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Algebra

Summary

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 interested in polynomial equations and seeks to discover which values solve them.
Abstract algebra studies algebraic structures, like the ring of integers given by the set of integers (${\displaystyle \mathbb {Z} }$) together with operations of addition (${\displaystyle +}$) and multiplication (${\displaystyle \times }$).

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.

Definition and etymology

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]

Major branches

Elementary algebra

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 ${\displaystyle 2+5=7}$ .[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 ${\displaystyle 2\times 3=3\times 2}$  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 the principle of commutativity expressed in the equation ${\displaystyle a\times b=b\times a}$ .[22]

Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention, the lowercase letters ${\displaystyle x}$ , ${\displaystyle y}$  and ${\displaystyle z}$  represent variables. In some cases, subscripts are added to distinguish variables, as in ${\displaystyle x_{1}}$ , ${\displaystyle x_{2}}$ , and ${\displaystyle x_{3}}$ . The lowercase letters ${\displaystyle a}$ , ${\displaystyle b}$ , and ${\displaystyle c}$  are usually used for constants and coefficients.[d] For example, the expression ${\displaystyle 5x+3}$  is an algebraic expression created by multiplying the number 5 with the variable ${\displaystyle x}$  and adding the number 3 to the result. Other examples of algebraic expressions are ${\displaystyle 32xyz}$  and ${\displaystyle 64x_{1}^{2}+7x_{2}-13}$ .[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 (${\displaystyle =}$ ), as in ${\displaystyle 5x^{2}+6x=3y+4}$ . Inequations are formed with symbols like the less-than sign (${\displaystyle <}$ ), the greater-than sign (${\displaystyle >}$ ), and the inequality sign (${\displaystyle \neq }$ ). 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 ${\displaystyle x^{2}=4}$  is true if ${\displaystyle x}$  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 ${\displaystyle 2x+5x=7x}$ . Conditional equations are only true for some values. For example, the equation ${\displaystyle x+4=9}$  is only true if ${\displaystyle x}$  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 ${\displaystyle x-7=4}$  can be solved for ${\displaystyle x}$  by adding 7 to both sides, which isolates ${\displaystyle x}$  on the left side and results in the equation ${\displaystyle x=11}$ .[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 ${\displaystyle 7x-3x}$  can be replaced with the expression ${\displaystyle 4x}$ .[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 ${\displaystyle x^{2}-3x-10}$  can be factorized as ${\displaystyle (x+2)(x-5)}$ . The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if ${\displaystyle x}$  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 ${\displaystyle y=3x}$  then one can simplify the expression ${\displaystyle 7xy}$  to arrive at ${\displaystyle 21x^{2}}$ . 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.[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 ${\displaystyle x}$  is set to zero in the equation ${\displaystyle y=0.5x-1}$  then ${\displaystyle y}$  has to be −1 for the equation to be true. This means that the ${\displaystyle x}$ -${\displaystyle y}$ -pair ${\displaystyle (0,-1)}$  is part of the graph of the equation. The ${\displaystyle x}$ -${\displaystyle y}$ -pair ${\displaystyle (0,7)}$ , by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of all ${\displaystyle x}$ -${\displaystyle y}$ -pairs that solve the equation.[32]

Linear algebra

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 ${\displaystyle 0.25x-4=y}$  and ${\displaystyle x_{1}-7x_{2}+3x_{3}=0}$  are linear while the equations ${\displaystyle x^{2}=y}$  and ${\displaystyle 3x_{1}x_{2}+15=0}$  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) ${\displaystyle 9x_{1}+3x_{2}-13x_{3}=0}$
(b) ${\displaystyle 2.3x_{1}+7x_{3}=9}$
(c) ${\displaystyle -5x_{1}-17x_{2}=-3}$

can be written as

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

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 ${\displaystyle 3x_{1}-x_{2}=0}$  and ${\displaystyle x_{1}+x_{2}=8}$  then using the values 1 and 3 for ${\displaystyle x_{1}}$  and ${\displaystyle x_{2}}$  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 ${\displaystyle x_{1}-3x_{2}=0}$  and ${\displaystyle x_{1}-3x_{2}=7}$  contradict each other since no values of ${\displaystyle x_{1}}$  and ${\displaystyle x_{2}}$  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 ${\displaystyle -x+7y=3}$  and ${\displaystyle 2x-7y=10}$ , it is possible to eliminate ${\displaystyle y}$  by adding the first to the second equation, thereby revealing that ${\displaystyle x}$  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

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 ${\displaystyle \langle \mathbb {N} ,+\rangle }$  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 ${\displaystyle (a\circ b)\circ c}$  is the same as ${\displaystyle a\circ (b\circ c)}$  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 ${\displaystyle a\circ e=e\circ a=a}$ . An operation admits inverse elements if for any element ${\displaystyle a}$  there exists a reciprocal element ${\displaystyle a^{-1}}$  that reverses its effects. If an element is linked to its inverse then the result is the neutral element e, expressed formally as ${\displaystyle a\circ a^{-1}=a^{-1}\circ a=e}$ . Every algebraic structure that fulfills these requirements is a group.[52] For example, ${\displaystyle \langle \mathbb {Z} ,+\rangle }$  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 ${\displaystyle a}$  is ${\displaystyle -a}$ .[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 (${\displaystyle \circ }$  and ${\displaystyle \star }$ ) 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 is commutative, meaning that ${\displaystyle a\circ b=b\circ a}$  is true for all elements. The axiom of distributivity governs how the two operations interact with each other. It states that ${\displaystyle a\star (b\circ c)=(a\star b)\circ (a\star c)}$  and ${\displaystyle (b\circ c)\star a=(b\star a)\circ (c\star a)}$ .[l][57] The ring of integers is the ring denoted by ${\displaystyle \langle \mathbb {Z} ,+,\times \rangle }$ .[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 ${\displaystyle 7}$  is ${\displaystyle {\tfrac {1}{7}}}$ , 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

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 ${\displaystyle a\circ b}$  is identical to ${\displaystyle b\circ a}$  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 ${\displaystyle \langle A,\circ \rangle }$  and ${\displaystyle \langle B,\star \rangle }$  then the function ${\displaystyle h:A\to B}$  is a homomorphism if it fulfills the following requirement: ${\displaystyle h(x\circ y)=h(x)\star h(y)}$ . The existence of a homomorphism reveals that the operation ${\displaystyle \star }$  in the second algebraic structure plays the same role as the operation ${\displaystyle \circ }$  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 ${\displaystyle \langle A,\circ \rangle }$  is a subalgebra of ${\displaystyle \langle B,\circ \rangle }$  then the set ${\displaystyle A}$  is a subset of ${\displaystyle B}$ .[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 ${\displaystyle A}$ , meaning that they only produce elements that belong to ${\displaystyle A}$ .[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]

History

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]

François Viète and René Descartes invented a symbolic notation to express equations in an abstract and concise manner.

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]

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 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]

Other branches of mathematics

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 ${\displaystyle y=3x-7}$  describes a line in two-dimensional space while the equation ${\displaystyle x^{2}+y^{2}+z^{2}=1}$  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 ${\displaystyle y=x+1}$  intersects with the circle described by ${\displaystyle x^{2}+y^{2}=25}$  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

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]

Education

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 (${\displaystyle 2x+x=12}$ ) and to determine how many apples Naomi has (${\displaystyle x=4}$ ).[118]

References

Notes

1. ^ 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 ${\displaystyle \omega :A^{n}\to A}$ . 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 ${\displaystyle +:\mathbb {R} ^{2}\to \mathbb {R} }$ .[1]
2. ^ 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]
3. ^ 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]
4. ^ Constants represent fixed magnitudes that, unlike variables, cannot change.
5. ^ 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 ${\displaystyle 3x^{2}-7}$  and ${\displaystyle 5x^{3}y+4yz}$ .[28]
6. ^ A matrix is a table of numbers, such as
${\displaystyle {\begin{bmatrix}3&-7&19\\0.3&7&-4\end{bmatrix}}}$

7. ^ A vector is an array of numbers or a matrix with only one column, such as
${\displaystyle {\begin{bmatrix}2.1\\0\\-1\end{bmatrix}}}$

8. ^ 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]
9. ^ 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]
10. ^ 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]
11. ^ Symbols like ${\displaystyle \circ }$  and ${\displaystyle \star }$  are often used in abstract algebra to represent any operation that may or may not resemble arithmetic operations.
12. ^ Some definitions additionally require that the second operation is associative.[56]
13. ^ For the second operation, there is usually one element, corresponding to 0, that does not require an inverse element.[59]
14. ^ The conditions take the form of a Horn clause.
15. ^ This means that all the elements of ${\displaystyle A}$  are also elements of ${\displaystyle B}$  but ${\displaystyle B}$  may contain elements that are not found in ${\displaystyle A}$ .
16. ^ According to some definitions, it is also possible for a subalgebra to have fewer operations.[73]
17. ^ The exact date is disputed.
18. ^ Some historians consider him the "father of algebra" while others reserve this title for Diophantus.[79]
19. ^ 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 ${\displaystyle ax^{2}+bx+c=0}$  is ${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}}$ . The absence of general solutions does not mean that there are no numerical solutions.[89][90]
20. ^ Algebraic varieties studied in geometry are different from the more general varieties studied in universal algebra.

Citations

1. ^ Baranovich 2023, Lead Section
2. ^
3. ^
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.
5. ^ Higham 2019, p. 296.
6. ^ Library of Congress, p. 3.
7. ^
8. ^
9. ^ a b
10. ^
11. ^
12. ^
13. ^
14. ^
15. ^
16. ^ EoM Staff 2017
17. ^
18. ^ a b
19. ^ a b
20. ^
21. ^
22. ^ a b
23. ^
24. ^
25. ^
26. ^
27. ^
28. ^
29. ^
30. ^
31. ^
32. ^
33. ^
34. ^
35. ^
36. ^
37. ^
38. ^
39. ^
40. ^
41. ^
42. ^
43. ^
44. ^
45. ^
46. ^ Tanton 2005, p. 460
47. ^ a b Ovchinnikov 2015, p. 27
48. ^ Grillet 2007, p. 247
49. ^
50. ^
51. ^
52. ^
53. ^
54. ^
55. ^ Tanton 2005, p. 242
56. ^ Weisstein 2024c
57. ^
58. ^ Terr & Weisstein
59. ^ Weisstein 2024b
60. ^
61. ^
62. ^
• 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.
63. ^ Cooper 2011, p. 60
64. ^
65. ^
66. ^
67. ^
68. ^
69. ^
70. ^
71. ^
72. ^ a b
73. ^ a b Indurkhya 2013, pp. 217–218
74. ^
75. ^
76. ^
77. ^ Higgins 2015, p. 89
78. ^
79. ^
80. ^
81. ^
82. ^
83. ^
84. ^
85. ^
86. ^
87. ^
88. ^
89. ^ Igarashi et al. 2014, p. 103.
90. ^ Sun & Zhang 2020, p. 94.
91. ^ a b
92. ^ Corry 2024, § Applications of Group Theory
93. ^
94. ^
95. ^
96. ^
97. ^
98. ^
99. ^
100. ^
101. ^
102. ^
103. ^ Sterling 2021, p. 645
104. ^
105. ^
106. ^
107. ^ Hintikka 2019, Lead Section, § Nature and Varieties of Logic
108. ^
109. ^ Andréka, Németi & Sain 2001, pp. 133–134
110. ^
111. ^ McGrath & Frank 2023, Lead Section
112. ^
113. ^
114. ^
115. ^ a b Gardella & DeLucia 2020, pp. 19–22
116. ^
117. ^
118. ^

Sources

• Abas, Syed Jan; Salman, Amer Shaker (1994). Symmetries Of Islamic Geometrical Patterns. World Scientific. ISBN 978-981-4502-21-4.
• Aleskerov, Fuad; Ersel, Hasan; Piontkovski, Dmitri (18 August 2011). Linear Algebra for Economists. Springer Science & Business Media. ISBN 978-3-642-20570-5.
• Andréka, H.; Németi, I.; Sain, I. (2001). "Algebraic Logic". Handbook of Philosophical Logic. Springer Netherlands. doi:10.1007/978-94-017-0452-6_3. ISBN 978-94-017-0452-6. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Andréka, H.; Madarász, J. X.; Németi, I. (2020). "Algebraic Logic". Encyclopedia of Mathematics. Springer. Archived from the original on 24 January 2024. Retrieved 23 October 2023.
• Andrilli, Stephen; Hecker, David (2022). Elementary Linear Algebra. Academic Press. ISBN 978-0-323-98426-3. Archived from the original on 2024-01-17. Retrieved 2024-01-18.
• Anton, Howard; Rorres, Chris (2013). Elementary Linear Algebra: Applications Version. John Wiley & Sons. ISBN 978-1-118-47422-8. Archived from the original on 2024-01-17. Retrieved 2024-01-18.
• Anton, Howard (2013). Elementary Linear Algebra. John Wiley & Sons. ISBN 978-1-118-67730-8. Archived from the original on 2024-01-18. Retrieved 2024-01-18.
• Arcavi, Abraham; Drijvers, Paul; Stacey, Kaye (2016). The Learning and Teaching of Algebra: Ideas, Insights and Activities. Routledge. ISBN 978-1-134-82077-1. Archived from the original on 2024-01-23. Retrieved 2024-01-24.
• Artamonov, V. A. (2003). "Quasivarieties". In Hazewinkel, M. (ed.). Handbook of Algebra. Elsevier. ISBN 978-0-08-053297-4. Archived from the original on 2024-01-29. Retrieved 2024-01-21.
• Atanasiu, Dragu; Mikusinski, Piotr (8 April 2019). A Bridge To Linear Algebra. World Scientific. ISBN 978-981-12-0024-3.
• Baranovich, T. M. (2023). "Algebraic Operation". Encyclopedia of Mathematics. Springer. Archived from the original on 23 August 2023. Retrieved 11 January 2023.
• Barrera-Mora, Fernando (2023). Linear Algebra: A Minimal Polynomial Approach to Eigen Theory. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-113591-5. Archived from the original on 2024-01-16. Retrieved 2024-01-18.
• Benson, Donald C. (2003). A Smoother Pebble: Mathematical Explorations. Oxford University Press. ISBN 978-0-19-514436-9. Archived from the original on 2024-01-16. Retrieved 2024-01-16.
• Berggren, John L. (2015). "Elementary Algebra". Encyclopædia Britannica. Archived from the original on 14 January 2024. Retrieved 14 January 2024.
• Boschini, Cecilia; Hansen, Arne; Wolf, Stefan (2022). Discrete Mathematics. vdf Hochschulverlag ETH Zürich. ISBN 978-3-7281-4110-1. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Boyer, Carl B.; Merzbach, Uta C. (2011). A History of Mathematics. John Wiley & Sons. ISBN 978-0-470-63056-3. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Brody, Boruch A. (2006). Encyclopedia of Philosophy. Vol. 5. Donald M. Borchert (2nd ed.). Thomson Gale/Macmillan Reference US. pp. 535–536. ISBN 978-0-02-865780-6. OCLC 61151356.
• Bukhshtab, A. A.; Pechaev, V. I. (2020). "Arithmetic". Encyclopedia of Mathematics. Springer. Archived from the original on 4 October 2009. Retrieved 23 October 2023.
• Burgin, Mark (2022). Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective. World Scientific. ISBN 978-981-12-3685-3. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Burris, Stanley; Legris, Javier (2021). "The Algebra of Logic Tradition". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 29 January 2024. Retrieved 22 January 2024.
• Buthusiem, Gregory; Toth, Gabor (2020). Precalculus. Linus Learning. ISBN 978-1-60797-798-8. Archived from the original on 2024-01-14. Retrieved 2024-01-16.
• Chang, C. C.; Keisler, H. J. (1990). Model Theory. Elsevier. ISBN 978-0-08-088007-5. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Cooper, Ellis D. (2011). Mathematical Mechanics: From Particle to Muscle. World Scientific. ISBN 978-981-4289-70-2. Archived from the original on 2024-01-20. Retrieved 2024-01-20.
• Corrochano, Eduardo Bayro; Sobczyk, Garret (2011). Geometric Algebra with Applications in Science and Engineering. Springer Science & Business Media. ISBN 978-1-4612-0159-5. Archived from the original on 2024-01-23. Retrieved 2024-01-24.
• Corry, Leo (2024). "Algebra". Encyclopædia Britannica. Archived from the original on 19 January 2024. Retrieved 25 January 2024.
• Cox, David A.; Little, John; O'Shea, Donal (2015). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer. ISBN 978-3-319-16721-3. Archived from the original on 2024-01-29. Retrieved 2024-01-21.
• Cresswell, Julia (2010). Oxford Dictionary of Word Origins. OUP Oxford. ISBN 978-0-19-954793-7. Archived from the original on 2024-01-27. Retrieved 2024-01-27.
• Danilov, V. I. (2006). "II. Algebraic Varieties and Schemes". Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes. Springer Science & Business Media. ISBN 978-3-540-51995-9. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Dekker, Truus; Dolk, Maarten (2011). "3. From Arithmetic to Algebra". In Drijvers, Paul (ed.). Secondary Algebra Education: Revisiting Topics and Themes and Exploring the Unknown. Springer Science & Business Media. ISBN 978-94-6091-334-1. Archived from the original on 2024-01-23. Retrieved 2024-01-24.
• Derbyshire, John (2006). "2. The Father of Algebra". Unknown Quantity: A Real and Imaginary History of Algebra. National Academies Press. ISBN 978-0-309-09657-7. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Dominich, Sándor (2008). The Modern Algebra of Information Retrieval. Springer Science & Business Media. ISBN 978-3-540-77659-8. Archived from the original on 2024-01-20. Retrieved 2024-01-20.
• Drijvers, Paul; Goddijn, Aad; Kindt, Martin (2011). "1. Algebra Education: Exploring Topics and Themes". In Drijvers, Paul (ed.). Secondary Algebra Education: Revisiting Topics and Themes and Exploring the Unknown. Springer Science & Business Media. ISBN 978-94-6091-334-1. Archived from the original on 2024-01-23. Retrieved 2024-01-24.
• Edwards, C. H. (2012). Advanced Calculus of Several Variables. Courier Corporation. ISBN 978-0-486-13195-5. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Emch, Gerard G.; Sridharan, R.; Srinivas, M. D. (2005). Contributions to the History of Indian Mathematics. Springer. ISBN 978-93-86279-25-5. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• EoM Staff (2017). "Algebra". Encyclopedia of Mathematics. Springer. Archived from the original on 29 November 2022. Retrieved 11 January 2023.
• Fiche, Georges; Hebuterne, Gerard (2013). Mathematics for Engineers. John Wiley & Sons. ISBN 978-1-118-62333-6. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Franks, Curtis (2023). "Propositional Logic". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 29 January 2024. Retrieved 22 January 2024.
• Gardella, Francis; DeLucia, Maria (2020). Algebra for the Middle Grades. IAP. ISBN 978-1-64113-847-5. Archived from the original on 2024-01-23. Retrieved 2024-01-24.
• Gilbert, William J.; Nicholson, W. Keith (2004). Modern Algebra with Applications. John Wiley & Sons. ISBN 978-0-471-46989-6. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Golan, Jonathan S. (1995). "Algebras Over A Field". Foundations of Linear Algebra. Kluwer Texts in the Mathematical Sciences. Vol. 11. Springer Netherlands. pp. 219–227. doi:10.1007/978-94-015-8502-6_18. ISBN 978-94-015-8502-6. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Goodman, A. W. (4 September 2001). Algebra From A To Z - Volume 1. World Scientific. ISBN 978-981-310-266-8.
• Grätzer, George (2008). Universal Algebra (2 ed.). Springer Science & Business Media. ISBN 978-0-387-77487-9. Archived from the original on 2024-01-27. Retrieved 2024-01-27.
• Grillet, Pierre Antoine (2007). "Universal Algebra". Abstract Algebra. Graduate Texts in Mathematics. Vol. 242. Springer. pp. 559–580. doi:10.1007/978-0-387-71568-1_15. ISBN 978-0-387-71568-1. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Häberle, L. (14 October 2009). "On Classification of Molecules and Species of Representation Rings". In Fink, Andreas; Lausen, Berthold; Seidel, Wilfried; Ultsch, Alfred (eds.). Advances in Data Analysis, Data Handling and Business Intelligence. Springer Science & Business Media. ISBN 978-3-642-01044-6.
• Halmos, Paul R. (1956). "The Basic Concepts of Algebraic Logic". The American Mathematical Monthly. 63 (6): 363–387. doi:10.2307/2309396. ISSN 0002-9890. JSTOR 2309396.
• Harrison, Michael; Waldron, Patrick (2011). Mathematics for Economics and Finance. Routledge. ISBN 978-1-136-81921-6. Archived from the original on 2024-01-17. Retrieved 2024-01-18.
• Hazewinkel, Michiel (1994). Encyclopaedia of Mathematics (Set). Springer Science & Business Media. ISBN 978-1-55608-010-4. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• HC Staff (2022). "Arithmetic". American Heritage Dictionary. HarperCollins. Archived from the original on 8 November 2023. Retrieved 19 October 2023.
• Higgins, Peter M. (2015). Algebra: A Very Short Introduction. OUP Oxford. ISBN 978-0-19-104746-6. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Higham, Nicholas J. (16 December 2019). Handbook of Writing for the Mathematical Sciences (3 ed.). SIAM. ISBN 978-1-61197-610-6.
• Hintikka, Jaakko J. (2019). "Philosophy of Logic". Encyclopædia Britannica. Archived from the original on 28 April 2015. Retrieved 21 November 2021.
• Hoad, T. F. (1993). The Concise Oxford Dictionary of English Etymology. Oxford University Press. ISBN 978-0-19-283098-2.
• Houston, Stephen D. (9 December 2004). The First Writing: Script Invention as History and Process. Cambridge University Press. ISBN 978-0-521-83861-0.
• Igarashi, Yoshihide; Altman, Tom; Funada, Mariko; Kamiyama, Barbara (2014). Computing: A Historical and Technical Perspective. CRC Press. ISBN 978-1-4822-2741-3. Archived from the original on 2024-01-29. Retrieved 2024-01-29.
• Indurkhya, Bipin (2013). "6.5 Algebras and Structures". Metaphor and Cognition: An Interactionist Approach. Springer Science & Business Media. ISBN 978-94-017-2252-0. Archived from the original on 2024-01-29. Retrieved 2024-01-21.
• Insall, Matt; Sakharov, Alex. "Universal Algebra". Wolfram MathWorld. Wolfram. Archived from the original on 10 January 2024. Retrieved 12 January 2024.
• Irving, Ronald S. (2004). Integers, Polynomials, and Rings: A Course in Algebra. Springer Science & Business Media. ISBN 978-0-387-40397-7. Archived from the original on 2024-01-19. Retrieved 2024-01-20.
• Ivanova, O. A. (2016). "Ring". Encyclopedia of Mathematics. Springer. Archived from the original on 1 January 2023. Retrieved 11 January 2023.
• Ivanova, O. A.; Smirnov, D. M. (2012). "Isomorphism". Encyclopedia of Mathematics. Springer.
• Jansana, Ramon (2022). "Algebraic Propositional Logic". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 20 December 2016. Retrieved 22 January 2024.
• Jarvis, Frazer (2014). Algebraic Number Theory. Springer. ISBN 978-3-319-07545-7. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Jenkins, Everett (2010). The Muslim Diaspora (Volume 1, 570-1500): A Comprehensive Chronology of the Spread of Islam in Asia, Africa, Europe and the Americas. McFarland. ISBN 978-0-7864-4713-8. Archived from the original on 2024-01-29. Retrieved 2024-01-28.
• Kachroo, Pushkin; Özbay, Kaan M. A. (2018). Feedback Control Theory for Dynamic Traffic Assignment. Springer. ISBN 978-3-319-69231-9. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Kargapolov, M. I.; Merzlyakov, Yu. I. (2016). "Group". Encyclopedia of Mathematics. Springer. Archived from the original on 5 December 2022. Retrieved 11 January 2023.
• Khattar, Dinesh; Agrawal, Neha (2023). Group Theory. Springer Nature. ISBN 978-3-031-21307-6. Archived from the original on 2024-01-19. Retrieved 2024-01-20.
• Kilty, Joel; McAllister, Alex (2018). Mathematical Modeling and Applied Calculus. Oxford University Press. ISBN 978-0-19-255813-8. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Kleiner, Israel (2007). A History of Abstract Algebra. Springer Science & Business Media. ISBN 978-0-8176-4685-1. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Knoebel, Arthur (2011). Sheaves of Algebras Over Boolean Spaces. Springer Science & Business Media. ISBN 978-0-8176-4218-1. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Kramer, Jürg; Pippich, Anna-Maria von (2017). From Natural Numbers to Quaternions. Springer. ISBN 978-3-319-69429-0. Archived from the original on 2024-01-19. Retrieved 2024-01-20.
• Kvasz, L. (2006). "The History of Algebra and the Development of the Form of Its Language". Philosophia Mathematica. 14 (3): 287–317. doi:10.1093/philmat/nkj017.
• Lamagna, Edmund A. (2019). Computer Algebra: Concepts and Techniques. CRC Press. ISBN 978-1-351-60583-0. Archived from the original on 2024-01-14. Retrieved 2024-01-16.
• Library of Congress. Library of Congress Classification: Class Q - Science (PDF). Library of Congress.
• Maddocks, J. R. (2008). "Algebra". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). The Gale Encyclopedia of Science (4th ed.). Thompson Gale. ISBN 978-1-4144-2877-2. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Majewski, Miroslaw (2004). MuPAD Pro Computing Essentials (2 ed.). Springer. ISBN 978-3-540-21943-9.
• Mal’cev, A. I. (1973). "Quasivarieties". Algebraic Systems. Springer. pp. 210–266. doi:10.1007/978-3-642-65374-2_5. ISBN 978-3-642-65374-2. Archived from the original on 2018-06-18. Retrieved 2024-01-21.
• Mancosu, Paolo (1999). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford University Press. ISBN 978-0-19-513244-1. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Markushevich, A. I. (2015). "Polynomial". Encyclopedia of Mathematics. Springer. Retrieved 11 January 2023.
• Maxwell, E. A. (2009). Algebraic Structure and Matrices Book 2. Syracuse University Press. ISBN 978-0-521-10905-5. Archived from the original on 2024-01-19. Retrieved 2024-01-20.
• McGrath, Matthew; Frank, Devin (2023). "Propositions". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 29 January 2024. Retrieved 22 January 2024.
• McKeague, Charles P. (2014). Intermediate Algebra: A Text/Workbook. Academic Press. ISBN 978-1-4832-1417-7. Archived from the original on 2024-01-14. Retrieved 2024-01-16.
• McWeeny, R. (2002). Symmetry: An Introduction to Group Theory and Its Applications. Courier Corporation. ISBN 978-0-486-42182-7. Archived from the original on 2024-01-19. Retrieved 2024-01-20.
• Menini, Claudia; Oystaeyen, Freddy Van (2017). Abstract Algebra: A Comprehensive Treatment. CRC Press. ISBN 978-1-4822-5817-2. Archived from the original on 2024-01-27. Retrieved 2024-01-27.
• Merzlyakov, Yu. I.; Shirshov, A. I. (2020). "Algebra(2)". Encyclopedia of Mathematics. Springer. Archived from the original on 7 April 2023. Retrieved 11 January 2023.
• Mortensen, C. E. (2013). Inconsistent Mathematics. Springer Science & Business Media. ISBN 978-94-015-8453-1. Archived from the original on 2024-01-17. Retrieved 2024-01-18.
• Musielak, Dora (2020). Sophie Germain: Revolutionary Mathematician. Springer Nature. ISBN 978-3-030-38375-6. Archived from the original on 2024-01-25. Retrieved 2024-01-27.
• Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach. John Wiley & Sons. ISBN 978-1-118-48700-6.
• MW Staff (2023). "Definition of Arithmetic". Merriam-Webster. Archived from the original on 14 November 2023. Retrieved 19 October 2023.
• Nakahara, Mikio (2018). Geometry, Topology and Physics. Taylor & Francis. ISBN 978-1-4200-5694-5. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Negro, Luca Dal (2022). Waves in Complex Media. Cambridge University Press. ISBN 978-1-107-03750-2. Archived from the original on 2024-01-29. Retrieved 2024-01-21.
• Neri, Ferrante (2019). Linear Algebra for Computational Sciences and Engineering. Springer. ISBN 978-3-030-21321-3. Archived from the original on 2024-01-23. Retrieved 2024-01-24.
• Olver, Peter J. (13 January 1999). Classical Invariant Theory. Cambridge University Press. ISBN 978-0-521-55821-1.
• OUP Staff. "Algebra". Lexico. Oxford University Press. Archived from the original on 2013-11-20.
• Ovchinnikov, Sergei (2015). Number Systems. American Mathematical Society. ISBN 978-1-4704-2018-5. Archived from the original on 2024-01-19. Retrieved 2024-01-20.
• Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Company, Inc. ISBN 978-1-4027-5796-9. Archived from the original on 2024-01-28. Retrieved 2024-01-28.
• Plotkin, B. (2012). Universal Algebra, Algebraic Logic, and Databases. Springer Science & Business Media. ISBN 978-94-011-0820-1. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Pratt, Vaughan (2022). "Algebra". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Archived from the original on 29 January 2024. Retrieved 11 January 2024.
• Rabadan, Raul; Blumberg, Andrew J. (2019). Topological Data Analysis for Genomics and Evolution: Topology in Biology. Cambridge University Press. ISBN 978-1-107-15954-9. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Renze, John; Weisstein, Eric W. "Algebra". Wolfram MathWorld. Wolfram. Archived from the original on 12 January 2024. Retrieved 12 January 2024.
• Rohde, Ulrich L.; Jain, G. C.; Poddar, Ajay K.; Ghosh, A. K. (2012). Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners. John Wiley & Sons. ISBN 978-1-118-13014-8. Archived from the original on 2024-01-15. Retrieved 2024-01-16.
• Romanowski, Perry (2008). "Arithmetic". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). The Gale Encyclopedia of Science (4th ed.). Thompson Gale. ISBN 978-1-4144-2877-2. Archived from the original on 2023-11-01. Retrieved 2024-01-13.
• Rowland, Todd; Weisstein, Eric W. "Group". Wolfram MathWorld. Wolfram. Archived from the original on 17 January 2024. Retrieved 12 January 2024.
• Seshadri, C. S. (2010). Studies in the History of Indian Mathematics. Springer. ISBN 978-93-86279-49-1. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Sialaros, Michalis (2018). Revolutions and Continuity in Greek Mathematics. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-056527-0. Archived from the original on 2024-01-25. Retrieved 2024-01-27.
• Silvia, M. T.; Robinson, E. A. (1979). Deconvolution of Geophysical Time Series in the Exploration for Oil and Natural Gas. Elsevier. ISBN 978-0-08-086864-6. Archived from the original on 2024-01-29. Retrieved 2024-01-21.
• Smorynski, Craig (2007). History of Mathematics: A Supplement. Springer Science & Business Media. ISBN 978-0-387-75481-9. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Sorell, Tom (12 October 2000). Descartes: A Very Short Introduction. OUP Oxford. ISBN 978-0-19-285409-4.
• Sterling, Mary Jane (26 May 2016). Algebra I For Dummies. John Wiley & Sons. ISBN 978-1-119-29756-7.
• Sterling, Mary Jane (18 November 2021). Algebra I All-in-One For Dummies. John Wiley & Sons. ISBN 978-1-119-84306-1.
• Straffin, Philip D. (1980). "Linear Algebra in Geography: Eigenvectors of Networks". Mathematics Magazine. 53 (5): 269–276. doi:10.2307/2689388. ISSN 0025-570X. JSTOR 2689388.
• Sullivan, Michael (2010). Finite Mathematics: An Applied Approach. John Wiley & Sons. ISBN 978-0-470-87639-8. Archived from the original on 2024-01-17. Retrieved 2024-01-18.
• Sun, Shuyu; Zhang, Tao (2020). Reservoir Simulations: Machine Learning and Modeling. Gulf Professional Publishing. ISBN 978-0-12-820962-2. Archived from the original on 2024-01-29. Retrieved 2024-01-29.
• Tan, Kiat Shi; Steeb, Willi-Hans; Hardy, Yorick (2012). SymbolicC++:An Introduction to Computer Algebra Using Object-Oriented Programming: An Introduction to Computer Algebra Using Object-Oriented Programming. Springer Science & Business Media. ISBN 978-1-4471-0405-6. Archived from the original on 2024-01-14. Retrieved 2024-01-16.
• Tanton, James (2005). Encyclopedia of Mathematics. Facts On File. ISBN 978-0-8160-5124-3.
• Terr, David; Weisstein, Eric W. "Ring of Integers". Wolfram MathWorld. Wolfram. Archived from the original on 10 January 2024. Retrieved 12 January 2024.
• Viterbo, Emanuele; Hong, Yi (2011). "3.4 Algebraic Number Theory". In Hlawatsch, Franz; Matz, Gerald (eds.). Wireless Communications Over Rapidly Time-Varying Channels. Academic Press. ISBN 978-0-08-092272-0. Archived from the original on 2024-01-24. Retrieved 2024-01-24.
• Voitsekhovskii, M. I. (2011). "Linear Equation". Encyclopedia of Mathematics. Springer. Archived from the original on 23 November 2023. Retrieved 10 January 2024.
• Waerden, Bartel L. van der (2013). A History of Algebra: From al-Khwārizmī to Emmy Noether. Springer Science & Business Media. ISBN 978-3-642-51599-6. Archived from the original on 2024-01-26. Retrieved 2024-01-27.
• Wagner, Sigrid; Kieran, Carolyn (2018). Research Issues in the Learning and Teaching of Algebra: The Research Agenda for Mathematics Education, Volume 4. Routledge. ISBN 978-1-135-43421-2. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Walz, Guido (2016). "Algebra". Lexikon der Mathematik: Band 1: A bis Eif (in German). Springer-Verlag. ISBN 978-3-662-53498-4. Archived from the original on 2024-01-12. Retrieved 2024-01-13.
• Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics (2nd ed.). Chapman & Hall/CRC. ISBN 978-1-58488-347-0.
• Weisstein, Eric W. (2024a). "Field". Wolfram MathWorld. Wolfram. Archived from the original on 16 January 2024. Retrieved 12 January 2024.
• Weisstein, Eric W. (2024b). "Field Axioms". Wolfram MathWorld. Wolfram. Archived from the original on 24 December 2023. Retrieved 12 January 2024.
• Weisstein, Eric W. (2024c). "Ring". Wolfram MathWorld. Wolfram. Archived from the original on 21 January 2024. Retrieved 12 January 2024.
• Williams, Gareth (2007). Linear Algebra with Applications. Jones & Bartlett Learning. ISBN 978-0-7637-5753-3. Archived from the original on 2024-01-17. Retrieved 2024-01-18.
• Williams, G. Arnell (23 August 2022). Algebra the Beautiful: An Ode to Math's Least-Loved Subject. Basic Books. ISBN 978-1-5416-0070-6.
• Young, Cynthia Y. (2010). Precalculus. John Wiley & Sons. ISBN 978-0-471-75684-2. Archived from the original on 2024-01-13. Retrieved 2024-01-16.
• Young, Cynthia Y. (2023). Precalculus. John Wiley & Sons. ISBN 978-1-119-86940-5. Archived from the original on 2024-01-17. Retrieved 2024-01-18.
• zbMATH Open (2024). "Classification". zbMATH Open. Mathematical Reviews and zbMATH Open. Retrieved 17 March 2024.
• Zill, Dennis; Dewar, Jacqueline (2011). Algebra and Trigonometry. Jones & Bartlett Publishers. ISBN 978-0-7637-5461-7. Archived from the original on 2024-01-14. Retrieved 2024-01-16.
• Zwillinger, Daniel (2002). CRC Standard Mathematical Tables and Formulae. CRC Press. ISBN 978-1-4200-3534-6. Archived from the original on 2024-01-26. Retrieved 2024-01-27.