive Guide'', Publisher Course Technology PTR, 2008, ISBN 1598631586, 9781598631586, [http://books.google.co.uk/books?id=754knV_fyf8C&lpg=PA46&ots=8fEi1F-H8-&dq=python%20asterisk%20exponentiation&pg=PA46#v=onepage&q=double%20asterisk%20exponentiation&f=false page 46]</ref> and [[Ruby (programming language)|Ruby]],<ref>Kevin C. Baird, ''Ruby by Example: Concepts and Code'', Publisher No Starch Press, 2007, ISBN 1593271484, 9781593271480, [http://books.google.co.uk/books?id=kq2dBNdAl3IC&lpg=PA72&ots=0UU3k-Pvh8&dq=ruby%20asterisk%20exponentiation&pg=PA72#v=onepage&q=double%20asterisk%20exponentiation&f=false page 72]</ref> a double asterisk is used, so ''x''<sup>2</sup> is written as ''x'' ** 2.
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| The [[plus-minus sign]], ±, is used as a shorthand notation for two expressions written as one, representing one expression with a plus sign, the other with a minus sign. For example ''y'' = ''x'' ± 1 represents the two equations ''y'' = ''x'' + 1 and ''y'' = ''x'' − 1. Sometimes it is used for denoting positive-or-negative term such as ±''x''.
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| ==Arithmetic vs algebraic operations==
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| Algebraic operations work in the same way as [[Arithmetic operation#Arithmetic operations|arithmetic operations]], as can be seen in the table below.
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| {| class="wikitable"
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| |-
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| !Operation
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| !Arithmetic<br>{{nobold|Example}}
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| !Algebra'''<br>{{nobold|Example}}
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| !Comments'''<br>{{nobold|≡ – means "equivalent to"<br>≢ – means "not equivalent to"}}
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| |- align="center"
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| | [[Addition]]
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| |<math>(5 \times 5) + 5 + 5 + 3</math>
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| equivalent to:
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| <math>5^2 + (2 \times 5) + 3</math>
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| |<math>(b \times b) + b + b + a</math>
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| equivalent to:
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| <math>b^2 + 2b + a</math>
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| |<math>\begin{align} 2 \times b & \equiv 2b\\
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| b + b + b & \equiv 3b\\
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| b \times b & \equiv b^2 \end{align}</math>
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| |- align="center"
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| | [[Subtraction]]
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| |<math>(7 \times 7) - 7 - 5</math>
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| equivalent to:
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| <math>7^2 - 7 - 5</math>
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| |<math>(b \times b) - b - a</math>
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| equivalent to:
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| <math>b^2 - b - a</math>
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| |<math>\begin{align}b^2 - b & \not\equiv b\\
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| 3b - b & \equiv 2b\\
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| b^2 - b & \equiv b(b-1)\end{align}</math>
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| |- align="center"
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| | [[Multiplication]]
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| |<math>3 \times 5</math> or
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| <math>3 \ .\ 5</math> or <math>3 \cdot 5</math>
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| or <math>(3)(5)</math>
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| |<math>a \times b</math> or
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| <math>a . b</math> or <math>a \cdot b</math>
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| or <math>ab</math>
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| |<math>a \times a \times a</math> is the same as <math>a^3</math>
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| |- align="center"
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| | [[Division (mathematics)|Division]]|| <math>12 \div 4</math> or
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| <math>12 / 4</math> or
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| <math>\frac {12}{4}</math>
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| | <math>b \div a</math> or
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| <math>b / a</math> or
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| <math>\frac {b}{a}</math>
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| |<math>\frac{(a+b)}{3} \equiv \tfrac{1}{3} \times (a+b)</math>
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| |- align="center"
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| | [[Exponentiation]]
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| | <math>3^{\frac{1}{2}}</math><br /> <math>2^3</math>
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| | <math>a^{\frac{1}{2}}</math><br /> <math>a^3</math>|| <math>a^{\frac{1}{2}}</math> is the same as <math>\sqrt a</math><br />
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| <math>a^3</math> is the same as <math>a \times a \times a</math>
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| |}
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| Note: the use of the letters <math>a</math> and <math>b</math> is arbitrary, and the examples would be equally valid if we had used <math>x</math> and <math>y</math>.
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| ==Properties of arithmetic and algebraic operations==
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| {| class="wikitable"
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| |-
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| !Property
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| !Arithmetic<br>{{nobold|Example}}
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| !Algebra<br>{{nobold|Example}}
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| !Comments<br>{{nobold|≡ – means "equivalent to"<br>≢ – means "not equivalent to"}}
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| |- align="center"
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| | [[Commutative property|Commutativity]]
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| |<math>(3 + 5) = (5 + 3)</math><br /><math>(3 \times 5) = (5 \times 3)</math>
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| |<math>(a + b) = (b + a)</math><br /><math>(a \times b) = (b \times a)</math>
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| | rowspan=2 |Addition and multiplication are<br>commutative and associative<ref name="larson2007p7">Ron Larson, Robert Hostetler, Bruce H. Edwards, ''Algebra And Trigonometry: A Graphing Approach'', Publisher: Cengage Learning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, [http://books.google.co.uk/books?id=5iXVZHhkjAgC&lpg=PA6&ots=iwrSrCrrOb&dq=operations%20addition%2C%20subtraction%2C%20multiplication%2C%20division%20exponentiation.&pg=PA7#v=onepage&q=associative%20property&f=false page 7]</ref> <br />Subtraction and division are not<br />
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| e.g. <math>(a - b) \not\equiv (b - a)</math><br>(except when <math>a=b</math>)
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| |- align="center"
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| | [[Associative property|Associativity]]
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| |<math>(3 + 5) + 7 = 3 + (5 + 7)</math><br /><math>(3 \times 5) \times 7 = 3 \times (5 \times 7)</math>
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| |<math>(a + b) + c = a + (b + c)</math><br /><math>(a \times b) \times c = a \times (b \times c)</math>
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| |}
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| ==References==
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| {{Reflist}}
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| ==See also==
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| *[[Elementary algebra]]
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| *[[Order of operations]]
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| [[Category:Elementary algebra]]
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| [[Category:Elementary mathematics]]
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