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{{collapse top| What is Algebra}}▼
==What is algebra?==▼
Algebra is a multifaceted branch of mathematics with many parts that are vastly different from others. Essentially, algebra is manipulation of symbols based on given properties about them.<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.▼
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]]▼
:<math>ax^2+bx+c=0,</math>▼
<math>a, b, c</math> can be any numbers whatsoever (except that <math>a</math> cannot be <math>0</math>), and the [[quadratic formula]] can be used to quickly and easily find the 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, while others form subjects such as [[linear 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 short, the study of algebra involves any set of items which share properties. As long as it is possible to distill the similarities into similar sets that relate to one another in different ways, it is a part of algebra.▼
Today, algebra has grown until it 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''. Today algebra 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 extensively in 11-[[Number theory]] and 14-[[Algebraic geometry]].▼
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==Antiassociative algebra==
An algebra antiassociative if (xy)z = -x(yz) for every case of x,y, and z.
https://books.google.com/books?id=_PEWt18egGgC&pg=PA235&lpg=PA235&dq=%22antiassociative%22+algebra+aplications&source=bl&ots=Atxm0cdUVs&sig=OQjjF3ig6NYCQwP6O9P8fLgwSDE&hl=en&sa=X&ved=2ahUKEwix94P66LPdAhVIu1MKHckzBNQQ6AEwCHoECAYQAQ#v=onepage&q=%22antiassociative%22%20algebra%20aplications&f=false
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{{collapse top | Ugandan knuckles}}
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The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’s ‘Ugandan Knuckles’ Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>
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▲{{collapse top| What is Algebra}}
▲==What is algebra?==
▲Algebra is a multifaceted branch of mathematics with many parts that are vastly different from others. Essentially, algebra is manipulation of symbols based on given properties about them.<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.
▲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]]
▲:<math>ax^2+bx+c=0,</math>
▲<math>a, b, c</math> can be any numbers whatsoever (except that <math>a</math> cannot be <math>0</math>), and the [[quadratic formula]] can be used to quickly and easily find the 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, while others form subjects such as [[linear 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 short, the study of algebra involves any set of items which share properties. As long as it is possible to distill the similarities into similar sets that relate to one another in different ways, it is a part of algebra.
▲Today, algebra has grown until it 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''. Today algebra 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 extensively in 11-[[Number theory]] and 14-[[Algebraic geometry]].
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