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**Algebra** is the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.

Elementary algebra is 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 utilizes different methods of transforming equations to isolate variables. Linear algebra is a closely related field investigating variables that appear in several linear equations, so-called systems of linear equations. It tries to discover the values that solve all equations at the same time.

Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several binary operations defined 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 as groups, rings, and fields, based on the number of operations they use and the laws they follow. Universal algebra constitutes 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 the ancient period to solve specific problems in fields like geometry. 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 a theory of equations to cover diverse types of algebraic operations and algebraic structures. Algebra is relevant to many branches of mathematics, like geometry, topology, number theory, and calculus, and other fields of inquiry, like logic and the empirical sciences.

Algebra is the branch of mathematics that studies algebraic operations^{[a]} and algebraic structures.^{[2]} An algebraic structure is a non-empty set of mathematical objects, such as the real numbers, together with algebraic operations defined on that set, such as addition and multiplication.^{[3]} Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it studies the use of variables in equations and how to manipulate these equations.^{[4]}^{[b]}

Algebra is often understood as a generalization of arithmetic.^{[8]} Arithmetic studies arithmetic operations, like addition, subtraction, multiplication, and division, in a specific domain of numbers, like the real numbers.^{[9]} Elementary algebra constitutes 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.^{[10]} 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, like groups and rings. These algebraic structures are not restricted to typical arithmetic operations and cover other binary operations besides them.^{[11]} 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.^{[12]}

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.^{[14]} When used as a countable noun, an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation.^{[15]} Depending on the context, "algebra" can also refer to other algebraic structures, like a Lie algebra or an associative algebra.^{[16]}

The word *algebra* comes from the Arabic term الجبر (*al-jabr*) and originally referred to the surgical treatment of bonesetting. In the 9th century, the term received a mathematical meaning when the Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to describe a method of solving equations and used it as the title of a treatise on algebra, also known by the name *The Compendious Book on Calculation by Completion and Balancing*. The word entered the English language in the 16th century from Italian, Spanish, and medieval Latin.^{[17]} Initially, the meaning of the term was restricted to the theory of equations, that is, to the art of manipulating polynomial equations in view of solving them. This changed in the course of the 19th century^{[c]} when the scope of algebra broadened to cover the study of diverse types of algebraic operations and algebraic structures together with their underlying axioms.^{[20]}

Elementary algebra, also referred to as school algebra, college algebra, and classical algebra,^{[21]} is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on the use of variables and examines how mathematical statements may be transformed.^{[22]}

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using arithmetic operations like addition, subtraction, multiplication, and division. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in .^{[9]}

Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables are symbols for 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, the equation 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 the principle of commutativity expressed in the equation .^{[22]}

Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention, the lowercase letters , and represent variables. In some cases, subscripts are added to distinguish variables, as in , , and . The lowercase letters , , and are usually used for constants and coefficients.^{[d]} For example, the expression is an algebraic expression created by multiplying the number 5 with the variable and adding the number 3 to the result. Other examples of algebraic expressions are and .^{[23]}

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 an equals sign ( ), as in . Inequations are formed with symbols like the less-than sign ( ), the greater-than sign ( ), and the inequality sign ( ). 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 is true if is either 2 or −2 and false otherwise.^{[24]} 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 . Conditional equations are only true for some values. For example, the equation is only true if is 5.^{[25]}

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 as solving the equation for that variable. For example, the equation can be solved for by adding 7 to both sides, which isolates on the left side and results in the equation .^{[26]}

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 can be replaced with the expression .^{[27]} Factorization is used to rewrite an expression as a product of several factors. This technique is common for polynomials^{[e]} to determine for which values the expression is zero. For example, the polynomial can be factorized as . The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if is either −2 or 5.^{[29]} For statements with several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that then one can simplify the expression to arrive at . 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.^{[30]}

Elementary algebra has applications in many branches of mathematics, the sciences, business, and everyday life.^{[31]} An important application in the field of geometry concerns the use of algebraic equations to describe geometric figures in the form of a graph. To do so, the different variables in the equation are interpreted as coordinates and the values that solve the equation are interpreted as points of the graph. For example, if is set to zero in the equation then has to be −1 for the equation to be true. This means that the - -pair is part of the graph of the equation. The - -pair , by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of all - -pairs that solve the equation.^{[32]}

Linear algebra employs the methods of elementary algebra to study systems of linear equations.^{[33]} An equation is linear if no variable is multiplied with another variable and no operations like exponentiation, extraction of roots, and logarithm are applied to variables. For example, the equations and are linear while the equations and are non-linear. Several equations form a system of equations if they all rely on the same set of variables.^{[34]}

Systems of linear equations are often expressed through matrices^{[f]} and vectors^{[g]} to 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 the coefficients of the equations and multiplying it with the column vector made up of the variables.^{[35]} For example, the system of equations

- (a)
- (b)
- (c)

can be written as

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 and then using the values 1 and 3 for and does not solve the system of equations because it only solves the first but not the second equation.^{[36]}

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 called consistent. 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 and contradict each other since no values of and exist that solve both equations at the same time.^{[37]}

Whether a consistent system of equations has a unique solution depends on the number of variables and the number of independent 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.^{[38]}

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.^{[39]} 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 and , it is possible to eliminate by adding the first to the second equation, thereby revealing that is 13. In some cases, the equation has to be multiplied by a constant before adding it to another equation.^{[40]} Many advanced techniques implement algorithms based on matrix calculations, such as Cramer's rule, the Gauss–Jordan elimination, and LU Decomposition.^{[41]}

On a geometric level, systems of equations can be interpreted as geometric figures. For systems that have two variables, each equation represents a line in two-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.^{[42]} 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 to planes in three-dimensional space and the points where all planes intersect solve the system of equations.^{[43]}

Abstract algebra, also called modern algebra,^{[44]} studies different types of algebraic structures. An algebraic structure is a framework for understanding operations on mathematical 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 as groups, rings, and fields.^{[45]}

On a formal level, an algebraic structure is a set^{[h]} of mathematical objects, called the underlying set, together with one or several operations.^{[i]} Abstract algebra usually restricts itself to binary operations^{[j]} that take any two objects from the underlying set as inputs and map them to another object from this set as output.^{[49]} For example, the algebraic structure has the natural numbers as the underlying set and addition as its binary operation.^{[47]} The underlying set can contain mathematical objects other than numbers and the operations are not restricted to regular arithmetic operations.^{[50]} For instance, the underlying set of the symmetry group of a geometric object is made up of the geometric transformations, such as rotations, under which the object remains unchanged. Its binary operation is function composition, which takes two transformations as input and has the transformation resulting from applying the first transformation followed by the second as its output.^{[51]}

Abstract algebra classifies algebraic structures based on the laws or axioms that 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 is associative and has an identity element and inverse elements. An operation^{[k]} is associative if the order of several applications does not matter, i.e., if is the same as for all elements. An operation has an identity element or a neutral element if one element *e* exists that does not change the value of any other element, i.e., if . An operation admits inverse elements if for any element there exists a reciprocal element that reverses its effects. If an element is linked to its inverse then the result is the neutral element *e*, expressed formally as . Every algebraic structure that fulfills these requirements is a group.^{[52]} For example, is a group formed by the set of integers together with the operation of addition. The neutral element is 0 and the inverse element of any number is .^{[53]} The natural numbers, by contrast, do not form a group since they contain only positive numbers and therefore lack inverse elements.^{[54]} Group theory is the subdiscipline of abstract algebra studying groups.^{[55]}

A ring is an algebraic structure with two operations ( and ) 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 is commutative, meaning that is true for all elements. The axiom of distributivity governs how the two operations interact with each other. It states that and .^{[l]}^{[57]} The ring of integers is the ring denoted by .^{[58]} 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.^{[m]}^{[60]} The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of is , which is not part of the integers. The rational numbers, the real numbers, and the complex numbers each form a field with the operations addition and multiplication.^{[61]}

Besides groups, rings, and fields, there are many other algebraic structures studied by abstract algebra. They include magmas, semigroups, monoids, abelian groups, commutative rings, modules, lattices, vector spaces, and algebras 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.^{[62]} For example, a magma becomes a semigroup if its operation is associative.^{[63]}

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 as ternary 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.^{[64]} One of those structural features concerns the identities that are true in different algebraic structures. In this context, an identity is a universal equation or an equation that is true for all elements of the underlying set. For example, commutativity is a universal equation that states that is identical to for all elements.^{[65]} Two algebraic structures that share all their identities are said to belong to the same variety.^{[66]} For instance, the ring of integers and the ring of polynomials form 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.^{[67]}

Besides identities, universal algebra is also interested in structural features associated with quasi-identities. A quasi-identity is an identity that only needs to be present under certain conditions.^{[n]} 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 same quasivariety.^{[68]}

Homomorphisms are a tool in universal algebra to examine structural features by comparing two algebraic structures.^{[69]} 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 and then the function is a homomorphism if it fulfills the following requirement: . The existence of a homomorphism reveals that the operation in the second algebraic structure plays the same role as the operation does in the first algebraic structure.^{[70]} Isomorphisms are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a bijective homomorphism, 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.^{[71]}

Another tool of comparison is the relation between an algebraic structure and its subalgebra.^{[72]} If is a subalgebra of then the set is a subset of .^{[o]} A subalgebra has to use the same operations as the algebraic structure^{[p]} and they have to follow the same axioms. This includes the requirement that all operations in the subalgebra are closed in , meaning that they only produce elements that belong to .^{[72]} For example, the set of even integers together 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.^{[73]}

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 as Babylonia, Egypt, Greece, China, and India. One of the earliest documents is the Rhind Papyrus from ancient Egypt, which was written around 1650 BCE^{[q]} and discusses how to solve linear 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 and quadratic polynomial equations, such as the method of completing the square.^{[74]}

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 in Pythagoras' formulation of the difference of two squares method and later in Euclid's *Elements*.^{[75]} In the 3rd century BCE, Diophantus provided a detailed treatment of how to solve algebraic equations in a series of books called *Arithmetica*. He was the first to experiment with symbolic notation to express polynomials.^{[76]} In ancient China, the book *The Nine Chapters on the Mathematical Art* explored various techniques for solving algebraic equations, including the use of matrix-like constructs.^{[77]}

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.^{[78]} This changed with the Persian mathematician al-Khwarizmi,^{[r]} who published his *The Compendious Book on Calculation by Completion and Balancing* in 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.^{[80]} Other influential contributions to algebra came from the Arab mathematician Thābit ibn Qurra in the 9th century and the Persian mathematician Omar Khayyam in the 11th and 12th centuries.^{[81]}

In India, Brahmagupta investigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his other innovations were the use of zero and negative numbers in algebraic equations.^{[82]} The Indian mathematicians Mahāvīra in the 9th century and Bhāskara II in the 12th century further refined Brahmagupta's methods and concepts.^{[83]} In 1247, the Chinese mathematician Qin Jiushao wrote the *Mathematical Treatise in Nine Sections*, which includes an algorithm for the numerical evaluation of polynomials, including polynomials of higher degrees.^{[84]}

The Italian mathematician Fibonacci brought al-Khwarizmi's ideas and techniques to Europe in books like his *Liber Abaci*.^{[85]} In 1545, the Italian polymath Gerolamo Cardano published his book *Ars Magna*, which covered many topics in algebra and was the first to present general methods for solving cubic and quartic equations.^{[86]} In the 16th and 17th centuries, the French mathematicians François Viète and René Descartes introduced 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.^{[87]} 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.^{[88]}

Many attempts in the 17th and 18th centuries to find general solutions^{[s]} to polynomials of degree five and higher failed.^{[91]} At the end of the 18th century, the German mathematician Carl Friedrich Gauss proved the fundamental theorem of algebra, which describes the existence of zeros of polynomials of any degree without providing a general solution.^{[18]} At the beginning of the 19th century, the Italian mathematician Paolo Ruffini and the Norwegian mathematician Niels Henrik Abel were able to show that no general solution exists for polynomials of degree five and higher.^{[91]} In response to and shortly after their findings, the French mathematician Évariste Galois developed what came later to be known as Galois theory, which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation of group theory.^{[19]} Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory.^{[92]}

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 of abstract algebra. This approach explored the axiomatic basis of arbitrary algebraic operations.^{[93]} The invention of new algebraic systems based on different operations and elements accompanied this development, such as Boolean algebra, vector algebra, and matrix algebra.^{[94]} Influential early developments in abstract algebra were made by the German mathematicians David Hilbert, Ernst Steinitz, Emmy Noether, and Emil Artin. They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, such as groups, rings, and fields.^{[95]} The idea of the even more general approach associated with universal algebra was conceived by the English mathematician Alfred North Whitehead in his 1898 book *A Treatise on Universal Algebra*. Starting in the 1930s, the American mathematician Garrett Birkhoff expanded these ideas and developed many of the foundational concepts of this field.^{[96]} Closely related developments were the formulation of model theory, category theory, topological algebra, homological algebra, Lie algebras, free algebras, and homology groups.^{[97]}

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 in physics and related disciplines to express scientific laws and solve equations.^{[98]} They are also used in fields like engineering, economics, computer science, and geography to express relationships, solve problems, and model systems.^{[99]}

The algebraization of mathematics is the process of applying algebraic methods and principles to other branches 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.^{[100]} This is possible because the abstract patterns studied by algebra have many concrete applications in fields like geometry, topology, number theory, and calculus.^{[101]}

Geometry is interested in geometric figures, which can be described with algebraic statements. For example, the equation describes a line in two-dimensional space while the equation corresponds to a sphere in three-dimensional space. Of special interest to algebraic geometry are algebraic varieties,^{[t]} which are solutions to systems of polynomial equations that can be used to describe more complex geometric figures.^{[102]} Algebraic reasoning can also be used to solve geometric problems. For example, one can determine whether and where the line described by intersects with the circle described by by solving the system of equations made up of these two equations.^{[103]} Topology studies the properties of geometric figures or topological spaces that are preserved under operations of continuous deformation. Algebraic topology relies on algebraic theories like group theory to classify topological spaces. For example, homotopy groups classify topological spaces based on the existence of loops or holes in them.^{[104]} Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods to this field of inquiry, for example, by using algebraic expressions to describe laws, such as Fermat's Last Theorem, and by analyzing how numbers form algebraic structures, such as the ring of integers.^{[105]} The insights of algebra are also relevant to calculus, which utilizes mathematical expressions to examine rates of change and accumulation. It relies on algebra to understand how these expressions can be transformed and what role variables play in them.^{[106]}

Logic is the study of correct reasoning.^{[107]} Algebraic logic employs algebraic methods to describe and analyze the structures and patterns that underlie logical reasoning.^{[108]} One part of it is interested in understanding the mathematical structures themselves without regard for the concrete consequences they have on the activity of drawing inferences. 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.^{[109]}

Boolean algebra is an influential device in algebraic logic to describe propositional logic.^{[110]} Propositions are statements that can be true or false.^{[111]} Propositional logic uses logical connectives to 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 the truth value of a complex proposition depends on the truth values of its constituents.^{[112]} 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.^{[113]} Algebraic logic is also interested in how more complex systems of logic can be described through algebraic structures and which varieties and quasivarities these algebraic structures belong to.^{[114]}

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 until secondary education since it requires mastery of the fundamentals of arithmetic while posing new cognitive challenges associated abstract reasoning and generalization.^{[116]} 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.^{[117]}

The use of balance scales to 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.^{[115]} The use of word problems is 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 ( ) and to determine how many apples Naomi has ( ).^{[118]}

**^**When understood in the widest sense, an algebraic operation is a function from a Cartesian power of a set into that set, expressed formally as . Addition of real numbers is an example of an algebraic operations: it takes two numbers as input and produces one number as output. It has the form .^{[1]}**^**Algebra is covered by division 512 in the Dewey Decimal Classification^{[5]}and subclass QA 150-272.5 in the Library of Congress Classification.^{[6]}It encompasses several areas in the Mathematics Subject Classification.^{[7]}**^**These changes were in part triggered by discoveries that solved many of the older problems of algebra. For example, the proof of the fundamental theorem of algebra demonstrated the existence of complex solutions of polynomials^{[18]}and the introduction of Galois theory characterized the polynomials that have general solutions.^{[19]}**^**Constants represent fixed magnitudes that, unlike variables, cannot change.**^**A polynomial is an expression consisting of one or more terms that are added or subtracted from each other. Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive-integer power. Examples are and .^{[28]}**^**A matrix is a table of numbers, such as**^**A vector is an array of numbers or a matrix with only one column, such as**^**A set is an unordered collection of distinct elements, such as numbers, vectors, or other sets. Set theory describes the laws and properties of sets.^{[46]}**^**According to some definitions, algebraic structures include a distinguished element as an additional component, such as the identity element in the case of multiplication.^{[47]}**^**Some of the algebraic structures studied by abstract algebra include unary operation in addition to binary operations. For example, normed vector spaces have a norm, which is a unary operation often used to associate a vector with its length.^{[48]}**^**Symbols like and are often used in abstract algebra to represent any operation that may or may not resemble arithmetic operations.**^**Some definitions additionally require that the second operation is associative.^{[56]}**^**For the second operation, there is usually one element, corresponding to 0, that does not require an inverse element.^{[59]}**^**The conditions take the form of a Horn clause.**^**This means that all the elements of are also elements of but may contain elements that are not found in .**^**According to some definitions, it is also possible for a subalgebra to have fewer operations.^{[73]}**^**The exact date is disputed.**^**Some historians consider him the "father of algebra" while others reserve this title for Diophantus.^{[79]}**^**A general solution or a solution in radicals is a closed form algebraic equation that isolates the variable on one side. For example, the general solution to quadratic equations of the form is . The absence of general solutions does not mean that there are no numerical solutions.^{[89]}^{[90]}**^**Algebraic varieties studied in geometry are different from the more general varieties studied in universal algebra.

**^**Baranovich 2023, Lead Section**^**- Merzlyakov & Shirshov 2020, Lead Section
- Gilbert & Nicholson 2004, p. 4

**^**- Fiche & Hebuterne 2013, p. 326
- Merzlyakov & Shirshov 2020, § The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.
- Gilbert & Nicholson 2004, p. 4

**^**- Pratt 2022, Lead Section, § 1. Elementary Algebra, § 2. Abstract Algebra, § 3. Universal Algebra
- Merzlyakov & Shirshov 2020, § The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.

**^**Higham 2019, p. 296.**^**Library of Congress, p. 3.**^**zbMATH Open 2024.**^**- Maddocks 2008, p. 129
- Burgin 2022, p. 45

- ^
^{a}^{b} **^**- Maddocks 2008, pp. 129–130
- Pratt 2022, Lead Section, § 1. Elementary Algebra
- Wagner & Kieran 2018, p. 225

**^**- Maddocks 2008, pp. 131–132
- Pratt 2022, Lead Section, § 2. Abstract Algebra
- Wagner & Kieran 2018, p. 225

**^**- Pratt 2022, § 3. Universal Algebra
- Grillet 2007, p. 559

**^**- Cresswell 2010, p. 11
- OUP Staff
- Menini & Oystaeyen 2017, p. 722

**^**- Weisstein 2003, p. 46
- Renze & Weisstein
- Walz 2016, Algebra

**^**- Weisstein 2003, p. 46
- Renze & Weisstein
- Golan 1995, pp. 219–227

**^**EoM Staff 2017**^**- Cresswell 2010, p. 11
- OUP Staff
- Menini & Oystaeyen 2017, p. 722
- Hoad 1993, p. 10

- ^
^{a}^{b}- Tanton 2005, p. 10
- Kvasz 2006, p. 308
- Corry 2024, § The Fundamental Theorem of Algebra

- ^
^{a}^{b}- Kvasz 2006, pp. 314–345
- Merzlyakov & Shirshov 2020, § Historical Survey
- Corry 2024, § Galois Theory, § Applications of Group Theory

**^**- Tanton 2005, p. 10
- Corry 2024, § Structural Algebra
- Hazewinkel 1994, pp. 73–74

**^**- Renze & Weisstein
- Benson 2003, pp. 111–112

- ^
^{a}^{b}- Maddocks 2008, p. 129
- Berggren 2015, Lead Section
- Pratt 2022, § 1. Elementary Algebra
- Merzlyakov & Shirshov 2020, § 1. Historical Survey

**^**- Maddocks 2008, pp. 129–130
- Young 2010, p. 999
- Majewski 2004, p. 347
- Buthusiem & Toth 2020, pp. 24–28
- Pratt 2022, § 1. Elementary Algebra
- Sterling 2016, p. 13
- Sorell 2000, p. 19

**^**- Maddocks 2008, pp. 129–130
- Buthusiem & Toth 2020, pp. 24–28

**^****^**- Maddocks 2008, p. 130
- Buthusiem & Toth 2020, pp. 25–28
- Pratt 2022, § 1. Elementary Algebra
- Merzlyakov & Shirshov 2020, § 1. Historical Survey

**^**- Tan, Steeb & Hardy 2012, p. 306
- Lamagna 2019, p. 150

**^**Markushevich 2015.**^**- Buthusiem & Toth 2020, pp. 24–28
- Berggren 2015, § Algebraic Expressions, § Solving Algebraic Equations

**^**- Zill & Dewar 2011, p. 529
- Berggren 2015, § Solving Systems of Algebraic Equations
- McKeague 2014, p. 386

**^**- Maddocks 2008, pp. 130–131
- Walz 2016, Algebra

**^**- Maddocks 2008, pp. 130–131
- Rohde et al. 2012, p. 89
- Walz 2016, Algebra

**^**- Maddocks 2008, p. 131
- Barrera-Mora 2023, pp. ix, 1–2,

**^**- Anton & Rorres 2013, pp. 2–3
- Maddocks 2008, p. 131
- Voitsekhovskii 2011

**^**- Barrera-Mora 2023, pp. ix, 1, 12–13
- Young 2010, pp. 726–727
- Anton & Rorres 2013, pp. 32–34

**^**- Maddocks 2008, p. 131
- Andrilli & Hecker 2022, pp. 57–58

**^**- Anton & Rorres 2013, pp. 3–7
- Mortensen 2013, pp. 73–74
- Williams 2007, pp. 4–5
- Young 2023, pp. 714–715

**^**- Maddocks 2008, p. 131
- Harrison & Waldron 2011, p. 464
- Anton 2013, p. 255

**^**- Young 2010, pp. 697–698
- Maddocks 2008, p. 131
- Sullivan 2010, pp. 53–54

**^**- Anton & Rorres 2013, pp. 7–8
- Sullivan 2010, pp. 55–56
- Atanasiu & Mikusinski 2019, p. 75

**^**- Maddocks 2008, p. 131
- Anton & Rorres 2013, pp. 7–8, 11, 491

**^**- Anton & Rorres 2013, pp. 3–5
- Young 2010, pp. 696–697
- Williams 2007, pp. 4–5

**^**- Anton & Rorres 2013, pp. 3–5
- Young 2010, p. 713
- Williams 2007, pp. 4–5

**^**- Gilbert & Nicholson 2004, p. 1
- Dominich 2008, p. 19

**^**- Maddocks 2008, pp. 131–132
- Pratt 2022, Lead Section, § 2. Abstract Algebra
- Gilbert & Nicholson 2004, pp. 1–3
- Dominich 2008, p. 19

**^**Tanton 2005, p. 460- ^
^{a}^{b}Ovchinnikov 2015, p. 27 **^**Grillet 2007, p. 247**^**- Ovchinnikov 2015, p. 27
- Fiche & Hebuterne 2013, p. 326
- Gilbert & Nicholson 2004, p. 4
- Pratt 2022, Lead Section, § 2. Abstract Algebra

**^**- Maddocks 2008, pp. 131–132
- Pratt 2022, Lead Section, § 2. Abstract Algebra

**^**- Olver 1999, pp. 55–56
- Abas & Salman 1994, pp. 58–59
- Häberle 2009, p. 640

**^**- Rowland & Weisstein
- Kargapolov & Merzlyakov 2016, § Definition
- Khattar & Agrawal 2023, pp. 4–6
- Maddocks 2008, pp. 131–132
- Pratt 2022, Lead Section, § 2. Abstract Algebra

**^**- Khattar & Agrawal 2023, pp. 6–7
- Maddocks 2008, pp. 131–132

**^**- McWeeny 2002, p. 6
- Kramer & Pippich 2017, p. 49

**^**Tanton 2005, p. 242**^**Weisstein 2024c**^**- Weisstein 2024c
- Ivanova 2016
- Maxwell 2009, pp. 73–74
- Pratt 2022, § 2.3 Rings

**^**Terr & Weisstein**^**Weisstein 2024b**^**- Weisstein 2024a
- Weisstein 2024b
- Pratt 2022, § 2.4 Fields

**^**- Weisstein 2024a
- Irving 2004, p. 236
- Negro 2022, p. 365

**^**- Pratt 2022, Lead Section, § 2. Abstract Algebra
- Merzlyakov & Shirshov 2020, The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.

**^**Cooper 2011, p. 60**^**- Pratt 2022, § 3. Universal Algebra
- Insall & Sakharov

**^**- Pratt 2022, § 3.2 Equational Logic
- Mal’cev 1973, pp. 210–211
- Insall & Sakharov

**^**- Pratt 2022, § 3. Universal Algebra
- Mal’cev 1973, pp. 210–211

**^****^**- Mal’cev 1973, pp. 210–211
- Pratt 2022, § 3. Universal Algebra
- Artamonov 2003, p. 873

**^**- Insall & Sakharov
- Pratt 2022, § 3.3 Birkhoff’s Theorem
- Grätzer 2008, p. 34

**^**- Pratt 2022, § 3.3 Birkhoff’s Theorem
- Insall & Sakharov
- Silvia & Robinson 1979, p. 82

**^**- Neri 2019, pp. 278–279
- Ivanova & Smirnov 2012

- ^
^{a}^{b}- Indurkhya 2013, pp. 217–218
- Pratt 2022, § 3.3 Birkhoff’s Theorem
- Grätzer 2008, p. 34

- ^
^{a}^{b}Indurkhya 2013, pp. 217–218 **^**- Tanton 2005, p. 9
- Kvasz 2006, p. 290
- Corry 2024, § Problem Solving in Egypt and Babylon

**^**- Tanton 2005, p. 9
- Kvasz 2006, p. 290
- Corry 2024, § The Pythagoreans and Euclid

**^**- Merzlyakov & Shirshov 2020, § Historical Survey
- Sialaros 2018, p. 55
- Musielak 2020, p. 36
- Corry 2024, § Diophantus

**^**Higgins 2015, p. 89**^**- Kvasz 2006, pp. 290–291
- Sialaros 2018, p. 55
- Boyer & Merzbach 2011, p. 161
- Derbyshire 2006, p. 31

**^**- Boyer & Merzbach 2011, p. 161
- Derbyshire 2006, p. 31

**^**- Tanton 2005, p. 10
- Kvasz 2006, pp. 291–293
- Merzlyakov & Shirshov 2020, § Historical Survey

**^**- Waerden 2013, pp. 3, 15–16, 24–25
- Jenkins 2010, p. 82
- Pickover 2009, p. 90

**^**- Tanton 2005, pp. 9–10
- Corry 2024, § The Equation in India and China

**^**- Seshadri 2010, p. 156
- Emch, Sridharan & Srinivas 2005, p. 20

**^**- Smorynski 2007, p. 137
- Zwillinger 2002, p. 812

**^**- Waerden 2013, pp. 32–35
- Tanton 2005, p. 10
- Kvasz 2006, p. 293

**^**- Tanton 2005, p. 10
- Kvasz 2006, p. 293
- Corry 2024, § Cardano and the Solving of Cubic and Quartic Equations

**^**- Tanton 2005, p. 10
- Kvasz 2006, pp. 291–292, 297–298, 302
- Merzlyakov & Shirshov 2020, § Historical Survey
- Corry 2024, § Viète and the Formal Equation, § Analytic Geometry

**^**- Hazewinkel 1994, p. 73
- Merzlyakov & Shirshov 2020, § Historical Survey

**^**Igarashi et al. 2014, p. 103.**^**Sun & Zhang 2020, p. 94.- ^
^{a}^{b}- Tanton 2005, p. 10
- Merzlyakov & Shirshov 2020, § Historical Survey
- Corry 2024, § Impasse with Radical Methods

**^**Corry 2024, § Applications of Group Theory**^**- Merzlyakov & Shirshov 2020, § Historical Survey
- Tanton 2005, p. 10
- Corry 2024, § Structural Algebra
- Hazewinkel 1994, pp. 73–74

**^**- Merzlyakov & Shirshov 2020, § Historical Survey
- Tanton 2005, p. 10
- Corry 2024, § Matrices, § Quaternions and Vectors

**^**- Merzlyakov & Shirshov 2020, § Historical Survey
- Corry 2024, § Hilbert and Steinitz, § Noether and Artin
- Hazewinkel 1994, pp. 73–74

**^**- Grätzer 2008, p. vii
- Chang & Keisler 1990, p. 603
- Knoebel 2011, p. 5
- Hazewinkel 1994, pp. 74–75

**^**- Hazewinkel 1994, pp. 74–75
- Grätzer 2008, p. 338
- Pratt 2022, § 6. Free Algebras

**^**- Houston 2004, p. 319
- Corrochano & Sobczyk 2011, p. xvii
- Neri 2019, p. xii

**^**- Corrochano & Sobczyk 2011, p. xvii
- Neri 2019, p. xii
- Aleskerov, Ersel & Piontkovski 2011, pp. 1–9
- Straffin 1980, p. 269

**^**- Mancosu 1999, pp. 84–85
- Kleiner 2007, p. 100
- Pratt 2022, § 5. Algebraization of Mathematics

**^**- Kleiner 2007, p. 100
- Pratt 2022, § 5. Algebraization of Mathematics
- Maddocks 2008, p. 130

**^**- Pratt 2022, § 5.1 Algebraic Geometry
- Danilov 2006, pp. 172, 174

**^**Sterling 2021, p. 645**^**- Pratt 2022, § 5.3 Algebraic Topology
- Rabadan & Blumberg 2019, pp. 49–50
- Nakahara 2018, p. 121
- Weisstein 2003, pp. 52–53

**^**- Pratt 2022, § 5.2 Algebraic Number Theory
- Jarvis 2014, p. 1
- Viterbo & Hong 2011, p. 127

**^**- Kilty & McAllister 2018, pp. x, 347, 589
- Edwards 2012, pp. ix–x

**^**Hintikka 2019, Lead Section, § Nature and Varieties of Logic**^**- Halmos 1956, p. 363
- Burris & Legris 2021, § 1. Introduction

**^**Andréka, Németi & Sain 2001, pp. 133–134**^**- Andréka, Madarász & Németi 2020, § Concrete Algebraic Logic
- Pratt 2022, § 5.4 Algebraic Logic
- Plotkin 2012, pp. 155–156
- Jansana 2022, Lead Section

**^**McGrath & Frank 2023, Lead Section**^**- Boschini, Hansen & Wolf 2022, p. 21
- Brody 2006, pp. 535–536
- Franks 2023, Lead Section

**^**- Andréka, Madarász & Németi 2020, § Concrete Algebraic Logic
- Plotkin 2012, pp. 155–156
- Kachroo & Özbay 2018, pp. 176–177

**^**- Andréka, Madarász & Németi 2020, § Abstract Algebraic Logic
- Jansana 2022, § 4. Algebras

- ^
^{a}^{b}Gardella & DeLucia 2020, pp. 19–22 **^**- Arcavi, Drijvers & Stacey 2016, p. xiii
- Dekker & Dolk 2011, p. 69

**^**- Arcavi, Drijvers & Stacey 2016, pp. 2–5
- Drijvers, Goddijn & Kindt 2011, pp. 8–10, 16–18

**^**- Arcavi, Drijvers & Stacey 2016, pp. 58–59
- Drijvers, Goddijn & Kindt 2011, p. 13

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