User:IntegralPython/sandbox: Difference between revisions

Revision as of 12:57, 25 October 2018

Revision as of 12:59, 25 October 2018

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The most simple parts of algebra begin with computations similar to those of [[arithmetic]] but with variables that take on the properties of numbers.<ref name=citeboyer /> This allows proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]

:<math>ax^2+bx+c=0,</math>

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>

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects 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.

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Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of [[real number| real]] or [[complex number]]s. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, 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 in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of complex geometric structures such as [[manifoldconstructible number]]s and [[singularity theory|singularities]]. Reducing properties of geometric structures into algebraic structures has created subjects such as [[algebraic geometry]], [[geometric algebra]], 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]].