User:IntegralPython/sandbox: Difference between revisions

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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>

Algebra isalso later extended to considerconsiders 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.

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>