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A simpler expression than this is generally desired, and simplification is needed when working with general expressions.
This simplification is normally done through [[rewriting|rewriting rules]].<ref>{{cite book |last1=Buchberger |first1=Bruno |first2=Rüdiger |last2=Loos |chapter=Algebraic simplification |chapter-url=https://www.risc.jku.at/people/buchberg/papers/1982-00-00-B.pdf | editor1-last = Buchberger | editor1-first = Bruno | editor2-first = George Edwin | editor3-first = Rüdiger | editor4-first = Rudolf |title= Computer Algebra: Symbolic and Algebraic Computation|series=Computing Supplementa |year=1983 |volume=4 |pages=11–43 |doi=10.1007/978-3-7091-7551-4_2 |isbn=978-3-211-81776-6 }}</ref> There are several classes of rewriting rules to be considered. The simplest are rules that always reduce the size of the expression, like {{math|''E'' − ''E'' → 0}} or {{math|sin(0) → 0}}. They are systematically applied in computer algebra systems.
A difficulty occurs with [[associative operation]]s like addition and multiplication. The standard way to deal with associativity is to consider that addition and multiplication have an arbitrary number of operands, that is that {{math|''a'' + ''b'' + ''c''}} is represented as {{math|"+"(''a'', ''b'', ''c'')}}. Thus {{math|''a'' + (''b'' + ''c'')}} and {{math|(''a'' + ''b'') + ''c''}} are both simplified to {{math|"+"(''a'', ''b'', ''c'')}}, which is displayed {{math|''a'' + ''b'' + ''c''}}. In the case of expressions such as {{math|''a'' − ''b'' + ''c''}}, the simplest way is to systematically rewrite {{math|−''E''}}, {{math|''E'' − ''F''}}, {{math|''E''/''F''}} as, respectively, {{math|(−1)⋅''E''}}, {{math|''E'' + (−1)⋅''F''}}, {{math|''E''⋅''F''<sup>−1</sup>}}. In other words, in the internal representation of the expressions, there is no subtraction nor division nor unary minus, outside the representation of the numbers.
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==History==
At the beginning of computer algebra, circa 1970, when the long-known [[algorithm]]s were first put on computers, they turned out to be highly inefficient.<ref>{{cite book | editor1-last = Buchberger | editor1-first = Bruno | editor2-first = George Edwin | editor3-first = Rüdiger | editor4-first = Rudolf |title=Computer Algebra: Symbolic and Algebraic Computation |pages=95–113 |doi=10.1007/978-3-7091-7551-4_8 |first1=Erich|last1=Kaltofen|chapter=Factorization of polynomials |series=Computing Supplementa |year=1983 |volume=4 |isbn=978-3-211-81776-6 |s2cid=18352153 }}</ref> Therefore, a large part of the work of the researchers in the field consisted in revisiting classical [[algebra]] in order to make it [[Computable function|effective]] and to discover [[algorithmic efficiency|efficient algorithms]] to implement this effectiveness. A typical example of this kind of work is the computation of [[polynomial greatest common divisor]]s, which is required to simplify fractions. Surprisingly, the classical [[Euclid's algorithm]] turned out to be inefficient for polynomials over infinite fields, and thus new algorithms needed to be developed. The same was also true for the classical algorithms from [[linear algebra]].
== See also ==
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*{{cite book|first1=Joachim|last1=von zur Gathen|first2=Jürgen|last2=Gerhard|title=Modern computer algebra|edition=2nd|publisher=Cambridge University Press|year=2003|isbn = 0-521-82646-2}}
*{{Cite book | last1 = Geddes | first1 = K. O. | last2 = Czapor | first2 = S. R. | last3 = Labahn | first3 = G. | doi = 10.1007/b102438 | title = Algorithms for Computer Algebra | year = 1992 | bibcode = 1992afca.book.....G | isbn = 978-0-7923-9259-0 | url-access = registration | url = https://archive.org/details/algorithmsforcom0000gedd }}
*{{Cite book | editor1-last = Buchberger | editor1-first = Bruno | editor2-first = George Edwin | editor3-first = Rüdiger | editor4-first = Rudolf | doi = 10.1007/978-3-7091-7551-4 | title = Computer Algebra: Symbolic and Algebraic Computation | series = Computing Supplementa | volume = 4 | year = 1983 | isbn = 978-3-211-81776-6 | s2cid = 5221892 | editor2-last = Collins | editor3-last = Loos | editor4-last = Albrecht | url-access = registration | url = https://archive.org/details/computeralgebras0000unse }}
{{Computer algebra systems}}
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