Algebra

For the kind of algebraic structure, seeAlgebra over a field.For other uses, seeAlgebra (disambiguation).

Elementary algebrais interested inpolynomial equationsand seeks to discover which valuessolve them.

Abstract algebrastudiesalgebraic structures,like thering of integersgiven by the set ofintegers($\mathbb {Z}$) together withoperationsofaddition($+$) andmultiplication($\times$).

Algebrais the branch ofmathematicsthat studiesalgebraic structuresand the manipulation of statements within those structures. It is a generalization ofarithmeticthat introducesvariablesandalgebraic operationsother than the standard arithmetic operations such asadditionandmultiplication.

Elementary algebrais the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables.Linear algebrais a closely related field investigating variables that appear in severallinear equations,so-calledsystems of linear equations.It tries to discover the values that solve all equations at the same time.

Abstract algebrastudies algebraic structures, which consist of asetofmathematical objectstogether with one or severalbinary operationsdefined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such asgroups,rings,andfields,based on the number of operations they use and thelaws they follow.Universal algebraconstitutes a further level of generalization that is not limited to binary operations and investigates more abstract patterns that characterize different classes of algebraic structures.

Algebraic methods were first studied in theancient periodto solve specific problems in fields likegeometry.Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They relied on verbal descriptions of problems and solutions until the 16th and 17th centuries, when a rigorous mathematical formalism was developed. In the mid-19th century, the scope of algebra broadened beyond atheory of equationsto cover diverse types of algebraic operations and algebraic structures. Algebra is relevant to many branches of mathematics, like geometry,topology,number theory,andcalculus,and other fields of inquiry, likelogicand theempirical sciences.

Definition and etymology

Algebra is the branch of mathematics that studiesalgebraic operationsandalgebraic structures.An algebraic structure is a non-emptysetofmathematical objects,such as thereal numbers,together with algebraic operations defined on that set, such asadditionandmultiplication.Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it studies the use ofvariablesinequationsand how to manipulate these equations.

Algebra is often understood as a generalization ofarithmetic.Arithmetic studies arithmetic operations, like addition,subtraction,multiplication, anddivision,in a specific domain of numbers, like the real numbers.Elementary algebraconstitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in the form of variables in addition to numbers.A higher level of abstraction is achieved in abstract algebra, which is not limited to a specific domain and studies different classes of algebraic structures, likegroupsandrings.These algebraic structures are not restricted to typical arithmetic operations and cover other binary operations besides them.Universal algebra is still more abstract in that it is not limited to binary operations and not interested in specific classes of algebraic structures but investigates the characteristics of algebraic structures in general.

The wordalgebracomes from the title ofal-Khwarizmi's bookAl-Jabr.

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.When used as a countable noun, an algebra is aspecific type of algebraic structurethat involves avector spaceequipped witha certain type of binary operation.Depending on the context, "algebra" can also refer to other algebraic structures, like aLie algebraor anassociative algebra.

The wordalgebracomes from the Arabic termالجبر(al-jabr), which originally referred to the surgical treatment ofbonesetting.In the 9th century, the term received a mathematical meaning when the Persian mathematicianMuhammad ibn Musa al-Khwarizmiemployed it to describe a method of solving equations and used it as the title of a treatise on algebra, also known by the nameThe Compendious Book on Calculation by Completion and Balancing.The word entered the English language in the 16th century from Italian, Spanish, and medieval Latin.Initially, the meaning of the term was restricted to thetheory of equations,that is, to the art of manipulatingpolynomial equationsin view of solving them. This changed in the course of the 19th centurywhen the scope of algebra broadened to cover the study of diverse types of algebraic operations and algebraic structures together with their underlying axioms.

Major branches

Elementary algebra

Main article:Elementary algebra

Algebraic expression notation:
1 – power (exponent)
2 – coefficient
3 – term
4 – operator
5 – constant term
$x$$y$$c$– variables/constants

Elementary algebra, also referred to as school algebra, college algebra, and classical algebra,is the oldest and most basic form of algebra. It is a generalization ofarithmeticthat relies on the use ofvariablesand examines how mathematicalstatementsmay be transformed.

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using arithmetic operations likeaddition,subtraction,multiplication,anddivision.For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in$2+5=7$.

Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables aresymbolsfor unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, theequation$2\times 3=3\times 2$belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combinations of numbers, like theprinciple of commutativityexpressed in the equation$a\times b=b\times a$.

Algebraic expressionsare formed by using arithmetic operations to combine variables and numbers. By convention, the lowercase letters$x$,$y$and$z$represent variables. In some cases, subscripts are added to distinguish variables, as in$x_{1}$,$x_{2}$,and$x_{3}$.The lowercase letters$a$,$b$,and$c$are usually used forconstantsandcoefficients.For example, the expression$5x+3$is an algebraic expression created by multiplying the number 5 with the variable$x$and adding the number 3 to the result. Other examples of algebraic expressions are$32xyz$and$64x_{1}^{2}+7x_{2}-13$.

Algebraic expressions are used to construct statements that relate two expressions to one another. An equation is a statement formed by comparing two expressions with anequals sign($=$), as in$5x^{2}+6x=3y+4$.Inequationsare formed with symbols like theless-than sign($<$), thegreater-than sign($>$), and the inequality sign ($\neq$). Unlike mere expressions, statements can be true or false and their truth value usually depends on the values of the variables. For example, the statement$x^{2}=4$is true if$x$is either 2 or −2 and false otherwise.Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, like the equation$2x+5x=7x$.Conditional equations are only true for some values. For example, the equation$x+4=9$is only true if$x$is 5.

The main objective of elementary algebra is to determine for which values a statement is true. Techniques to transform and manipulate statements are used to achieve this. A key principle guiding this process is that whatever operation is applied to one side of an equation also needs to be done to the other side of the equation. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side of the equation to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known assolving the equationfor that variable. For example, the equation$x-7=4$can be solved for$x$by adding 7 to both sides, which isolates$x$on the left side and results in the equation$x=11$.

There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression$7x-3x$can be replaced with the expression$4x$.Factorizationis used to rewrite an expression as a product of several factors. This technique is common forpolynomialsto determine for which values the expression iszero.For example, the polynomial$x^{2}-3x-10$can be factorized as$(x+2)(x-5)$.The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if$x$is either −2 or 5.For statements with several variables,substitutionis a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that$y=3x$then one can simplify the expression$7xy$to arrive at$21x^{2}$.In a similar way, if one knows the exact value of one variable one may be able to use it to determine the value of other variables.

Algebraic equations can be used to describe geometric figures. All values for$x$and$y$that solve the equation are interpreted as points and drawn as a red line.

Elementary algebra has applications in many branches of mathematics, the sciences, business, and everyday life.An important application in the field ofgeometryconcerns the use of algebraic equations to describegeometric figuresin the form of agraph.To do so, the different variables in the equation are interpreted ascoordinatesand the values that solve the equation are interpreted as points of the graph. For example, if$x$is set to zero in the equation$y=0.5x-1$then$y$has to be −1 for the equation to be true. This means that the$x$-$y$-pair$(0,-1)$is part of the graph of the equation. The$x$-$y$-pair$(0,7)$,by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of all$x$-$y$-pairs that solve the equation.

Linear algebra

Main article:Linear algebra

Linear algebra employs the methods of elementary algebra to studysystems of linear equations.Anequation is linearif no variable is multiplied with another variable and no operations likeexponentiation,extraction ofroots,andlogarithmare applied to variables. For example, the equations$0.25x-4=y$and$x_{1}-7x_{2}+3x_{3}=0$are linear while the equations$x^{2}=y$and$3x_{1}x_{2}+15=0$arenon-linear.Several equations form a system of equations if they all rely on the same set of variables.

Systems of linear equations are often expressed throughmatricesandvectorsto represent the whole system in a single equation. This can be done by moving the variables to the left side of each equation and moving the constant terms to the right side. The system is then expressed by formulating a matrix that contains all thecoefficientsof the equations andmultiplyingit with thecolumn vectormade up of the variables.For example, the system of equations

(a)$9x_{1}+3x_{2}-13x_{3}=0$

(b)$2.3x_{1}+7x_{3}=9$

(c)$-5x_{1}-17x_{2}=-3$

can be written as

$${\begin{bmatrix}9&3&-13\\2.3&0&7\\-5&-17&0\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}0\\9\\-3\end{bmatrix}}$$

Like elementary algebra, linear algebra is interested in manipulating and transforming equations to solve them. It goes beyond elementary algebra by dealing with several equations at once and looking for the values for which all equations are true at the same time. For example, if the system is made of the two equations$3x_{1}-x_{2}=0$and$x_{1}+x_{2}=8$then using the values 1 and 3 for$x_{1}$and$x_{2}$does not solve the system of equations because it only solves the first but not the second equation.

Two central questions in linear algebra are whether a system of equations has any solutions and, if so, whether it has a unique solution. A system of equations that has solutions is calledconsistent.This is the case if the equations do not contradict each other. If two or more equations contradict each other, the system of equations is inconsistent and has no solutions. For example, the equations$x_{1}-3x_{2}=0$and$x_{1}-3x_{2}=7$contradict each other since no values of$x_{1}$and$x_{2}$exist that solve both equations at the same time.

Whether a consistent system of equations has a unique solution depends on the number of variables and the number ofindependent equations.Several equations are independent of each other if they do not provide the same information and cannot be derived from each other. A unique solution exists if the number of variables is the same as the number of independent equations.Underdetermined systems,by contrast, have more variables than equations and have an infinite number of solutions if they are consistent.

Linear equations with two variables can be interpreted geometrically as lines. The solution of a system of linear equations is where the lines intersect.

Many of the techniques employed in elementary algebra to solve equations are also applied in linear algebra. The substitution method starts with one equation and isolates one variable in it. It proceeds to the next equation and replaces the isolated variable with the found expression, thereby reducing the number of unknown variables by one. It applies the same process again to this and the remaining equations until the values of all variables are determined.The elimination method creates a new equation by adding one equation to another equation. This way, it is possible to eliminate one variable that appears in both equations. For a system that contains the equations$-x+7y=3$and$2x-7y=10$,it is possible to eliminate$y$by adding the first to the second equation, thereby revealing that$x$is 13. In some cases, the equation has to be multiplied by a constant before adding it to another equation.Many advanced techniques implement algorithms based on matrix calculations, such asCramer's rule,theGauss–Jordan elimination,andLU Decomposition.

On a geometric level, systems of equations can be interpreted as geometric figures. For systems that have two variables, each equation represents alineintwo-dimensional space.The point where the two lines intersect is the solution. For inconsistent systems, the two lines run parallel, meaning that there is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to graphically look for solutions by plotting the equations and determining where they intersect.The same principles also apply to systems of equations with more variables, with the difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond toplanesinthree-dimensional spaceand the points where all planes intersect solve the system of equations.

Abstract algebra

Main article:Abstract algebra

Abstract algebra, also called modern algebra,studies different types ofalgebraic structures.An algebraic structure is a framework for understandingoperationsonmathematical objects,like the addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such asgroups,rings,andfields.

Many algebraic structures rely on binary operations, which take two objects as input and combine them into a single object as output, like addition and multiplication do.

On a formal level, an algebraic structure is asetof mathematical objects, called the underlying set, together with one or several operations.Abstract algebra usually restricts itself tobinary operationsthat take any two objects from the underlying set as inputs and map them to another object from this set as output.For example, the algebraic structure$\langle \mathbb {N},+\rangle$has thenatural numbersas the underlying set and addition as its binary operation.The underlying set can contain mathematical objects other than numbers and the operations are not restricted to regular arithmetic operations.For instance, the underlying set of thesymmetry groupof a geometric object is made up of thegeometric transformations,such asrotations,under which the object remainsunchanged.Its binary operation isfunction composition,which takes two transformations as input and has the transformation resulting from applying the first transformation followed by the second as its output.

Abstract algebra classifies algebraic structures based on the laws oraxiomsthat its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation isassociativeand has anidentity elementandinverse elements.An operationis associative if the order of several applications does not matter, i.e., if$(a\circ b)\circ c$is the same as$a\circ (b\circ c)$for all elements. An operation has an identity element or a neutral element if one elementeexists that does not change the value of any other element, i.e., if$a\circ e=e\circ a=a$.An operation admits inverse elements if for any element$a$there exists a reciprocal element$a^{-1}$that reverses its effects. If an element is linked to its inverse then the result is the neutral elemente,expressed formally as$a\circ a^{-1}=a^{-1}\circ a=e$.Every algebraic structure that fulfills these requirements is a group.For example,$\langle \mathbb {Z},+\rangle$is a group formed by the set ofintegerstogether with the operation of addition. The neutral element is 0 and the inverse element of any number$a$is$-a$.The natural numbers, by contrast, do not form a group since they contain only positive numbers and therefore lack inverse elements.Group theoryis the subdiscipline of abstract algebra studying groups.

Diagram of relations between some algebraic structures

A ring is an algebraic structure with two operations ($\circ$and$\star$) that work similarly to addition and multiplication. All the requirements of groups also apply to the first operation: it is associative and has an identity element and inverse elements. Additionally, it iscommutative,meaning that$a\circ b=b\circ a$is true for all elements. The axiom ofdistributivitygoverns how the two operations interact with each other. It states that$a\star (b\circ c)=(a\star b)\circ (a\star c)$and$(b\circ c)\star a=(b\star a)\circ (c\star a)$.Thering of integersis the ring denoted by$\langle \mathbb {Z},+,\times \rangle$.A ring becomes a field if both operations follow the axioms of associativity, commutativity, and distributivity and if both operations have an identity element and inverse elements.The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of$7$is${\tfrac {1}{7}}$,which is not part of the integers. Therational numbers,thereal numbers,and thecomplex numberseach form a field with the operations addition and multiplication.

Besides groups, rings, and fields, there are many other algebraic structures studied by abstract algebra. They includemagmas,semigroups,monoids,abelian groups,commutative rings,modules,lattices,vector spaces,andalgebras over a field.They differ from each other in regard to the types of objects they describe and the requirements that their operations fulfill. Many of them are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements.For example, a magma becomes a semigroup if its operation is associative.

Universal algebra

Main article:Universal algebra

Universal algebra is the study of algebraic structures in general. It is a generalization of abstract algebra that is not limited to binary operations and allows operations with more inputs as well, such asternary operations.Universal algebra is not interested in the specific elements that make up the underlying sets and instead investigates what structural features different algebraic structures have in common.One of those structural features concerns theidentitiesthat are true in different algebraic structures. In this context, an identity is auniversalequation or an equation that is true for all elements of the underlying set. For example, commutativity is a universal equation that states that$a\circ b$is identical to$b\circ a$for all elements.Two algebraic structures that share all their identities are said to belong to the samevariety.For instance, the ring of integers and thering of polynomialsform part of the same variety because they have the same identities, such as commutativity and associativity. The field of rational numbers, by contrast, does not belong to this variety since it has additional identities, such as the existence of multiplicative inverses.

Besides identities, universal algebra is also interested in structural features associated withquasi-identities.A quasi-identity is an identity that only needs to be present under certain conditions.It is a generalization of identity in the sense that every identity is a quasi-identity but not every quasi-identity is an identity. Algebraic structures that share all their quasi-identities have certain structural characteristics in common, which is expressed by stating that they belong to the samequasivariety.

Homomorphismsare a tool in universal algebra to examine structural features by comparing two algebraic structures.A homomorphism is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form$\langle A,\circ \rangle$and$\langle B,\star \rangle$then the function$h:A\to B$is a homomorphism if it fulfills the following requirement:$h(x\circ y)=h(x)\star h(y)$.The existence of a homomorphism reveals that the operation$\star$in the second algebraic structure plays the same role as the operation$\circ$does in the first algebraic structure.Isomorphismsare a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is abijectivehomomorphism, meaning that it establishes a one-to-one relationship between the elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure.

Subalgebrasrestrict their operations to a subset of the underlying set of the original algebraic structure.

Another tool of comparison is the relation between an algebraic structure and itssubalgebra.If$\langle A,\circ \rangle$is a subalgebra of$\langle B,\circ \rangle$then the set$A$is asubsetof$B$.A subalgebra has to use the same operations as the algebraic structureand they have to follow the same axioms. This includes the requirement that all operations in the subalgebra areclosedin$A$,meaning that they only produce elements that belong to$A$.For example, the set ofeven integerstogether with addition is a subalgebra of the full set of integers together with addition. This is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra since adding two odd numbers produces an even number, which is not part of the chosen subset.

History

Main articles:History of algebraandTimeline of algebra

TheRhind Papyrusfromancient Egypt,dated around 1650 BCE, is one of the earliest documents discussing algebraic problems.

The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities. These developments happened in the ancient period in diverse regions such asBabylonia,Egypt,Greece,China,andIndia.One of the earliest documents is theRhind Papyrusfrom ancient Egypt, which was written around 1650 BCEand discusses how to solvelinear equations,as expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?" Babylonian clay tablets from around the same time explain methods to solve linear andquadratic polynomial equations,such as the method ofcompleting the square.

Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main interest was geometry rather than algebra, but they employed algebraic methods to solve geometric problems. For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified inPythagoras' formulation of thedifference of two squaresmethod and later inEuclid'sElements.In the 3rd century CE,Diophantusprovided a detailed treatment of how to solve algebraic equations in a series of books calledArithmetica.He was the first to experiment with symbolic notation to express polynomials.In ancient China, the bookThe Nine Chapters on the Mathematical Artexplored various techniques for solving algebraic equations, including the use of matrix-like constructs.

Al-Khwarizmi'sThe Compendious Book on Calculation by Completion and Balancingprovided a general account of how linear and quadratic equations can be solved through the methods of "reducing" and "balancing".

It is controversial to what extent these early developments should be considered part of algebra proper rather than precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications.This changed with the Persian mathematicianal-Khwarizmi,who published hisThe Compendious Book on Calculation by Completion and Balancingin 825 CE. It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides.Other influential contributions to algebra came from the Arab mathematicianThābit ibn Qurrain the 9th century and the Persian mathematicianOmar Khayyamin the 11th and 12th centuries.

In India,Brahmaguptainvestigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his other innovations were the use ofzeroand negative numbers in algebraic equations.The Indian mathematiciansMahāvīrain the 9th century andBhāskara IIin the 12th century further refined Brahmagupta's methods and concepts.In 1247, the Chinese mathematicianQin Jiushaowrote theMathematical Treatise in Nine Sections,which includesan algorithmfor thenumerical evaluation of polynomials,including polynomials of higher degrees.

François VièteandRené Descartesinvented a symbolic notation to express equations in an abstract and concise manner.

The Italian mathematicianFibonaccibrought al-Khwarizmi's ideas and techniques to Europe in books like hisLiber Abaci.In 1545, the Italian polymathGerolamo Cardanopublished his bookArs Magna,which covered many topics in algebra and was the first to present general methods for solvingcubicandquartic equations.In the 16th and 17th centuries, the French mathematiciansFrançois VièteandRené Descartesintroduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner. Their predecessors had relied on verbal descriptions of problems and solutions.Some historians see this development as a key turning point in the history of algebra and consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation.

Garrett Birkhoffdeveloped many of the foundational concepts of universal algebra.

Many attempts in the 17th and 18th centuries to find general solutionsto polynomials of degree five and higher failed.At the end of the 18th century, the German mathematicianCarl Friedrich Gaussproved thefundamental theorem of algebra,which describes the existence ofzerosof polynomials of any degree without providing a general solution.At the beginning of the 19th century, the Italian mathematicianPaolo Ruffiniand the Norwegian mathematicianNiels Henrik Abelwereable to showthat no general solution exists for polynomials of degree five and higher.In response to and shortly after their findings, the French mathematicianÉvariste Galoisdeveloped what came later to be known asGalois theory,which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation ofgroup theory.Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory.

Starting in the mid-19th century, interest in algebra shifted from the study of polynomials associated with elementary algebra towards a more general inquiry into algebraic structures, marking the emergence ofabstract algebra.This approach explored the axiomatic basis of arbitrary algebraic operations.The invention of new algebraic systems based on different operations and elements accompanied this development, such asBoolean algebra,vector algebra,andmatrix algebra.Influential early developments in abstract algebra were made by the German mathematiciansDavid Hilbert,Ernst Steinitz,Emmy Noether,andEmil Artin.They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, such as groups, rings, and fields.The idea of the even more general approach associated with universal algebra was conceived by the English mathematicianAlfred North Whiteheadin his 1898 bookA Treatise on Universal Algebra.Starting in the 1930s, the American mathematicianGarrett Birkhoffexpanded these ideas and developed many of the foundational concepts of this field.Closely related developments were the formulation ofmodel theory,category theory,topological algebra,homological algebra,Lie algebras,free algebras,andhomology groups.

Applications

The influence of algebra is wide reaching and includes many branches of mathematics as well as the empirical sciences. Algebraic notation and algebraic principles play a key role inphysicsand related disciplines to expressscientific lawsand solve equations.They are also used in fields likeengineering,economics,computer science,andgeographyto express relationships, solve problems, and model systems.

Other branches of mathematics

The algebraization of mathematics is the process of applying algebraic methods and principles to otherbranches of mathematics.This involves employing symbols in the form of variables to express mathematical insights on a more general level and the use of algebra to develop mathematical models describing how objects interact and relate to each other.This is possible because the abstract patterns studied by algebra have many concrete applications in fields likegeometry,topology,number theory,andcalculus.

The algebraic equation$x^{2}+y^{2}+z^{2}=1$describes asphereat theoriginwith a radius of 1.

Geometry is interested in geometric figures, which can be described with algebraic statements. For example, the equation$y=3x-7$describes a line in two-dimensional space while the equation$x^{2}+y^{2}+z^{2}=1$corresponds to aspherein three-dimensional space. Of special interest toalgebraic geometryarealgebraic varieties,which are solutions tosystems of polynomial equationsthat can be used to describe more complex geometric figures.Algebraic reasoning can also be used to solve geometric problems. For example, one can determine whether and where the line described by$y=x+1$intersects with the circle described by$x^{2}+y^{2}=25$by solving the system of equations made up of these two equations.Topology studies the properties of geometric figures ortopological spacesthat are preserved under operations ofcontinuous deformation.Algebraic topologyrelies on algebraic theories likegroup theoryto classify topological spaces. For example,homotopy groupsclassify topological spaces based on the existence ofloopsorholesin them.Number theory is concerned with the properties of and relations between integers.Algebraic number theoryapplies algebraic methods to this field of inquiry, for example, by using algebraic expressions to describe laws, such asFermat's Last Theorem,and by analyzing how numbers form algebraic structures, such as thering of integers.The insights of algebra are also relevant to calculus, which utilizes mathematical expressions to examinerates of changeandaccumulation.It relies on algebra to understand how these expressions can be transformed and what role variables play in them.

Logic

Logicis the study of correct reasoning.Algebraic logicemploys algebraic methods to describe and analyze the structures and patterns that underlielogical reasoning.One part of it is interested in understanding the mathematical structures themselves without regard for the concrete consequences they have on the activity of drawinginferences.Another part investigates how the problems of logic can be expressed in the language of algebra and how the insights obtained through algebraic analysis affect logic.

Boolean algebrais an influential device in algebraic logic to describepropositional logic.Propositionsare statements that can be true or false.Propositional logic useslogical connectivesto combine two propositions to form a complex proposition. For example, the connective "if... then "can be used to combine the propositions" it rains "and" the streets are wet "to form the complex proposition" if it rains then the streets are wet ". Propositional logic is interested in how thetruth valueof a complex proposition depends on the truth values of its constituents.With Boolean algebra, this problem can be addressed by interpreting truth values as numbers: 0 corresponds to false and 1 corresponds to true. Logical connectives are understood as binary operations that take two numbers as input and return the output that corresponds to the truth value of the complex proposition.Algebraic logic is also interested in how more complexsystems of logiccan be described through algebraic structures and which varieties and quasivarities these algebraic structures belong to.

Education

Main article:Mathematics education

Balance scalesare used in algebra education to help students understand how equations can be transformed to determine unknown values.

Algebra education mostly focuses on elementary algebra, which is one of the reasons why it is referred to as school algebra. It is usually not introduced untilsecondary educationsince it requires mastery of the fundamentals of arithmetic while posing new cognitive challenges associated abstract reasoning and generalization.It aims to familiarize students with the abstract side of mathematics by helping them understand mathematical symbolism, for example, how variables can be used to represent unknown quantities. An additional difficulty for students lies in the fact that, unlike arithmetic calculations, algebraic expressions often cannot be directly solved. Instead, students need to learn how to transform them according to certain laws, often with the goal of determining an unknown quantity.

The use ofbalance scalesto represent equations is a pictorial approach to introduce students to the basic problems of algebra. The mass of some objects on the scale is unknown and represents variables. Solving an equation corresponds to adding and removing objects on both sides in such a way that the sides stay in balance until the only object remaining on one side is the object of unknown mass.The use ofword problemsis another tool to show how algebra is applied to real-life situations. For example, students may be presented with a situation in which Naomi's brother has twice as many apples as Naomi. Given that both together have twelve apples, students are then asked to find an algebraic equation that describes this situation ($2x+x=12$) and to determine how many apples Naomi has ($x=4$).

See also

• Algebra over a set
• Algebra tile
• Algebraic combinatorics
• C*-algebra
• Composition algebra
• Computer algebra
• Exterior algebra
• F-algebra
• F-coalgebra
• Heyting algebra
• Hopf algebra
• Non-associative algebra
• Outline of algebra
• Relational algebra
• Sigma-algebra
• Symmetric algebra
• T-algebra
• Tensor algebra

References

Sources

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