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| In [[mathematics]], an '''algebraic expression''' is an [[Expression (mathematics)|expression]] built up from [[constant (mathematics)|constant]]s, [[Variable (mathematics)|variables]], and a finite number of [[algebraic operation]]s ([[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]] and [[exponentiation]] by an exponent that is a [[rational number]]).<ref>{{cite book |last1=Morris |first1=Christopher G. |title=Academic Press dictionary of science and technology |page=74 |year=1992 |url=http://books.google.co.uk/books?id=nauWlPTBcjIC&lpg=PA74&dq=algebraic%20expression%20over%20a%20field&pg=PA74#v=onepage&q&f=false}}</ref> For example, <math style="margin-bottom:8px">3x^2 - 2xy + c</math> is an algebraic expression. Since taking the [[square root]] is the same as raising to the power <math>\tfrac{1}{2}</math>,
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| :<math>\sqrt{\frac{1-x^2}{1+x^2}}</math>
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| is also an algebraic expression. In math algebraic expression are very useful tools.
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| A '''rational expression''' is an [[expression (mathematics)|expression]] that may be rewritten to a [[rational fraction]] by using the properties of the arithmetic operations ([[commutative property|commutative properties]] and [[associative property|associative properties]] of addition and multiplication, [[distributive property]] and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of [[arithmetic]]. Thus, <math>3x^2 - 2xy + c</math> is a rational expression, whereas <math>\sqrt{\frac{1-x^2}{1+x^2}}</math> is not.
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| A '''rational equation''' is an equation in which two [[rational fraction]]s (or [[rational expression]]s) of the form <math> \frac{P(x)}{Q(x)}</math> are set equal to each other. These expressions obey the same rules as [[fraction (mathematics)|fractions]]. The equations can be solved by [[cross-multiplication|cross-multiplying]]. Division by zero is undefined, so that a solution causing formal division by zero is rejected.
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| ==Terminology==
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| Algebra has its own terminology to describe parts of an expression:
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| <center> | |
| [[File:algebraic equation notation.svg|256px]]<br>
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| 1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, <math>x, y</math> - variables
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| </center>
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| ==Conventions==
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| ===Variables===
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| By convention, letters at the beginning of the alphabet (e.g. <math>a, b, c</math>) are typically used to represent [[Mathematical constant|constant]]s, and those toward the end of the alphabet (e.g. <math>x, y</math> and <math>z</math>) are used to represent [[Variable (mathematics)|variable]]s.<ref>William L. Hosch (editor), ''The Britannica Guide to Algebra and Trigonometry'', Britannica Educational Publishing, The Rosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, [http://books.google.co.uk/books?id=ad0P0elU1_0C&lpg=PA71&dq=elementary%20algebra%20letters%20alphabet%20constants%20variables&pg=PA71#v=onepage&q=letters&f=false page 71]</ref> They are usually written in italics.<ref>James E. Gentle, ''Numerical Linear Algebra for Applications in Statistics'', Publisher: Springer, 1998, ISBN 0387985425, 9780387985428, 221 pages, [James E. Gentle page 183]</ref>
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| ===Exponents===
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| By convention, terms with the highest power ([[Exponentiation|exponent]]), are written on the left, for example, <math style="margin-bottom:8px">x^2</math> is written to the left of <math>x</math>. When a coefficient is one, it is usually omitted (e.g. <math style="margin-bottom:8px">1x^2</math> is written <math style="margin-bottom:8px">x^2</math>).<ref>David Alan Herzog, ''Teach Yourself Visually Algebra'', Publisher John Wiley & Sons, 2008, ISBN 0470185597, 9780470185599, 304 pages, [http://books.google.co.uk/books?id=Igs6t_clf0oC&lpg=PA72&ots=Excnhf1AgW&dq=algebra%20coefficient%20one&pg=PA72#v=onepage&q=coefficient%20of%201&f=false page 72]</ref> Likewise when the exponent (power) is one, (e.g. <math style="margin-bottom:8px">3x^1</math> is written <math style="margin-bottom:8px">3x</math>),<ref>John C. Peterson, ''Technical Mathematics With Calculus'', Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, [http://books.google.co.uk/books?id=PGuSDjHvircC&lpg=PA31&ots=NKrtZZ1KDE&dq=%22when%20the%20exponent%20is%201%22&pg=PA32#v=onepage&q=%22when%20the%20exponent%20is%201%22&f=false page 31]</ref> and, when the exponent is zero, the result is always 1 (e.g. <math style="margin-bottom:8px">x^0</math> is always <math>1</math>).<ref>Jerome E. Kaufmann, Karen L. Schwitters, ''Algebra for College Students'', Publisher Cengage Learning, 2010, ISBN 0538733543, 9780538733540, 803 pages, [http://books.google.co.uk/books?id=-AHtC0IYMhYC&lpg=PP1&ots=kL8erjajyR&dq=algebra%20exponents%20zero%20one&pg=PA222#v=onepage&q=exponents%20&f=false page 222]</ref>
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| ===Author===
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| The word "algebra" first appeared in "Hisab aljabr wal-muqabala" by Al-Khwarizmi around AD 825. This book was regarded as the first book relating to algebra. He introduced algebra as a means of systemic way to solve problems.
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| ==Rational expressions==
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| A ''rational algebraic expression'' (or ''rational expression'') is an algebraic expression that can be written as a [[quotient]] of [[polynomial]]s, such as {{math|''x''<sup>2</sup> + 4''x'' + 4}}. An ''irrational algebraic expression'' is one that is not rational, such as {{math|{{radical|''x'' + 4}}}}.
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| ==See also==
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| *[[Algebraic equation]]
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| *[[Analytical expression]]
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| *[[Arithmetic expression]]
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| ==References==
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| {{Reflist}}
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| *{{cite book |last1=James |first1=Robert Clarke |last2=James |first2=Glenn |title=Mathematics dictionary |page=8 |year=1992 |url=http://books.google.co.uk/books?id=UyIfgBIwLMQC&lpg=PA8&dq=algebraic%20expression%20over%20a%20field&pg=PA8#v=onepage&q&f=false}}
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| ==External links==
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| *{{MathWorld|title=Algebraic Expression|id=AlgebraicExpression}}
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| [[Category:Elementary algebra]]
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