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{{collapse top| What is Algebra}}
==What is algebra?==
Algebra is
The most simple parts of algebra begin with computations similar to those of [[arithmetic]] but with variables standing for numbers.<ref name=citeboyer /> This allowed proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
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where <math>a, b, c</math> are any given numbers (except that <math>a</math> cannot be <math>0</math>), the [[quadratic formula]] can be used to find the two unique values of the unknown quantity <math>x</math> which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the [[quadratic equation]] above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.<ref>{{cite book |last=Gattengo |first=Caleb |year=2010 |title=The Common Sense of Teaching Mathematics |publisher=Educational Solutions Inc. |isbn=978-0878252206 }}</ref>
Because algebra is simply the manipulation of entities, there is no rule that states that only numbers and variables that stand for numbers are allowed. In this way, algebra is extended to consider entities that do not stand for just one number, such as [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra
Even though algebra had already expanded into manipulations of many numbers in the defined topics above, it is possible to define entities that are unlike any familiar numbers. These entities are created using only their properties, and involve strict definitions that create a set of entities that work together with their properties. The entities form [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. Abstract algebra is the study of these entities and more.<ref>http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018</ref>
In geometry, algebra can be used to help in the manipulation of geometric properties; reducing properties of geometric structures into algebraic structures has created subjects such as [[algebraic geometry]] and [[algebraic topology]].
Today, the study of algebra includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|publisher=|accessdate=5 October 2014}}</ref> where none of the first level areas (two digit entries) is called ''algebra''. Algebra instead includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebra]]s, 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used in 14-[[Algebraic geometry]] and 11-[[Number theory]] via [[algebraic number theory]].
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