View source for Algebra

{{Short description|Branch of mathematics}}
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|footer=[[Elementary algebra]] is interested in polynomial equations and seeks to discover which values [[Equation solving|solve them]] (top image). [[Abstract algebra]] studies [[Algebraic structure|algebraic structure]]s, like the [[Ring of integers|ring of integers]] given by the set of [[Integer|integer]]s (<math>\Z</math>) together with [[Algebraic operation|operations]] of [[Addition|addition]] (<math>+</math>) and [[Multiplication|multiplication]] (<math>\times</math>) (bottom image).
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'''Algebra''' is the branch of [[Mathematics|mathematics]] that studies algebraic systems and the manipulation of [[Equation|equation]]s within those systems. It is a generalization of [[Arithmetic|arithmetic]] that includes [[Variable (mathematics)|variables]] besides regular numbers and [[Algebraic operation|algebraic operation]]s other than the standard arithmetic operations like [[Addition|addition]] and [[Multiplication|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 [[Equation solving|isolate variables]]. [[Linear algebra]] is a closely related field investigating variables that appear in several [[Linear equation|linear equation]]s, 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 (mathematics)|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 [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]], based on the number of operations they use and the [[Axiom|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 algebraic structures.

Algebraic methods were first studied in the ancient period to solve specific problems in fields like [[Geometry|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|theory of equations]] to cover diverse types of algebraic operations and algebraic structures.

Algebra is relevant to many branches of mathematics, like geometry, [[Topology|topology]], [[Number theory|number theory]], and [[Calculus|calculus]], and other fields of inquiry, like [[Logic|logic]] and the empirical sciences.

== Definition and etymology ==
Algebra is the branch of mathematics that studies algebraic operations{{efn|When understood in the widest sense, an algebraic operation is mapping from a [[Cartesian product#n-ary Cartesian power|Cartesian power of a set into that set]], expressed formally as <math>\omega: A^n \to A</math>. 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 <math>+: \R^2 \to \R</math>.<ref>{{harvnb|EoM Staff|2023|loc=Lead Section}}</ref>}} and algebraic structures.<ref>{{multiref | {{harvnb|EoM Staff|2020|loc=Lead Section}} | {{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 4]}} }}</ref> An algebraic structure is a non-empty [[Set (mathematics)|set]] of [[Philosophy:Mathematical object|mathematical object]]s, such as the real numbers, together algebraic operations defined on that set, such [[Addition|addition]] and [[Multiplication|multiplication]].<ref>{{multiref | {{harvnb|Fiche|Hebuterne|2013|p=[https://books.google.com/books?id=TqkckiuuXg8C&pg=PT326 326]}} | {{harvnb|EoM Staff|2020|loc=§ The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.}} | {{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 4]}} }}</ref> Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it studies the use of [[Variable (mathematics)|variables]] in [[Algebraic equation|equations]] and how to manipulate these equations.<ref>{{multiref | {{harvnb|Pratt|2022|loc=Lead Section, § 1. Elementary Algebra, § 2. Abstract Algebra, § 3. Universal Algebra}} | {{harvnb|EoM Staff|2020|loc=§ The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.}} }}</ref>

Algebra is often understood as a generalization of [[Arithmetic|arithmetic]].<ref>{{multiref | {{harvnb|Maddocks|2008|p=129}} | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA45 45]}} }}</ref> Arithmetic studies arithmetic operations, like addition, [[Subtraction|subtraction]], multiplication, and [[Division (mathematics)|division]], in a specific domain of numbers, like the real numbers.<ref name="auto1">{{multiref | {{harvnb|Romanowski|2008|pp=302–303}} | {{harvnb|HC Staff|2022}} | {{harvnb|MW Staff|2023}} | {{harvnb|EoM Staff|2020a}} }}</ref> [[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.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=129–130}} | {{harvnb|Pratt|2022|loc=Lead Section, § 1. Elementary Algebra}} | {{harvnb|Wagner|Kieran|2018|p=[https://books.google.com/books?id=uW4ECwAAQBAJ&pg=PT225 225]}} }}</ref> 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 [[Group (mathematics)|groups]] and [[Ring (mathematics)|rings]]. These algebraic structures are not restricted to typical arithmetic operations and cover other binary operations besides them.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead Section, § 2. Abstract Algebra}} | {{harvnb|Wagner|Kieran|2018|p=[https://books.google.com/books?id=uW4ECwAAQBAJ&pg=PT225 225]}} }}</ref> 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.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3. Universal Algebra}} | {{harvnb|Grillet|2007|p=[https://link.springer.com/chapter/10.1007/978-0-387-71568-1_15 559]}} }}</ref>

[[File:Muḥammad ibn Mūsā al-Khwārizmī.png|thumb|upright=0.8|alt=Stamp of al-Khwarizmi|The word ''algebra'' comes from the title of al-Khwarizmi's book ''[[Al-Jabr|Al-Jabr]]''.<ref>{{multiref | {{harvnb|Cresswell|2010|p=[https://books.google.com/books?id=J4i3zV4vnBAC&pg=PA11 11]}} | {{harvnb|OUP Staff}} | {{harvnb|Menini|Oystaeyen|2017|p=722}} }}</ref>]]

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.<ref>{{multiref | {{harvnb|Weisstein|2003|p=46}} | {{harvnb|Renze|Weisstein}} | {{harvnb|Walz|2016|loc=[https://www.spektrum.de/lexikon/mathematik/algebra/12062 Algebra]}} }}</ref> When used as a countable noun, an algebra is a [[Algebra over a field|specific type of algebraic structure]] that involves a [[Vector space|vector space]] equipped with [[Bilinear map|a certain type of binary operation]].<ref>{{multiref | {{harvnb|Weisstein|2003|p=46}} | {{harvnb|Renze|Weisstein}} | {{harvnb|Golan|1995|pp=[https://link.springer.com/chapter/10.1007/978-94-015-8502-6_18 219–227]}} }}</ref> Depending on the context, "algebra" can also refer to other algebraic structures, like a [[Lie algebra]] or an [[Associative algebra|associative algebra]].<ref>{{harvnb|EoM Staff|2017}}</ref>

The word ''algebra'' comes from the Arabic term {{lang|ar|الجبر}} (''al-jabr'') and originally referred to the surgical treatment of bonesetting. In the 9th century, the term received a mathematical meaning in the work ''The Compendious Book on Calculation by Completion and Balancing'', in which the Persian mathematician [[Biography:Muhammad ibn Musa al-Khwarizmi|Muhammad ibn Musa al-Khwarizmi]] used it to describe a method of solving equations. The word entered the English language in the 16th century from Italian, Spanish, and medieval Latin.<ref>{{multiref | {{harvnb|Cresswell|2010|p=[https://books.google.com/books?id=J4i3zV4vnBAC&pg=PA11 11]}} | {{harvnb|OUP Staff}} | {{harvnb|Menini|Oystaeyen|2017|p=722}} | {{harvnb|Hoad|1993|p=10}} }}</ref> Initially, the meaning of the term was restricted to the [[Theory of equations|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{{efn|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|fundamental theorem of algebra]] demonstrated the existence of complex solutions of polynomials<ref name="auto7">{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|p=308}} | {{harvnb|Corry|2024|loc=§ The Fundamental Theorem of Algebra}} }}</ref> and the introduction of [[Galois theory]] characterized the polynomials that have [[Solution in radicals|general solutions]].<ref name="auto2">{{multiref | {{harvnb|Kvasz|2006|pp=314–345}} | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Galois Theory, § Applications of Group Theory}} }}</ref>}} when the scope of algebra broadened to cover the study of diverse types of algebraic operations and algebraic structures together with their underlying axioms.<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Corry|2024|loc=§ Structural Algebra}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73–74]}} }}</ref>

== Major branches ==
=== Elementary algebra ===
{{main|Elementary algebra}}

[[File:algebraic equation notation.svg|thumb|right|alt=Diagram of an algebraic expression|Algebraic expression notation:<br />&nbsp; 1 – power (exponent)<br />&nbsp; 2 – coefficient<br />&nbsp; 3 – term<br />&nbsp; 4 – operator<br />&nbsp; 5 – constant term<br />&nbsp; ''x'' ''y'' ''c'' – variables/constants]]

Elementary algebra, also referred to as school algebra, college algebra, and classical algebra,<ref>{{multiref | {{harvnb|Renze|Weisstein}} | {{harvnb|Benson|2003|pp=[https://books.google.com/books?id=nNbnCwAAQBAJ&pg=PA111 111–112]}} }}</ref> is the oldest and most basic form of algebra. It is a generalization of [[Arithmetic|arithmetic]] that relies on the use of [[Variable (mathematics)|variables]] and examines how [[Formula|formula]]s may be transformed.<ref name="auto4">{{multiref | {{harvnb|Maddocks|2008|p=129}} | {{harvnb|Berggren|2015|loc=Lead Section}} | {{harvnb|Pratt|2022|loc=§ 1. Elementary Algebra}} | {{harvnb|EoM Staff|2020|loc=§ 1. Historical Survey}} }}</ref> 

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using arithmetic operations like [[Addition|addition]], [[Subtraction|subtraction]], [[Multiplication|multiplication]], and [[Division (mathematics)|division]]. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in <math>2 + 5=7</math>.<ref name="auto1"/> 

Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables are [[Symbol|symbol]]s 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|equation]] <math>2 \times 3 = 3 \times 2</math> 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, as in the equation <math>a \times b = b \times a</math>.<ref name="auto4"/>

Elementary algebra is interested in [[Algebraic expression|algebraic expression]]s, which are formed by using  arithmetic operations to combine variables and numbers. For example, the expression <math>5x + 3</math> is an algebraic expression created by multiplying the number 5 with the variable ''x'' and adding the number 3 to the result. Other examples of algebraic equations are <math>32xyz</math> and <math>64x^2 + 7y - 13</math>.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=129–130}} | {{harvnb|Young|2010|p=[https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA999 999]}} | {{harvnb|Majewski|2004|p=347}} | {{harvnb|Buthusiem|Toth|2020|pp=[https://books.google.com/books?id=a3QeEAAAQBAJ&pg=PA24 24–28]}} | {{harvnb|Pratt|2022|loc=§ 1. Elementary Algebra}} }}</ref>

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|equals sign]] (=), as in <math>5x^2 + 6x = 3y + 4</math>. [[Inequation]]s are formed with symbols like the [[Less-than sign|less-than sign]] (<) and the [[Greater-than sign|greater-than 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 <math>x^2 = 4</math> is true if ''x'' is either 2 or -2 and false otherwise.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=129–130}} | {{harvnb|Buthusiem|Toth|2020|pp=[https://books.google.com/books?id=a3QeEAAAQBAJ&pg=PA24 24–28]}} }}</ref>

The main objective of elementary algebra is to determine for which values a statement is true. To achieve this, it relies on different techniques used to transform and manipulate statements. A key principle guiding this process is that whatever is done 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 [[Equation solving|solving the equation]] for that variable. For example, the equation <math>x - 7 = 4</math> can be solved for ''x'' by adding 7 to both sides, which isolates ''x'' on the left side and results in the equation <math>x = 11</math>.<ref>{{multiref | {{harvnb|Maddocks|2008|p=130}} | {{harvnb|Buthusiem|Toth|2020|pp=[https://books.google.com/books?id=a3QeEAAAQBAJ&pg=PA25 25–28]}} | {{harvnb|Pratt|2022|loc=§ 1. Elementary Algebra}} | {{harvnb|EoM Staff|2020|loc=§ 1. Historical Survey}} }}</ref>

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 <math>7x - 3x</math> can be replaced with the expression <math>4x</math>.<ref>{{multiref | {{harvnb|Tan|Steeb|Hardy|2012|p=[https://books.google.com/books?id=UDb0BwAAQBAJ&pg=PA306 306]}} | {{harvnb|Lamagna|2019|p=[https://books.google.com/books?id=8PSDDwAAQBAJ&pg=PA150 150]}} }}</ref> [[Factorization]] is used to rewrite an expression as a product of several factors. This technique is common for [[Polynomial|polynomial]]s to determine for which values the expression is [[Zero of a function|zero]]. For example, the polynomial <math>x^2 - 3x - 10</math> can be factorized as <math>(x + 2)(x - 5)</math>. The polynomial as a whole is zero if one of its factors is zero, i.e., if ''x'' is either -2 or 5.<ref>{{multiref | {{harvnb|Buthusiem|Toth|2020|pp=[https://books.google.com/books?id=a3QeEAAAQBAJ&pg=PA24 24–28]}} | {{harvnb|Berggren|2015|loc=§ Algebraic Expressions, § Solving Algebraic Equations}} }}</ref> For statements with several variables, [[Philosophy:Substitution (logic)#Algebra|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 <math>y = 3x</math> then one can simplify the expression <math>7xy</math> to arrive at <math>21x^2</math>.<ref>{{multiref | {{harvnb|Zill|Dewar|2011|p=[https://books.google.com/books?id=MBAwrjc3gqMC&pg=PA529 529]}} | {{harvnb|Berggren|2015|loc=§ Solving Systems of Algebraic Equations}} | {{harvnb|McKeague|2014|p=[https://books.google.com/books?id=nI7iBQAAQBAJ&pg=PA386 386]}} }}</ref> Other techniques rely on [[Commutative property|commutative]], distributive, and [[Associative property|associative properties]].<ref>{{multiref | {{harvnb|Axler|2011|pp=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA7 7–11]}} | {{harvnb|Berggren|2015|loc=§ Algebraic Expressions, § Solving Algebraic Equations}} }}</ref>

[[File:Graph (y = 0.5x - 1).svg|thumb|alt=Graph of equation "y = 0.5x - 1"|Algebraic equations can be used to describe geometric figures. All values for x and y that solve the equation are interpreted as points and drawn as a red line.]]

Elementary algebra has applications in many branches of mathematics, the sciences, business, and everyday life.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=130–131}} | {{harvnb|Walz|2016|loc=[https://www.spektrum.de/lexikon/mathematik/algebra/12062 Algebra]}}}}</ref> An important application in the field of [[Geometry|geometry]] concerns the use of algebraic equations to describe geometric figures in the form of a [[Graph of a function|graph]]. To do so, the different variables in the equation are interpreted as [[Cartesian coordinate system|coordinates]] and the values that solve the equation are interpreted as points of the graph. For example, if ''x'' is set to zero in the equation <math>y=0.5x - 1</math> then ''y'' has to be −1 for the equation to be true. This means that the x-y-pair (0, −1) is part of the graph of the equation. The x-y-pair (0, 7), by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of all x-y-pairs that solve the equation.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=130–131}} | {{harvnb|Rohde|Jain|Poddar|Ghosh|2012|p=[https://books.google.com/books?id=vk2XbZpsBOwC&pg=PT89 89]}} | {{harvnb|Walz|2016|loc=[https://www.spektrum.de/lexikon/mathematik/algebra/12062 Algebra]}}}}</ref>

=== Linear algebra ===
{{main|Linear algebra}}

Linear algebra employs the methods of elementary algebra to study [[System of linear equations|systems of linear equations]].<ref>{{multiref | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Barrera-Mora|2023|pp=[https://books.google.com/books?id=Xmu8EAAAQBAJ&pg=PR9 ix, 1–2], }} }}</ref> An [[Linear equation|equation is linear]] if no variable is multiplied with another variable and no operations like [[Exponentiation|exponentiation]], extraction of [[Nth root|roots]], and [[Logarithm|logarithm]] are applied to variables. For example, the equations <math>0.25x - 4 = y</math> and <math>x_1 - 7x_2 + 3x_3 = 0</math> are linear while the equations <math>x^2=y</math> and <math>3x_1x_2 + 15 = 0</math> are [[Nonlinear system|non-linear]]. Several equations form a system of equations if they all rely on the same set of variables.<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=2–3}} | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|EoM Staff|2011}} }}</ref>

Systems of linear equations are often expressed through [[Matrix (mathematics)|matrices]]{{efn|A matrix is a table of numbers, such as <math display="block">\begin{bmatrix}3 & -7 & 19 \\ 0.3 & 7 & -4 \end{bmatrix}</math>}} and [[Vector (mathematics and physics)|vectors]]{{efn|A vector is an array of numbers or a matrix with only one column, such as <math display="block">\begin{bmatrix} 2.1 \\ 0 \\ -1 \end{bmatrix}</math>}} 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 [[Matrix multiplication|multiplying it with the vector]] made up of the variables.<ref>{{multiref | {{harvnb|Barrera-Mora|2023|pp=[https://books.google.com/books?id=Xmu8EAAAQBAJ&pg=PR9 ix, 1, 12–13]}} | {{harvnb|Young|2010|pp=[https://books.google.com/books?id=9HRLAn326zEC&pg=PA726 726–727]}} | {{harvnb|Anton|Rorres|2013|pp=32–34}} }}</ref> For example, the system of equations

# <math>9x_1 + 3x_2 - 13x_3 = 0</math>
# <math>2.3x_1 + 7x_3 = 9</math>
# <math>-5x_1 -17x_2 = -3</math>

can be written as

<math display="block">\begin{bmatrix}9 & 3 & -13 \\ 2.3 & 0 & 9 \\ -5 & -17 & 0 \end{bmatrix} \cdot \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix}0 \\ 9 \\ -3 \end{bmatrix}</math>

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 <math>3x_1 - x_2 = 0</math> and <math>x_1 + x_2 = 8</math> then using the values 1 and 3 for <math>x_1</math> and <math>x_2</math> does not solve the system of equations because it only solves the first but not the second equation.<ref>{{multiref | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Andrilli|Hecker|2022|pp=[https://books.google.com/books?id=WtpVEAAAQBAJ&pg=PA57 57–58]}} }}</ref>

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 and inconsistent equations|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 <math>x_1 - 3x_2 = 0</math> and <math>x_1 - 3x_2 = 7</math> contradict each other since no values of <math>x_1</math> and <math>x_2</math> exist that solve both equations at the same time.<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=3–7}} | {{harvnb|Mortensen|2013|pp=[https://books.google.com/books?id=KYDrCAAAQBAJ&pg=PA73 73–74]}} | {{harvnb|Williams|2007|pp=[https://books.google.com/books?id=HLQ9ocWuCzMC&pg=PA4 4–5]}} | {{harvnb|Young|2023|pp=[https://books.google.com/books?id=pMSZEAAAQBAJ&pg=PA714 714–715]}} }}</ref>

Whether a consistent system of equations has a unique solution depends on the number of variables and the number of [[Independent equation|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 system]]s, by contrast, have more variables than equations and have an infinite number of solutions if they are consistent.<ref>{{multiref | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Harrison|Waldron|2011|p=[https://books.google.com/books?id=_sisAgAAQBAJ&pg=PT464 464]}} | {{harvnb|Anton|2013|p=[https://books.google.com/books?id=neYGCwAAQBAJ&pg=PA255 255]}} }}</ref>

[[File:Linear Function Graph.svg|thumb|alt=Graph of two linear equations|Linear equations with two variables can be interpreted geometrically as lines. The solution of a system of linear equations is where the lines intersect.]]

Many of the techniques employed in elementary algebra to solve equations are also applied in linear algebra. The substitution method starts with one equation and isolates one variable in it. It proceeds to the next equation and replaces the isolated variable with the found expression, thereby reducing the number of unknown variables by one. It applies the same process again to this and the remaining equations until the values of all variables are determined.<ref>{{multiref | {{harvnb|Young|2010|pp=[https://books.google.com/books?id=9HRLAn326zEC&pg=PA697 697–698]}} | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Sullivan|2010|pp=[https://books.google.com/books?id=6NKaDwAAQBAJ&pg=PA53 53–54]}} }}</ref> 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 <math>-x + 7y = 3</math> and <math>2x - 7y = 10</math>, it is possible to eliminate ''y'' by adding the first to the second equation, thereby revealing that ''x'' is 13.{{efn|In some cases, an equation has to be multiplied by a constant before adding it to another equation.}}<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=7–8}} | {{harvnb|Sullivan|2010|pp=[https://books.google.com/books?id=6NKaDwAAQBAJ&pg=PA55 55–56]}} }}</ref> Many advanced techniques implement algorithms based on matrix calculations, such as [[Cramer's rule]], the Gauss–Jordan elimination, and LU Decomposition.<ref>{{multiref | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Anton|Rorres|2013|pp=7–8, 11, 491}} }}</ref>

On a geometric level, systems of equations can be interpreted as geometric figures. For systems that have two variables, each equation represents a [[Line (geometry)|line]] in [[Two-dimensional space|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.<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=3–5}} | {{harvnb|Young|2010|pp=[https://books.google.com/books?id=9HRLAn326zEC&pg=PA696 696–697]}} | {{harvnb|Williams|2007|pp=[https://books.google.com/books?id=HLQ9ocWuCzMC&pg=PA4 4–5]}} }}</ref> 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 [[Plane (mathematics)|planes]] in [[Three-dimensional space|three-dimensional space]] and the points where all planes intersect solve the system of equations.<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=3–5}} | {{harvnb|Young|2010|p=[https://books.google.com/books?id=9HRLAn326zEC&pg=PA713 713]}} | {{harvnb|Williams|2007|pp=[https://books.google.com/books?id=HLQ9ocWuCzMC&pg=PA4 4–5]}} }}</ref>

=== Abstract algebra ===
{{main|Abstract algebra}}

Abstract algebra, also called modern algebra,<ref>{{multiref | {{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA1 1]}} | {{harvnb|Dominich|2008|p=[https://books.google.com/books?id=uEedNKV3nlUC&pg=PA19 19]}} }}</ref> studies different types of algebraic structures. An algebraic structure is a framework for understanding [[Operation (mathematics)|operations]] on [[Philosophy:Mathematical object|mathematical object]]s, 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 [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].<ref>{{multiref | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead Section, § 2. Abstract Algebra}} | {{harvnb|Gilbert|Nicholson|2004|pp=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA1 1–3]}} | {{harvnb|Dominich|2008|p=[https://books.google.com/books?id=uEedNKV3nlUC&pg=PA19 19]}} }}</ref>

[[File:Binary operations as black box.svg|thumb|alt=Diagram of binary operation|Many algebraic structures rely on binary operations, which take two objects as input and combine them into a single object as output.]]

On a formal level, an algebraic structure is a [[Set (mathematics)|set]]{{efn|A set is a collection of elements, such as numbers, vectors, or other sets. [[Set theory]] describes the laws and properties of sets.<ref>{{harvnb|Tanton|2005|p=460}}</ref>}} of mathematical objects, called the underlying set, together with one or several operations.{{efn|According to some definitions, algebraic structures include a distinguished element as an additional component, such as the identity element in the case of multiplication.<ref name="auto5">{{harvnb|Ovchinnikov|2015|p=[https://books.google.com/books?id=UMbXBgAAQBAJ&pg=PA27 27]}}</ref>}} Abstract algebra usually restricts itself to [[Binary operation|binary operation]]s that take any two objects from the underlying set as inputs and map them to another object from this set as output.<ref>{{multiref | {{harvnb|Ovchinnikov|2015|p=[https://books.google.com/books?id=UMbXBgAAQBAJ&pg=PA27 27]}} | {{harvnb|Fiche|Hebuterne|2013|p=[https://books.google.com/books?id=TqkckiuuXg8C&pg=PT326 326]}} | {{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 4]}} | {{harvnb|Pratt|2022|loc=Lead Section, § 2. Abstract Algebra}} }}</ref> For example, the algebraic structure <math>\langle \N, + \rangle</math> has the natural numbers as the underlying set. Addition is its binary operation and takes two numbers as input to produce one number in the form of the sum as output.<ref name="auto5"/> The underlying set can contain mathematical objects other than numbers and the operations are not restricted to regular arithmetic operations.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead Section, § 2. Abstract Algebra}} }}</ref>

Abstract algebra classifies algebraic structures based on the laws or [[Axiom|axiom]]s 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|identity element]] and [[Inverse element|inverse elements]]. An operation{{efn|Symbols like <math>\circ</math> and <math>\star</math> are often used in abstract algebra to represent any operation that may or may not resemble arithmetic operations.}} is associative if the order of several applications does not matter, i.e., if <math>(a \circ b) \circ c</math> is the same as <math>a \circ (b \circ c)</math> 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 <math>a \circ e = e \circ a = a</math>. An operation admits inverse elements if for any element <math>a</math> there exists a reciprocal element <math>a^{-1}</math> that reverses its effects. If an element is linked to its inverse then the result is the neutral element ''e'', expressed formally as <math>a \circ a^{-1} = a^{-1} \circ a = e</math>. Every algebraic structure that fulfills these requirements is a group.<ref>{{multiref | {{harvnb|Rowland|Weisstein}} | {{harvnb|EoM Staff|2016|loc=§ Definition}} | {{harvnb|Khattar|Agrawal|2023|pp=[https://books.google.com/books?id=7-nIEAAAQBAJ&pg=PA4 4–6]}} | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead Section, § 2. Abstract Algebra}} }}</ref> For example, <math>\langle \Z, + \rangle</math> 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 <math>a</math> is <math>-a</math>.<ref>{{multiref | {{harvnb|Khattar|Agrawal|2023|pp=[https://books.google.com/books?id=7-nIEAAAQBAJ&pg=PA6 6–7]}} | {{harvnb|Maddocks|2008|pp=131–132}} }}</ref> The natural numbers, by contrast, do not form a group since they contain only positive numbers and therefore lack inverse elements.<ref>{{multiref | {{harvnb|McWeeny|2002|p=[https://books.google.com/books?id=x3fjIXY93TsC&pg=PA6 6]}} | {{harvnb|Kramer|Pippich|2017|p=[https://books.google.com/books?id=nvM-DwAAQBAJ&pg=PA49 49]}} }}</ref> [[Group theory]] is the subdiscipline of abstract algebra studying groups.<ref>{{harvnb|Tanton|2005|p=242}}</ref>

[[File:Magma to group4.svg|thumb|alt=Diagram of relations between some algebraic structures|Diagram of relations between some algebraic structures]]

A ring is an algebraic structure with two operations (<math>\circ</math> and <math>\star</math>) 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 <math>a \circ b = b \circ a</math> is true for all elements. The axiom of distributivity governs how the two operations interact with each other. It states that <math>a \star (b \circ c) = (a \star b) \circ (a \star c)</math> and <math>(b \circ c) \star a = (b \star a) \circ (c \star a)</math>.{{efn|Some definitions additionally require that the second operation is associative.<ref>{{harvnb|Weisstein|2024c}}</ref>}}<ref>{{multiref | {{harvnb|Weisstein|2024c}} | {{harvnb|EoM Staff|2016a}} | {{harvnb|Maxwell|2009|pp=[https://books.google.com/books?id=yD0irRUE_u4C&pg=PA73 73–74]}} | {{harvnb|Pratt|2022|loc=§ 2.3 Rings}} }}</ref> The [[Ring of integers|ring of integers]] is a ring of the form <math>\langle \Z, +, \times \rangle</math>.<ref>{{harvnb|Terr|Weisstein}}</ref> 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.{{efn|For the second operation, there is usually one element, corresponding to 0, that does not require an inverse element.<ref>{{harvnb|Weisstein|2024b}}</ref>}}<ref>{{multiref | {{harvnb|Weisstein|2024a}} | {{harvnb|Weisstein|2024b}} | {{harvnb|Pratt|2022|loc=§ 2.4 Fields}} }}</ref> The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of <math>7</math> is <math>\tfrac{1}{7}</math>, which is not part of the integers. The rational numbers, the real numbers, and the complex numbers each form a field.<ref>{{multiref | {{harvnb|Weisstein|2024a}} | {{harvnb|Irving|2004|p=236}} | {{harvnb|Negro|2022|p=[https://books.google.com/books?id=MIdoEAAAQBAJ&pg=PA365 365]}} }}</ref>

Besides groups, rings, and fields, there are many other algebraic structures studied by abstract algebra. They include [[Magma (algebra)|magmas]], [[Semigroup|semigroups]], [[Monoid|monoids]], [[Abelian group|abelian groups]], [[Commutative ring|commutative rings]], [[Module (mathematics)|modules]], [[Lattice (order)|lattices]], [[Vector space|vector spaces]], and [[Algebra over a field|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.<ref>{{multiref | {{harvnb|Pratt|2022|loc=Lead Section, § 2. Abstract Algebra}} | {{harvnb|EoM Staff|2020|loc=The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.}} }}</ref> For example, a magma becomes a semigroup if its operation is associative.<ref>{{harvnb|Cooper|2011|p=[https://books.google.com/books?id=Fybzl6QB62gC&pg=PA60 60]}}</ref>

=== Universal algebra ===
{{main|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 operation|ternary operation]]s. 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.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3. Universal Algebra}} | {{harvnb|Insall|Sakharov}} }}</ref> One of those structural features concerns the [[Identity (mathematics)#Logic and universal algebra|identities]] that are true in different algebraic structures. In this context, an identity is a [[Universal quantification|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 <math>a \circ b</math> is identical to <math>b \circ a</math> for all elements.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3.2 Equational Logic}} | {{harvnb|Mal’cev|1973|pp=210–211}} | {{harvnb|Insall|Sakharov}} }}</ref> Two algebraic structures that share all their identities are said to belong to the same [[Variety (universal algebra)|variety]].<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3. Universal Algebra}} | {{harvnb|Mal’cev|1973|pp=210–211}} }}</ref> For instance, the ring of integers and the [[Polynomial ring|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.<ref>{{multiref | {{harvnb|Cox|Little|O'Shea|2015|p=[https://books.google.com/books?id=yL7yCAAAQBAJ&pg=PA268 268]}} | {{harvnb|Insall|Sakharov}} | {{harvnb|Negro|2022|p=[https://books.google.com/books?id=MIdoEAAAQBAJ&pg=PA365 365]}} | {{harvnb|Weisstein|2024a}} }}</ref>

Besides identities, universal algebra is also interested in structural features associated with [[Quasi-identity|quasi-identities]]. A quasi-identity is an identity that only needs to be present under certain conditions.{{efn|The conditions take the form of a [[Horn clause]].}} 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|quasivariety]].<ref>{{multiref | {{harvnb|Mal’cev|1973|pp=210–211}} | {{harvnb|Pratt|2022|loc=§ 3. Universal Algebra}} | {{harvnb|Artamonov|2003|p=[https://books.google.com/books?id=sLDGY4Hk8V0C&pg=PA873 873]}} }}</ref>

[[File:Venn A subset B.svg|thumb|alt=Venn diagram of a set and its subset|[[Subalgebra|Subalgebra]]s restrict their operations to a subset of the underlying set of the original algebraic structure.]]

[[Homomorphism]]s are a tool in universal algebra to examine structural features by comparing two algebraic structures.<ref>{{multiref | {{harvnb|Insall|Sakharov}} | {{harvnb|Pratt|2022|loc=§ 3.3 Birkhoff’s Theorem}} | {{harvnb|Grätzer|2008|p=34}} }}</ref> A homomorphism is a function that takes the elements of the underlying set of one algebraic structure as input and has the elements of the underlying set of another algebraic structure as output. Its special feature is that it preserves certain structural characteristics of the algebraic structures. If the two algebraic structures use binary operations and have the form <math>\langle A, \circ \rangle</math> and <math>\langle B, \star \rangle</math> then the function <math>h: A \to B</math> is a homomorphism if it fulfills the following requirement: <math>h(x \circ y) = h(x) \star h(y)</math>. The existence of a homomorphism reveals that the operation <math>\star</math> in the second algebraic structure plays the same role as the operation <math>\circ</math> does in the first algebraic structure.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3.3 Birkhoff’s Theorem}} | {{harvnb|Insall|Sakharov}} | {{harvnb|Silvia|Robinson|1979|p=[https://books.google.com/books?id=Ecgfjh-MpU0C&pg=PA82 82]}} }}</ref>

Another tool of comparison is the relation between an algebraic structure and its [[Subalgebra#Subalgebras in universal algebra|subalgebra]].<ref name="auto8">{{multiref | {{harvnb|Indurkhya|2013|pp=[https://books.google.com/books?id=foTrCAAAQBAJ&pg=PA217 217–218]}} | {{harvnb|Pratt|2022|loc=§ 3.3 Birkhoff’s Theorem}} | {{harvnb|Grätzer|2008|p=34}} }}</ref> If <math>\langle A, \circ \rangle</math> is a subalgebra of <math>\langle B, \circ \rangle</math> then the set A is a [[Subset|subset]] of B.{{efn|This means that all the elements of A are also elements of B but B may contain elements that are not found in A.}} A subalgebra has to use the same operations as the algebraic structure{{efn|According to some definitions, it is also possible for a subalgebra to have fewer operations.<ref name="auto6">{{harvnb|Indurkhya|2013|pp=[https://books.google.com/books?id=foTrCAAAQBAJ&pg=PA217 217–218]}}</ref>}} and they have to follow the same axioms. This includes the requirement that all operations in the subalgebra are [[Closure (mathematics)|closed]] in A, meaning that they only produce elements that belong to A.<ref name="auto8"/> For example, the set of [[Parity (mathematics)|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.<ref name="auto6"/>

== History ==
{{main|History of algebra|Timeline of algebra}}

[[File:Rhind Mathematical Papyrus.jpg|thumb|alt=Rhind Papyrus|The Rhind Papyrus from ancient Egypt, dated around 1650 BCE, is one of the earliest documents discussing algebraic problems.]]

The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations, usually in the form of a [[Theory of equations|theory of equations]]. These developments happend in the ancient period in diverse regions such as [[History:Babylonia|Babylonia]], Egypt, {{wipe|Ancient Greece}}, China, and India. One of the earliest documents is the Rhind Papyrus from ancient Egypt, which was written around 1650 BCE{{efn|The exact date is disputed.}} 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 equation|quadratic polynomial equations]], such as the method of [[Completing the square|completing the square]].<ref>{{multiref | {{harvnb|Tanton|2005|p=9}} | {{harvnb|Kvasz|2006|p=290}} | {{harvnb|Corry|2024|loc=§ Problem Solving in Egypt and Babylon}} }}</ref>

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 [[Biography:Pythagoras|Pythagoras]]' formulation of the [[Difference of two squares|difference of two squares]] method and later in [[Euclid's Elements]].<ref>{{multiref | {{harvnb|Tanton|2005|p=9}} | {{harvnb|Kvasz|2006|p=290}} | {{harvnb|Corry|2024|loc=§ The Pythagoreans and Euclid}} }}</ref> In the 3rd century BCE, [[Biography:Diophantus|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.<ref>{{multiref | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} | {{harvnb|Sialaros|2018|p=[https://books.google.com/books?id=2PZYDwAAQBAJ&pg=PT55 55]}} | {{harvnb|Musielak|2020|p=[https://books.google.com/books?id=iqHYDwAAQBAJ&pg=PA36 36]}} | {{harvnb|Corry|2024|loc=§ Diophantus}} }}</ref> 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.<ref>{{harvnb|Higgins|2015|p=[https://books.google.com/books?id=QANiCgAAQBAJ&pg=PA89 89]}}</ref>

[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|upright=0.7|thumb|alt=Title page of The Compendious Book on Calculation by Completion and Balancing|al-Khwarizmi's The Compendious Book on Calculation by Completion and Balancing provided a general account of how linear and quadratic equations can be solved through the methods of "reducing" and "balancing".]]

It is controversial to what extent these early developments should be considered part of algebra proper rather than precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications.<ref>{{multiref | {{harvnb|Kvasz|2006|pp=290–291}} | {{harvnb|Sialaros|2018|p=[https://books.google.com/books?id=2PZYDwAAQBAJ&pg=PT55 55]}} | {{harvnb|Boyer|Merzbach|2011|p=[https://books.google.com/books?id=bR9HAAAAQBAJ&pg=PA161 161]}} | {{harvnb|Derbyshire|2006|p=[https://books.google.com/books?id=mLqaAgAAQBAJ&pg=PT39 31]}} }}</ref> This changed with the Persian mathematician al-Khwarizmi,{{efn|Some historians consider him the "father of algebra" while others reserve this title for Diophantus.<ref>{{multiref | {{harvnb|Boyer|Merzbach|2011|p=[https://books.google.com/books?id=bR9HAAAAQBAJ&pg=PA161 161]}} | {{harvnb|Derbyshire|2006|p=[https://books.google.com/books?id=mLqaAgAAQBAJ&pg=PT39 31]}} }}</ref> }} 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.<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|pp=291–293}} | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} }}</ref> Other influential contributions to algebra came from the Arab mathematician [[Biography:Thābit ibn Qurra|Thābit ibn Qurra]] in the 9th century and the Persian mathematician [[Biography:Omar Khayyam|Omar Khayyam]] in the 11th and 12th centuries.<ref>{{multiref | {{harvnb|Waerden|2013|pp=3, 15–16, 24–25}} | {{harvnb|Jr|2010|p=[https://books.google.com/books?id=giEkCQAAQBAJ&pg=PA82 82]}} | {{harvnb|Pickover|2009|p=[https://books.google.com/books?id=JrslMKTgSZwC&pg=PA90 90]}} }}</ref>

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.<ref>{{multiref | {{harvnb|Tanton|2005|pp=9–10}} | {{harvnb|Corry|2024|loc=§ The Equation in India and China}} }}</ref> The Indian mathematicians Mahāvīra in the 9th century and [[Biography:Bhāskara II|Bhāskara II]] in the 12th century further refined Brahmagupta's methods and concepts.<ref>{{multiref | {{harvnb|Seshadri|2010|p=[https://books.google.com/books?id=w_JdDwAAQBAJ&pg=PA156 156]}} | {{harvnb|Emch|Sridharan|Srinivas|2005|p=[https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA20 20]}} }}</ref> In 1247, the Chinese mathematician [[Biography:Qin Jiushao|Qin Jiushao]] wrote the Mathematical Treatise in Nine Sections, which includes [[Horner's method|an algorithm]] for the [[Polynomial evaluation|numerical evaluation of polynomials]], including polynomials of higher degrees.<ref>{{multiref | {{harvnb|Smorynski|2007|p=[https://books.google.com/books?id=qY657eFq7UgC&pg=PA137 137]}} | {{harvnb|Zwillinger|2002|p=[https://books.google.com/books?id=gE_MBQAAQBAJ&pg=PA812 812]}} }}</ref>

{{multiple image
|perrow=2
|total_width=350
|image1=Francois Viete.jpeg
|alt1=Drawing of François Viète
|link1=François Viète
|image2=Frans Hals - Portret van René Descartes (cropped).jpg
|alt2=Painting of René Descartes
|link2=René Descartes
|footer=[[Biography:François Viète|François Viète]] and [[Biography:René Descartes|René Descartes]] invented a symbolic notation to express equations as mathematical formulas.
}}

The Italian mathematician [[Biography:Fibonacci|Fibonacci]] brought al-Khwarizmi's ideas and techniques to Europe in books like his ''[[Liber Abaci]]''.<ref>{{multiref | {{harvnb|Waerden|2013|pp=32–35}} | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|p=293}} }}</ref> 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 equation|cubic]] and [[Quartic equation|quartic equations]].<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|p=293}} | {{harvnb|Corry|2024|loc=§ Cardano and the Solving of Cubic and Quartic Equations}} }}</ref> In the 16th and 17th centuries, the French mathematicians [[Biography:François Viète|François Viète]] and [[Biography:René Descartes|René Descartes]] introduced letters and symbols to denote variables and operations, making it possible to express equations as mathematical formulas. Their predecessors had relied on verbal descriptions of problems and solutions.<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|pp=291–292, 297–298, 302}} | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Viète and the Formal Equation, § Analytic Geometry}} }}</ref> 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.<ref>{{multiref | {{harvnb|Hazewinkel|1994|p=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73]}} | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} }}</ref>

[[File:Garrett Birkhoff.jpeg|thumb|alt=Photo of Garrett Birkhoff|Garrett Birkhoff developed many of the foundational concepts of universal algebra.]]

Many attempts in the 17th and 18th centuries to find general solutions{{efn|A general solution or a [[Solution in radicals|solution in radicals]] is a [[Closed-form expression|closed form]] algebraic equation that isolates the variable on one side. For example, the general solution to quadratic equations of the form <math>ax^2 + bx + c = 0</math> is  <math>x = \frac{-b \pm \sqrt {b^2-4ac\  }}{2a}</math>. The absence of general solutions does not mean that there are no numerical solutions.{{sfn|Igarashi|Altman|Funada|Kamiyama|2014|p=[https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA103 103]}}{{sfn|Sun|Zhang|2020|p=[https://books.google.com/books?id=SZ3hDwAAQBAJ&pg=PA94 94]}} }} to polynomials of degree five and higher failed.<ref name="auto3">{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Impasse with Radical Methods}} }}</ref> At the end of the 18th century, the German mathematician [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]] proved the [[Fundamental theorem of algebra|fundamental theorem of algebra]], which describes the existence of [[Zero of a function|zeros]] of polynomials of any degree without providing a general solution.<ref name="auto7"/> At the beginning of the 19th century, the Italian mathematician [[Biography:Paolo Ruffini|Paolo Ruffini]] and the Norwegian mathematician [[Biography:Niels Henrik Abel|Niels Henrik Abel]] were [[Abel–Ruffini theorem|able to show]] that no general solution exists for polynomials of degree five and higher.<ref name="auto3"/> In response to and shortly after their findings, the French mathematician [[Biography:Évariste Galois|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|group theory]].<ref name="auto2"/> Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory.<ref>{{harvnb|Corry|2024|loc=§ Applications of Group Theory}}</ref>

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|abstract algebra]]. This approach explored the axiomatic basis of arbitrary algebraic operations.<ref>{{multiref | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Corry|2024|loc=§ Structural Algebra}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73–74]}} }}</ref> The invention of new algebraic systems based on different operations and elements accompanied this development, such as [[Boolean algebra]], [[Vector algebra|vector algebra]], and matrix algebra.<ref>{{multiref | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Corry|2024|loc=§ Matrices, § Quaternions and Vectors}} }}</ref> Influential early developments in abstract algebra were made by the German mathematicians [[Biography:David Hilbert|David Hilbert]], [[Biography:Ernst Steinitz|Ernst Steinitz]], [[Biography:Emmy Noether|Emmy Noether]], and [[Biography:Emil Artin|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.<ref>{{multiref | {{harvnb|EoM Staff|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Hilbert and Steinitz, § Noether and Artin}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73–74]}} }}</ref> The basic idea of the even more general approach associated with universal algebra was first conceived by the English mathematician [[Biography:Alfred North Whitehead|Alfred North Whitehead]] in his 1898 book ''A Treatise on Universal Algebra''. Starting in the 1930s, the American mathematician [[Biography:Garrett Birkhoff|Garrett Birkhoff]] expanded these ideas and developed many of the foundational concepts of this field.<ref>{{multiref | {{harvnb|Grätzer|2008|p=[https://books.google.com/books?id=8lNkXPJas4wC&pg=PR7 vii]}} | {{harvnb|Chang|Keisler|1990|p=[https://books.google.com/books?id=uiHq0EmaFp0C&pg=PA603 603]}} | {{harvnb|Knoebel|2011|p=[https://books.google.com/books?id=VWS_sgO2uvgC&pg=PA5 5]}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA74 74–75]}} }}</ref> Closely related developments were the formulation of [[Model theory|model theory]], [[Category theory|category theory]], [[Topological algebra|topological algebra]], [[Homological algebra|homological algebra]], [[Lie algebra]]s, [[Free algebra|free algebra]]s, and homology groups.<ref>{{multiref | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA74 74–75]}} | {{harvnb|Grätzer|2008|p=338}} | {{harvnb|Pratt|2022|loc=§ 6. Free Algebras}} }}</ref>

{{clear}}

== In various fields ==
=== Other branches of mathematics ===
The algebraization of mathematics is the process of applying algebraic methods and principles to other branches of mathematics. This involves the use of symbols in the form of variables to express mathematical insights on a more general level. Another key aspect is to apply structures to model how different types of objects interact without the need to specify what the nature of these objects is besides their patterns of interaction.<ref>{{multiref | {{harvnb|Mancosu|1999|pp=[https://books.google.com/books?id=60qaEePdqcoC&pg=PA84 84–85]}} | {{harvnb|Kleiner|2007|p=[https://books.google.com/books?id=udj-1UuaOiIC&pg=PA100 100]}} | {{harvnb|Pratt|2022|loc=§ 5. Algebraization of Mathematics}} }}</ref> This is possible because the abstract patterns studied by algebra have many concrete applications in fields like [[Geometry|geometry]], [[Topology|topology]], [[Number theory|number theory]], and [[Calculus|calculus]].<ref>{{multiref | {{harvnb|Kleiner|2007|p=[https://books.google.com/books?id=udj-1UuaOiIC&pg=PA100 100]}} | {{harvnb|Pratt|2022|loc=§ 5. Algebraization of Mathematics}} | {{harvnb|Maddocks|2008|p=130}} }}</ref>

[[File:Sphere Quadric.png|thumb|alt=Rendered image of a sphere|The algebraic equation <math>x^2 + y^2 + z^2 = 1</math> describes a [[Sphere|sphere]] at the [[Origin (mathematics)|origin]] with a radius of 1.]]

Geometry is interested in geometric figures, which can be described with algebraic statements. For example, the equation <math>y = 3x - 7</math> describes a line in two-dimensional space while the equation <math>x^2 + y^2 + z^2 = 1</math> corresponds to a [[Sphere|sphere]] in three-dimensional space. Of special interest to [[Algebraic geometry|algebraic geometry]] are algebraic varieties,{{efn|Algebraic varieties studied in geometry are different from the more general varieties studied in universal algebra.}} which are solutions to systems of polynomial equations that can be used to describe more complex geometric figures.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 5.1 Algebraic Geometry}} | {{harvnb|Danilov|2006|pp=[https://books.google.com/books?id=-QMWR-x66XUC&pg=PA172 172, 174]}} }}</ref> Topology studies the properties of geometric figures or [[Topological space|topological space]]s that are preserved under operations of continuous deformation. [[Algebraic topology]] relies on algebraic theories like [[Group theory|group theory]] to classify topological spaces. For example, homotopy groups classify topological spaces based on the existence of [[Loop (topology)|loops]] or holes in them.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 5.3 Algebraic Topology}} | {{harvnb|Rabadan|Blumberg|2019|pp=[https://books.google.com/books?id=2967DwAAQBAJ&pg=PA49 49–50]}} | {{harvnb|Nakahara|2018|p=[https://books.google.com/books?id=p2C1DwAAQBAJ&pg=PA121 121]}} | {{harvnb|Weisstein|2003|pp=52–53}} }}</ref> 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|ring of integers]].<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 5.2 Algebraic Number Theory}} | {{harvnb|Jarvis|2014|p=[https://books.google.com/books?id=0j0qBAAAQBAJ&pg=PA1 1]}} | {{harvnb|Viterbo|Hong|2011|p=[https://books.google.com/books?id=d89QRR24jbMC&pg=PA127 127]}} }}</ref> The insights of algebra are also relevant to calculus, which utilizes mathematical expressions to examine rates of change and [[Integral|accumulation]]. It relies on algebra to understand how these expressions can be transformed and what role variables play in them.<ref>{{multiref | {{harvnb|Kilty|McAllister|2018|pp=x, 347, 589}} | {{harvnb|Edwards|2012|pp=[https://books.google.com/books?id=sZIFcJ8DJAIC&pg=PR9 ix–x]}} }}</ref> Because of its presence throughout mathematics, the influence of algebra extends to many sciences and related fields, including [[Physics:Physics|physics]], [[Computer science|computer science]], and [[Engineering:Engineering|engineering]].<ref>{{multiref | {{harvnb|Corrochano|Sobczyk|2011|p=[https://books.google.com/books?id=GUHhBwAAQBAJ&pg=PR17 xvii]}} | {{harvnb|Neri|2019|p=[https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PR12 xii]}} }}</ref>

=== Logic ===
[[Logic]] is the study of correct reasoning.<ref>{{harvnb|Hintikka|2019|loc=Lead Section, § Nature and Varieties of Logic}}</ref> [[Algebraic logic]] employs algebraic methods to describe and analyze the structures and patterns that underlie [[Philosophy:Logical reasoning|logical reasoning]].<ref>{{multiref | {{harvnb|Halmos|1956|p=363}} | {{harvnb|Burris|Legris|2021|loc=§ 1. Introduction}} }}</ref> 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 [[Philosophy:Inference|inference]]s. 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.<ref>{{harvnb|Andréka|Németi|Sain|2001|pp=[https://link.springer.com/chapter/10.1007/978-94-017-0452-6_3 133–134]}}</ref>

[[Boolean algebra]] is an influential device in algebraic logic to describe  propositional logic.<ref>{{multiref | {{harvnb|EoM Staff|2020b|loc=§ Concrete Algebraic Logic}} | {{harvnb|Pratt|2022|loc=§ 5.4 Algebraic Logic}} | {{harvnb|Plotkin|2012|pp=[https://books.google.com/books?id=-v3xCAAAQBAJ&pg=PA155 155–156]}} | {{harvnb|Jansana|2022|loc=Lead Section}} }}</ref> [[Proposition]]s are statements that can be true or false.<ref>{{harvnb|McGrath|Frank|2023|loc=Lead Section}}</ref> 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 [[Philosophy:Truth value|truth value]] of a complex proposition depends on the truth values of its constituents.<ref>{{multiref | {{harvnb|Boschini|Hansen|Wolf|2022|p=[https://books.google.com/books?id=huZtEAAAQBAJ&pg=PA21 21]}} | {{harvnb|Brody|2006|pp=535–536}} | {{harvnb|Franks|2023|loc=Lead Section}} }}</ref> 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.<ref>{{multiref | {{harvnb|EoM Staff|2020b|loc=§ Concrete Algebraic Logic}} | {{harvnb|Plotkin|2012|pp=[https://books.google.com/books?id=-v3xCAAAQBAJ&pg=PA155 155–156]}} | {{harvnb|Kachroo|Özbay|2018|pp=[https://books.google.com/books?id=qQNbDwAAQBAJ&pg=PA176 176–177]}} }}</ref> Algebraic logic is also interested in how more complex [[Logic#Systems of logic|systems of logic]] can be described through algebraic structures and which varieties and quasivarities these algebraic structures belong to.<ref>{{multiref | {{harvnb|EoM Staff|2020b|loc=§ Abstract Algebraic Logic}} | {{harvnb|Jansana|2022|loc=§ 4. Algebras}} }}</ref>

=== Education ===
{{main|Mathematics education}}

[[File:Balance scale.svg|thumb|upright=1.3|alt=Diagram of a balance scale|Balance scales are used in algebra education to help students understand how equations can be transformed to determine unknown values.<ref name="auto">{{harvnb|Gardella|DeLucia|2020|pp=[https://books.google.com/books?id=HBXFDwAAQBAJ&pg=PA19 19–22]}}</ref>]]

Algebra education mostly focuses on elementary algebra, which is one of the reasons why it is referred to as school algebra. It is usually introduced in secondary education after students have mastered the fundamentals of arithmetic.<ref>{{multiref | {{harvnb|Arcavi|Drijvers|Stacey|2016|p=xiii}} | {{harvnb|Dekker|Dolk|2011|p=[https://books.google.com/books?id=7sVFaMhwackC&pg=PA69 69]}} }}</ref> 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 until the unknown quantity can be determined.<ref>{{multiref | {{harvnb|Arcavi|Drijvers|Stacey|2016|pp=2–5}} | {{harvnb|Drijvers|Goddijn|Kindt|2011|pp=[https://books.google.com/books?id=7sVFaMhwackC&pg=PA8 8–10, 16–18]}} }}</ref>

A common example to introduce students to the basic problems of algebra is to use balance scales to represent equations. The mass of some weights on the scale is unknown and represents variables. Solving an equation corresponds to adding and removing weights on both sides in such a way that the sides stay in balance until the only weight remaining on one side is the weight of unknown mass.<ref name="auto"/> The use of [[Word problem (mathematics education)|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 has twice as many apples as her brother. 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.<ref>{{multiref | {{harvnb|Arcavi|Drijvers|Stacey|2016|pp=58–59}} | {{harvnb|Drijvers|Goddijn|Kindt|2011|p=[https://books.google.com/books?id=7sVFaMhwackC&pg=PA13 13]}} }}</ref>

== See also ==
{{div col|colwidth=30em}}
* Algebra over a set
* [[Algebra tile]]
* [[Algebraic combinatorics]]
* [[C*-algebra]]
* [[Composition algebra]]
* [[Computer algebra]]
* [[Exterior algebra]]
* [[F-algebra]]
* [[F-coalgebra]]
* [[Heyting algebra]]
* [[Hopf algebra]]
* [[Non-associative algebra]]
* [[Outline of algebra]]
* Relational algebra
* [[Sigma-algebra]]
* [[Symmetric algebra]]
* T-algebra
* [[Tensor algebra]]
{{div col end}}

== References ==
=== Notes ===
{{notelist}}

=== Citations ===
{{Reflist}}

=== Sources ===
{{Refbegin|30em}}
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* {{cite book |last1=Cooper |first1=Ellis D. |title=Mathematical Mechanics: From Particle to Muscle |date=2011 |publisher=World Scientific |isbn=978-981-4289-70-2 |url=https://books.google.com/books?id=Fybzl6QB62gC&pg=PA60 |language=en |access-date=2024-01-20 |archive-date=2024-01-20 |archive-url=https://web.archive.org/web/20240120093846/https://books.google.com/books?id=Fybzl6QB62gC&pg=PA60 |url-status=live }}
* {{cite book |last1=Corrochano |first1=Eduardo Bayro |last2=Sobczyk |first2=Garret |title=Geometric Algebra with Applications in Science and Engineering |date=2011 |publisher=Springer Science & Business Media |isbn=978-1-4612-0159-5 |url=https://books.google.com/books?id=GUHhBwAAQBAJ&pg=PR17 |language=en |access-date=2024-01-24 |archive-date=2024-01-23 |archive-url=https://web.archive.org/web/20240123185541/https://books.google.com/books?id=GUHhBwAAQBAJ&pg=PR17 |url-status=live }}
* {{cite web |last1=Corry |first1=Leo |title=Algebra |url=https://www.britannica.com/science/algebra |website=Encyclopædia Britannica |access-date=25 January 2024 |language=en |date=2024 |archive-date=19 January 2024 |archive-url=https://web.archive.org/web/20240119233613/https://www.britannica.com/science/algebra |url-status=live }}
* {{cite book |last1=Cox |first1=David A. |last2=Little |first2=John |last3=O'Shea |first3=Donal |title=Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra |date=2015 |publisher=Springer |isbn=978-3-319-16721-3 |url=https://books.google.com/books?id=yL7yCAAAQBAJ&pg=PA268 |language=en |access-date=2024-01-21 |archive-date=2024-01-29 |archive-url=https://web.archive.org/web/20240129081618/https://books.google.com/books?id=yL7yCAAAQBAJ&pg=PA268#v=onepage&q&f=false |url-status=live }}
* {{cite book |last1=Cresswell |first1=Julia |title=Oxford Dictionary of Word Origins |date=2010 |publisher=OUP Oxford |isbn=978-0-19-954793-7 |url=https://books.google.com/books?id=J4i3zV4vnBAC&pg=PA11 |language=en |access-date=2024-01-27 |archive-date=2024-01-27 |archive-url=https://web.archive.org/web/20240127130135/https://books.google.com/books?id=J4i3zV4vnBAC&pg=PA11 |url-status=live }}
* {{cite book |last1=Danilov |first1=V. I. |title=Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes |date=2006 |publisher=Springer Science & Business Media |isbn=978-3-540-51995-9 |url=https://books.google.com/books?id=-QMWR-x66XUC&pg=PA172 |language=en |chapter=II. Algebraic Varieties and Schemes |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094525/https://books.google.com/books?id=-QMWR-x66XUC&pg=PA172 |url-status=live }}
* {{cite book |last1=Dekker |first1=Truus |last2=Dolk |first2=Maarten |editor1-last=Drijvers |editor1-first=Paul |title=Secondary Algebra Education: Revisiting Topics and Themes and Exploring the Unknown |date=2011 |publisher=Springer Science & Business Media |isbn=978-94-6091-334-1 |url=https://books.google.com/books?id=7sVFaMhwackC&pg=PA5 |language=en |chapter=3. From Arithmetic to Algebra |access-date=2024-01-24 |archive-date=2024-01-23 |archive-url=https://web.archive.org/web/20240123185535/https://books.google.com/books?id=7sVFaMhwackC&pg=PA5 |url-status=live }}
* {{cite book |last1=Derbyshire |first1=John |title=Unknown Quantity: A Real and Imaginary History of Algebra |date=2006 |publisher=National Academies Press |isbn=978-0-309-09657-7 |url=https://books.google.com/books?id=mLqaAgAAQBAJ&pg=PT39 |language=en |chapter=2. The Father of Algebra |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126140355/https://books.google.com/books?id=mLqaAgAAQBAJ&pg=PT39 |url-status=live }}
* {{cite book |last1=Dominich |first1=Sándor |title=The Modern Algebra of Information Retrieval |date=2008 |publisher=Springer Science & Business Media |isbn=978-3-540-77659-8 |url=https://books.google.com/books?id=uEedNKV3nlUC&pg=PA19 |language=en |access-date=2024-01-20 |archive-date=2024-01-20 |archive-url=https://web.archive.org/web/20240120093845/https://books.google.com/books?id=uEedNKV3nlUC&pg=PA19 |url-status=live }}
* {{cite book |last1=Drijvers |first1=Paul |last2=Goddijn |first2=Aad |last3=Kindt |first3=Martin |editor1-last=Drijvers |editor1-first=Paul |title=Secondary Algebra Education: Revisiting Topics and Themes and Exploring the Unknown |date=2011 |publisher=Springer Science & Business Media |isbn=978-94-6091-334-1 |url=https://books.google.com/books?id=7sVFaMhwackC&pg=PA5 |language=en |chapter=1. Algebra Education: Exploring Topics and Themes |access-date=2024-01-24 |archive-date=2024-01-23 |archive-url=https://web.archive.org/web/20240123185535/https://books.google.com/books?id=7sVFaMhwackC&pg=PA5 |url-status=live }}
* {{cite book |last1=Edwards |first1=C. H. |title=Advanced Calculus of Several Variables |date=2012 |publisher=Courier Corporation |isbn=978-0-486-13195-5 |url=https://books.google.com/books?id=sZIFcJ8DJAIC&pg=PR9 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094526/https://books.google.com/books?id=sZIFcJ8DJAIC&pg=PR9 |url-status=live }}
* {{cite book |last1=Emch |first1=Gerard G. |last2=Sridharan |first2=R. |last3=Srinivas |first3=M. D. |title=Contributions to the History of Indian Mathematics |date=2005 |publisher=Springer |isbn=978-93-86279-25-5 |url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA20 |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126140357/https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA20 |url-status=live }}
* {{cite book |last1=Jr |first1=Everett Jenkins |title=The Muslim Diaspora (Volume 1, 570-1500): A Comprehensive Chronology of the Spread of Islam in Asia, Africa, Europe and the Americas |date=2010 |publisher=McFarland |isbn=978-0-7864-4713-8 |url=https://books.google.com/books?id=giEkCQAAQBAJ&pg=PA82 |language=en |access-date=2024-01-28 |archive-date=2024-01-29 |archive-url=https://web.archive.org/web/20240129081725/https://books.google.com/books?id=giEkCQAAQBAJ&pg=PA82#v=onepage&q&f=false |url-status=live }}
* {{cite web |author1=EoM Staff |title=Arithmetic |url=https://encyclopediaofmath.org/wiki/Arithmetic |website=Encyclopedia of Mathematics |publisher=Springer |access-date=23 October 2023 |date=2020a |archive-date=4 October 2009 |archive-url=https://web.archive.org/web/20091004230123/http://eom.springer.de/a/a013260.htm |url-status=live }}
* {{cite web |author1=EoM Staff |title=Algebra |url=https://encyclopediaofmath.org/wiki/Algebra |website=Encyclopedia of Mathematics |publisher=Springer |access-date=11 January 2023 |date=2017 |archive-date=29 November 2022 |archive-url=https://web.archive.org/web/20221129153630/https://encyclopediaofmath.org/wiki/Algebra |url-status=live }}
* {{cite web |author1=EoM Staff |title=Algebra(2) |url=https://encyclopediaofmath.org/wiki/Algebra(2) |website=Encyclopedia of Mathematics |publisher=Springer |access-date=11 January 2023 |date=2020 |archive-date=7 April 2023 |archive-url=https://web.archive.org/web/20230407165656/https://encyclopediaofmath.org/wiki/Algebra(2) |url-status=live }}
* {{cite web |author1=EoM Staff |title=Algebraic Operation |url=https://encyclopediaofmath.org/wiki/Algebraic_operation |website=Encyclopedia of Mathematics |publisher=Springer |access-date=11 January 2023 |date=2023 |archive-date=23 August 2023 |archive-url=https://web.archive.org/web/20230823194536/https://encyclopediaofmath.org/wiki/Algebraic_operation |url-status=live }}
* {{cite web |author1=EoM Staff |title=Linear Equation |url=https://encyclopediaofmath.org/wiki/Linear_equation |website=Encyclopedia of Mathematics |publisher=Springer |access-date=10 January 2024 |date=2011 |archive-date=23 November 2023 |archive-url=https://web.archive.org/web/20231123235759/https://encyclopediaofmath.org/wiki/Linear_equation |url-status=live }}
* {{cite web |author1=EoM Staff |title=Ring |url=https://encyclopediaofmath.org/wiki/Ring |website=Encyclopedia of Mathematics |publisher=Springer |access-date=11 January 2023 |date=2016a |archive-date=1 January 2023 |archive-url=https://web.archive.org/web/20230101135941/https://encyclopediaofmath.org/wiki/Ring |url-status=live }}
* {{cite web |author1=EoM Staff |title=Group |url=https://encyclopediaofmath.org/wiki/Group |website=Encyclopedia of Mathematics |publisher=Springer |access-date=11 January 2023 |date=2016 |archive-date=5 December 2022 |archive-url=https://web.archive.org/web/20221205014207/https://encyclopediaofmath.org/wiki/Group |url-status=live }}
* {{cite web |author1=EoM Staff |title=Algebraic Logic |url=https://encyclopediaofmath.org/wiki/Algebraic_logic |website=Encyclopedia of Mathematics |publisher=Springer |access-date=23 October 2023 |date=2020b |archive-date=24 January 2024 |archive-url=https://web.archive.org/web/20240124094606/https://encyclopediaofmath.org/wiki/Algebraic_logic |url-status=live }}
* {{cite book |last1=Fiche |first1=Georges |last2=Hebuterne |first2=Gerard |title=Mathematics for Engineers |date=2013 |publisher=John Wiley & Sons |isbn=978-1-118-62333-6 |url=https://books.google.com/books?id=TqkckiuuXg8C&pg=PT326 |language=en |access-date=2024-01-13 |archive-date=2024-01-12 |archive-url=https://web.archive.org/web/20240112102552/https://books.google.com/books?id=TqkckiuuXg8C&pg=PT326 |url-status=live }}
* {{cite web |last1=Franks |first1=Curtis |title=Propositional Logic |url=https://plato.stanford.edu/entries/logic-propositional/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=22 January 2024 |date=2023 |archive-date=29 January 2024 |archive-url=https://web.archive.org/web/20240129081715/https://plato.stanford.edu/entries/logic-propositional/ |url-status=live }}
* {{cite book |last1=Gardella |first1=Francis |last2=DeLucia |first2=Maria |title=Algebra for the Middle Grades |date=2020 |publisher=IAP |isbn=978-1-64113-847-5 |url=https://books.google.com/books?id=HBXFDwAAQBAJ&pg=PA19 |language=en |access-date=2024-01-24 |archive-date=2024-01-23 |archive-url=https://web.archive.org/web/20240123185518/https://books.google.com/books?id=HBXFDwAAQBAJ&pg=PA19 |url-status=live }}
* {{cite book |last1=Gilbert |first1=William J. |last2=Nicholson |first2=W. Keith |title=Modern Algebra with Applications |date=2004 |publisher=John Wiley & Sons |isbn=978-0-471-46989-6 |url=https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 |language=en |access-date=2024-01-13 |archive-date=2024-01-12 |archive-url=https://web.archive.org/web/20240112172510/https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 |url-status=live }}
* {{cite book |last1=Golan |first1=Jonathan S. |title=Foundations of Linear Algebra |date=1995 |publisher=Springer Netherlands |isbn=978-94-015-8502-6 |url=https://link.springer.com/chapter/10.1007/978-94-015-8502-6_18 |language=en |chapter=Algebras Over A Field |access-date=2024-01-13 |archive-date=2024-01-12 |archive-url=https://web.archive.org/web/20240112171825/https://link.springer.com/chapter/10.1007/978-94-015-8502-6_18 |url-status=live }}
* {{cite book |last1=Grätzer |first1=George |title=Universal Algebra |date=2008 |publisher=Springer Science & Business Media |isbn=978-0-387-77487-9 |url=https://books.google.com/books?id=8lNkXPJas4wC |language=en |edition=2 |access-date=2024-01-27 |archive-date=2024-01-27 |archive-url=https://web.archive.org/web/20240127120757/https://books.google.com/books?id=8lNkXPJas4wC |url-status=live }}
* {{cite book |last1=Grillet |first1=Pierre Antoine |title=Abstract Algebra |date=2007 |publisher=Springer |isbn=978-0-387-71568-1 |url=https://link.springer.com/chapter/10.1007/978-0-387-71568-1_15 |language=en |chapter=Universal Algebra |access-date=2024-01-13 |archive-date=2024-01-12 |archive-url=https://web.archive.org/web/20240112171841/https://link.springer.com/chapter/10.1007/978-0-387-71568-1_15 |url-status=live }}
* {{cite journal |last1=Halmos |first1=Paul R. |title=The Basic Concepts of Algebraic Logic |journal=The American Mathematical Monthly |date=1956 |volume=63 |issue=6 |doi=10.2307/2309396 |issn=0002-9890}}
* {{cite book |last1=Harrison |first1=Michael |last2=Waldron |first2=Patrick |title=Mathematics for Economics and Finance |date=2011 |publisher=Routledge |isbn=978-1-136-81921-6 |url=https://books.google.com/books?id=_sisAgAAQBAJ&pg=PT464 |language=en |access-date=2024-01-18 |archive-date=2024-01-17 |archive-url=https://web.archive.org/web/20240117144009/https://books.google.com/books?id=_sisAgAAQBAJ&pg=PT464 |url-status=live }}
* {{cite book |last1=Hazewinkel |first1=Michiel |title=Encyclopaedia of Mathematics (Set) |date=1994 |publisher=Springer Science & Business Media |isbn=978-1-55608-010-4 |url=https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126140353/https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 |url-status=live }}
* {{cite web |author1=HC Staff |title=Arithmetic |url=https://www.ahdictionary.com/word/search.html?q=arithmetic&submit.x=58&submit.y=14 |website=American Heritage Dictionary |publisher=HarperCollins |access-date=19 October 2023 |date=2022 |archive-date=8 November 2023 |archive-url=https://web.archive.org/web/20231108181459/https://www.ahdictionary.com/word/search.html?q=arithmetic&submit.x=58&submit.y=14 |url-status=live }}
* {{cite book |last1=Higgins |first1=Peter M. |title=Algebra: A Very Short Introduction |date=2015 |publisher=OUP Oxford |isbn=978-0-19-104746-6 |url=https://books.google.com/books?id=QANiCgAAQBAJ&pg=PA89 |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126140427/https://books.google.com/books?id=QANiCgAAQBAJ&pg=PA89 |url-status=live }}
* {{cite web |last1=Hintikka |first1=Jaakko J. |title=Philosophy of Logic |url=https://www.britannica.com/topic/philosophy-of-logic |website=Encyclopædia Britannica |access-date=21 November 2021 |language=en |archive-date=28 April 2015 |archive-url=https://web.archive.org/web/20150428101732/http://www.britannica.com/EBchecked/topic/346240/philosophy-of-logic |url-status=live |date=2019 }}
* {{cite book |last1=Hoad |first1=T. F. |title=The Concise Oxford Dictionary of English Etymology |date=1993 |publisher=Oxford University Press |isbn=978-0-19-283098-2 }}
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* {{cite book |last1=Indurkhya |first1=Bipin |title=Metaphor and Cognition: An Interactionist Approach |date=2013 |publisher=Springer Science & Business Media |isbn=978-94-017-2252-0 |url=https://books.google.com/books?id=foTrCAAAQBAJ&pg=PA217 |language=en |chapter=6.5 Algebras and Structures |access-date=2024-01-21 |archive-date=2024-01-29 |archive-url=https://web.archive.org/web/20240129081621/https://books.google.com/books?id=foTrCAAAQBAJ&pg=PA217#v=onepage&q&f=false |url-status=live }}
* {{cite web |last1=Insall |first1=Matt |last2=Sakharov |first2=Alex |title=Universal Algebra |url=https://mathworld.wolfram.com/UniversalAlgebra.html |website=Wolfram MathWorld |publisher=Wolfram |access-date=12 January 2024 |language=en |archive-date=10 January 2024 |archive-url=https://web.archive.org/web/20240110053330/https://mathworld.wolfram.com/UniversalAlgebra.html |url-status=live }}
* {{cite book |last1=Irving |first1=Ronald S. |title=Integers, Polynomials, and Rings: A Course in Algebra |date=2004 |publisher=Springer Science & Business Media |isbn=978-0-387-40397-7 |url=https://books.google.com/books?id=-7OM1OtOFJgC |language=en |access-date=2024-01-20 |archive-date=2024-01-19 |archive-url=https://web.archive.org/web/20240119180145/https://books.google.com/books?id=-7OM1OtOFJgC |url-status=live }}
* {{cite web |last1=Jansana |first1=Ramon |title=Algebraic Propositional Logic |url=https://plato.stanford.edu/entries/logic-algebraic-propositional/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=22 January 2024 |date=2022 |archive-date=20 December 2016 |archive-url=https://web.archive.org/web/20161220155433/https://plato.stanford.edu/entries/logic-algebraic-propositional/ |url-status=live }}
* {{cite book |last1=Jarvis |first1=Frazer |title=Algebraic Number Theory |date=2014 |publisher=Springer |isbn=978-3-319-07545-7 |url=https://books.google.com/books?id=0j0qBAAAQBAJ&pg=PA1 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094525/https://books.google.com/books?id=0j0qBAAAQBAJ&pg=PA1 |url-status=live }}
* {{cite book |last1=Kachroo |first1=Pushkin |last2=Özbay |first2=Kaan M. A. |title=Feedback Control Theory for Dynamic Traffic Assignment |date=2018 |publisher=Springer |isbn=978-3-319-69231-9 |url=https://books.google.com/books?id=qQNbDwAAQBAJ&pg=PA176 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094524/https://books.google.com/books?id=qQNbDwAAQBAJ&pg=PA176 |url-status=live }}
* {{cite book |last1=Khattar |first1=Dinesh |last2=Agrawal |first2=Neha |title=Group Theory |date=2023 |publisher=Springer Nature |isbn=978-3-031-21307-6 |url=https://books.google.com/books?id=7-nIEAAAQBAJ&pg=PA4 |language=en |access-date=2024-01-20 |archive-date=2024-01-19 |archive-url=https://web.archive.org/web/20240119141014/https://books.google.com/books?id=7-nIEAAAQBAJ&pg=PA4 |url-status=live }}
* {{cite book |last1=Kilty |first1=Joel |last2=McAllister |first2=Alex |title=Mathematical Modeling and Applied Calculus |date=2018 |publisher=Oxford University Press |isbn=978-0-19-255813-8 |url=https://books.google.com/books?id=YVRuDwAAQBAJ&pg=PA347 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094605/https://books.google.com/books?id=YVRuDwAAQBAJ&pg=PA347 |url-status=live }}
* {{cite book |last1=Kleiner |first1=Israel |title=A History of Abstract Algebra |date=2007 |publisher=Springer Science & Business Media |isbn=978-0-8176-4685-1 |url=https://books.google.com/books?id=udj-1UuaOiIC&pg=PA100 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094527/https://books.google.com/books?id=udj-1UuaOiIC&pg=PA100 |url-status=live }}
* {{cite book |last1=Knoebel |first1=Arthur |title=Sheaves of Algebras Over Boolean Spaces |date=2011 |publisher=Springer Science & Business Media |isbn=978-0-8176-4218-1 |url=https://books.google.com/books?id=VWS_sgO2uvgC&pg=PA5 |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126100049/https://books.google.com/books?id=VWS_sgO2uvgC&pg=PA5 |url-status=live }}
* {{cite book |last1=Kramer |first1=Jürg |last2=Pippich |first2=Anna-Maria von |title=From Natural Numbers to Quaternions |date=2017 |publisher=Springer |isbn=978-3-319-69429-0 |url=https://books.google.com/books?id=nvM-DwAAQBAJ&pg=PA49 |language=en |access-date=2024-01-20 |archive-date=2024-01-19 |archive-url=https://web.archive.org/web/20240119180140/https://books.google.com/books?id=nvM-DwAAQBAJ&pg=PA49 |url-status=live }}
* {{cite journal |last1=Kvasz |first1=L. |title=The History of Algebra and the Development of the Form of Its Language |journal=Philosophia Mathematica |date=2006 |volume=14 |issue=3 |doi=10.1093/philmat/nkj017}}
* {{cite book |last1=Lamagna |first1=Edmund A. |title=Computer Algebra: Concepts and Techniques |date=2019 |publisher=CRC Press |isbn=978-1-351-60583-0 |url=https://books.google.com/books?id=8PSDDwAAQBAJ&pg=PA150 |language=en |access-date=2024-01-16 |archive-date=2024-01-14 |archive-url=https://web.archive.org/web/20240114183412/https://books.google.com/books?id=8PSDDwAAQBAJ&pg=PA150 |url-status=live }}
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* {{cite book |last1=Majewski |first1=Miroslaw |title=MuPAD Pro Computing Essentials |date=2004 |publisher=Springer |isbn=978-3-540-21943-9 |edition=2}}
* {{cite book |last1=Mal’cev |first1=A. I. |title=Algebraic Systems |date=1973 |publisher=Springer |isbn=978-3-642-65374-2 |url=https://link.springer.com/chapter/10.1007/978-3-642-65374-2_5 |language=en |chapter=Quasivarieties |access-date=2024-01-21 |archive-date=2018-06-18 |archive-url=https://web.archive.org/web/20180618183135/https://link.springer.com/chapter/10.1007%2F978-3-642-65374-2_5 |url-status=live }}
* {{cite book |last1=Mancosu |first1=Paolo |title=Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century |date=1999 |publisher=Oxford University Press |isbn=978-0-19-513244-1 |url=https://books.google.com/books?id=60qaEePdqcoC&pg=PA84 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094524/https://books.google.com/books?id=60qaEePdqcoC&pg=PA84 |url-status=live }}
* {{cite book |last1=Maxwell |first1=E. A. |title=Algebraic Structure and Matrices Book 2 |date=2009 |publisher=Syracuse University Press |isbn=978-0-521-10905-5 |url=https://books.google.com/books?id=yD0irRUE_u4C&pg=PA73 |language=en |access-date=2024-01-20 |archive-date=2024-01-19 |archive-url=https://web.archive.org/web/20240119180157/https://books.google.com/books?id=yD0irRUE_u4C&pg=PA73 |url-status=live }}
* {{cite web |last1=McGrath |first1=Matthew |last2=Frank |first2=Devin |title=Propositions |url=https://plato.stanford.edu/entries/propositions/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=22 January 2024 |date=2023 |archive-date=29 January 2024 |archive-url=https://web.archive.org/web/20240129081718/https://plato.stanford.edu/entries/propositions/ |url-status=live }}
* {{cite book |last1=McKeague |first1=Charles P. |title=Intermediate Algebra: A Text/Workbook |date=2014 |publisher=Academic Press |isbn=978-1-4832-1417-7 |url=https://books.google.com/books?id=nI7iBQAAQBAJ&pg=PA386 |language=en |access-date=2024-01-16 |archive-date=2024-01-14 |archive-url=https://web.archive.org/web/20240114183639/https://books.google.com/books?id=nI7iBQAAQBAJ&pg=PA386 |url-status=live }}
* {{cite book |last1=McWeeny |first1=R. |title=Symmetry: An Introduction to Group Theory and Its Applications |date=2002 |publisher=Courier Corporation |isbn=978-0-486-42182-7 |url=https://books.google.com/books?id=x3fjIXY93TsC&pg=PA6 |language=en |access-date=2024-01-20 |archive-date=2024-01-19 |archive-url=https://web.archive.org/web/20240119180145/https://books.google.com/books?id=x3fjIXY93TsC&pg=PA6 |url-status=live }}
* {{cite book |last1=Menini |first1=Claudia |last2=Oystaeyen |first2=Freddy Van |url=https://books.google.com/books?id=3mlQDwAAQBAJ&pg=PA722 |title=Abstract Algebra: A Comprehensive Treatment |date=2017 |publisher=CRC Press |isbn=978-1-4822-5817-2 |language=en |access-date=2024-01-27 |archive-date=2024-01-27 |archive-url=https://web.archive.org/web/20240127130157/https://books.google.com/books?id=3mlQDwAAQBAJ&pg=PA722 |url-status=live }}
* {{cite book |last1=Mortensen |first1=C. E. |title=Inconsistent Mathematics |date=2013 |publisher=Springer Science & Business Media |isbn=978-94-015-8453-1 |url=https://books.google.com/books?id=KYDrCAAAQBAJ&pg=PA73 |language=en |access-date=2024-01-18 |archive-date=2024-01-17 |archive-url=https://web.archive.org/web/20240117144005/https://books.google.com/books?id=KYDrCAAAQBAJ&pg=PA73 |url-status=live }}
* {{cite book |last1=Musielak |first1=Dora |title=Sophie Germain: Revolutionary Mathematician |date=2020 |publisher=Springer Nature |isbn=978-3-030-38375-6 |url=https://books.google.com/books?id=iqHYDwAAQBAJ&pg=PA36 |language=en |access-date=2024-01-27 |archive-date=2024-01-25 |archive-url=https://web.archive.org/web/20240125151606/https://books.google.com/books?id=iqHYDwAAQBAJ&pg=PA36 |url-status=live }}
* {{cite web |author1=MW Staff |title=Definition of Arithmetic |url=https://www.merriam-webster.com/dictionary/arithmetic |website=Merriam-Webster |access-date=19 October 2023 |language=en |date=2023 |archive-date=14 November 2023 |archive-url=https://web.archive.org/web/20231114193352/https://www.merriam-webster.com/dictionary/arithmetic |url-status=live }}
* {{cite book |last1=Nakahara |first1=Mikio |title=Geometry, Topology and Physics |date=2018 |publisher=Taylor & Francis |isbn=978-1-4200-5694-5 |url=https://books.google.com/books?id=p2C1DwAAQBAJ&pg=PA121 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094525/https://books.google.com/books?id=p2C1DwAAQBAJ&pg=PA121 |url-status=live }}
* {{cite book |last1=Negro |first1=Luca Dal |title=Waves in Complex Media |date=2022 |publisher=Cambridge University Press |isbn=978-1-107-03750-2 |url=https://books.google.com/books?id=MIdoEAAAQBAJ&pg=PA365 |language=en |access-date=2024-01-21 |archive-date=2024-01-29 |archive-url=https://web.archive.org/web/20240129082117/https://books.google.com/books?id=MIdoEAAAQBAJ&pg=PA365#v=onepage&q&f=false |url-status=live }}
* {{cite book |last1=Neri |first1=Ferrante |title=Linear Algebra for Computational Sciences and Engineering |date=2019 |publisher=Springer |isbn=978-3-030-21321-3 |url=https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PR12 |language=en |access-date=2024-01-24 |archive-date=2024-01-23 |archive-url=https://web.archive.org/web/20240123185542/https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PR12 |url-status=live }}
* {{cite web |url=http://www.lexico.com/definition/algebra |archive-url=https://web.archive.org/web/20131120000000/http://www.lexico.com/definition/algebra |url-status=dead |archive-date=2013-11-20 |title=Algebra |website=Lexico |publisher=[[Organization:Oxford University Press|Oxford University Press]] |author=OUP Staff }}
* {{cite book |last1=Ovchinnikov |first1=Sergei |title=Number Systems |date=2015 |publisher=American Mathematical Society |isbn=978-1-4704-2018-5 |url=https://books.google.com/books?id=UMbXBgAAQBAJ&pg=PA27 |language=en |access-date=2024-01-20 |archive-date=2024-01-19 |archive-url=https://web.archive.org/web/20240119111232/https://books.google.com/books?id=UMbXBgAAQBAJ&pg=PA27 |url-status=live }}
* {{cite book |last1=Pickover |first1=Clifford A. |title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |date=2009 |publisher=Sterling Publishing Company, Inc. |isbn=978-1-4027-5796-9 |url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA90 |language=en |access-date=2024-01-28 |archive-date=2024-01-28 |archive-url=https://web.archive.org/web/20240128165823/https://books.google.com/books?id=JrslMKTgSZwC&pg=PA90 |url-status=live }}
* {{cite book |last1=Plotkin |first1=B. |title=Universal Algebra, Algebraic Logic, and Databases |date=2012 |publisher=Springer Science & Business Media |isbn=978-94-011-0820-1 |url=https://books.google.com/books?id=-v3xCAAAQBAJ&pg=PA155 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094523/https://books.google.com/books?id=-v3xCAAAQBAJ&pg=PA155 |url-status=live }}
* {{cite web |last1=Pratt |first1=Vaughan |title=Algebra |url=https://plato.stanford.edu/entries/algebra/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 January 2024 |date=2022 |archive-date=29 January 2024 |archive-url=https://web.archive.org/web/20240129081755/https://plato.stanford.edu/entries/algebra/ |url-status=live }}
* {{cite book |last1=Rabadan |first1=Raul |last2=Blumberg |first2=Andrew J. |title=Topological Data Analysis for Genomics and Evolution: Topology in Biology |date=2019 |publisher=Cambridge University Press |isbn=978-1-107-15954-9 |url=https://books.google.com/books?id=2967DwAAQBAJ&pg=PA49 |language=en |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094523/https://books.google.com/books?id=2967DwAAQBAJ&pg=PA49 |url-status=live }}
* {{cite web |last1=Renze |first1=John |last2=Weisstein |first2=Eric W. |title=Algebra |url=https://mathworld.wolfram.com/Algebra.html |website=Wolfram MathWorld |publisher=Wolfram |access-date=12 January 2024 |language=en |archive-date=12 January 2024 |archive-url=https://web.archive.org/web/20240112171819/https://mathworld.wolfram.com/Algebra.html |url-status=live }}
* {{cite book |last1=Rohde |first1=Ulrich L. |last2=Jain |first2=G. C. |last3=Poddar |first3=Ajay K. |last4=Ghosh |first4=A. K. |title=Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners |date=2012 |publisher=John Wiley & Sons |isbn=978-1-118-13014-8 |url=https://books.google.com/books?id=vk2XbZpsBOwC&pg=PT89 |language=en |access-date=2024-01-16 |archive-date=2024-01-15 |archive-url=https://web.archive.org/web/20240115181229/https://books.google.com/books?id=vk2XbZpsBOwC&pg=PT89 |url-status=live }}
* {{cite book |last1=Romanowski |first1=Perry |editor1-last=Lerner |editor1-first=Brenda Wilmoth |editor2-last=Lerner |editor2-first=K. Lee |title=The Gale Encyclopedia of Science |date=2008 |publisher=Thompson Gale |isbn=978-1-4144-2877-2 |edition=4th |chapter=Arithmetic |url=https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/arithmetic |access-date=2024-01-13 |archive-date=2023-11-01 |archive-url=https://web.archive.org/web/20231101124957/https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/arithmetic |url-status=live }}
* {{cite web |last1=Rowland |first1=Todd |last2=Weisstein |first2=Eric W. |title=Group |url=https://mathworld.wolfram.com/Group.html |website=Wolfram MathWorld |publisher=Wolfram |access-date=12 January 2024 |language=en |archive-date=17 January 2024 |archive-url=https://web.archive.org/web/20240117043547/https://mathworld.wolfram.com/Group.html |url-status=live }}
* {{cite book |last1=Seshadri |first1=C. S. |title=Studies in the History of Indian Mathematics |date=2010 |publisher=Springer |isbn=978-93-86279-49-1 |url=https://books.google.com/books?id=w_JdDwAAQBAJ&pg=PA156 |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126140422/https://books.google.com/books?id=w_JdDwAAQBAJ&pg=PA156 |url-status=live }}
* {{cite book |last1=Sialaros |first1=Michalis |title=Revolutions and Continuity in Greek Mathematics |date=2018 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-056527-0 |url=https://books.google.com/books?id=2PZYDwAAQBAJ&pg=PT55 |language=en |access-date=2024-01-27 |archive-date=2024-01-25 |archive-url=https://web.archive.org/web/20240125151607/https://books.google.com/books?id=2PZYDwAAQBAJ&pg=PT55 |url-status=live }}
* {{cite book |last1=Silvia |first1=M. T. |last2=Robinson |first2=E. A. |title=Deconvolution of Geophysical Time Series in the Exploration for Oil and Natural Gas |date=1979 |publisher=Elsevier |isbn=978-0-08-086864-6 |url=https://books.google.com/books?id=Ecgfjh-MpU0C&pg=PA82 |language=en |access-date=2024-01-21 |archive-date=2024-01-29 |archive-url=https://web.archive.org/web/20240129082232/https://books.google.com/books?id=Ecgfjh-MpU0C&pg=PA82#v=onepage&q&f=false |url-status=live }}
* {{cite book |last1=Smorynski |first1=Craig |title=History of Mathematics: A Supplement |date=2007 |publisher=Springer Science & Business Media |isbn=978-0-387-75481-9 |url=https://books.google.com/books?id=qY657eFq7UgC&pg=PA137 |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126140353/https://books.google.com/books?id=qY657eFq7UgC&pg=PA137 |url-status=live }}
* {{cite book |last1=Sullivan |first1=Michael |title=Finite Mathematics: An Applied Approach |date=2010 |publisher=John Wiley & Sons |isbn=978-0-470-87639-8 |url=https://books.google.com/books?id=6NKaDwAAQBAJ&pg=PA53 |language=en |access-date=2024-01-18 |archive-date=2024-01-17 |archive-url=https://web.archive.org/web/20240117143920/https://books.google.com/books?id=6NKaDwAAQBAJ&pg=PA53 |url-status=live }}
* {{cite book |last1=Sun |first1=Shuyu |last2=Zhang |first2=Tao |title=Reservoir Simulations: Machine Learning and Modeling |date=2020 |publisher=Gulf Professional Publishing |isbn=978-0-12-820962-2 |url=https://books.google.com/books?id=SZ3hDwAAQBAJ&pg=PA94 |language=en}}
* {{cite book |last1=Tan |first1=Kiat Shi |last2=Steeb |first2=Willi-Hans |last3=Hardy |first3=Yorick |title=SymbolicC++:An Introduction to Computer Algebra Using Object-Oriented Programming: An Introduction to Computer Algebra Using Object-Oriented Programming |date=2012 |publisher=Springer Science & Business Media |isbn=978-1-4471-0405-6 |url=https://books.google.com/books?id=UDb0BwAAQBAJ&pg=PA306 |language=en |access-date=2024-01-16 |archive-date=2024-01-14 |archive-url=https://web.archive.org/web/20240114180537/https://books.google.com/books?id=UDb0BwAAQBAJ&pg=PA306 |url-status=live }}
* {{cite book |last1=Tanton |first1=James |title=Encyclopedia of Mathematics |date=2005 |publisher=Facts On File |isbn=978-0-8160-5124-3}}
* {{cite web |last1=Terr |first1=David |last2=Weisstein |first2=Eric W. |title=Ring of Integers |url=https://mathworld.wolfram.com/RingofIntegers.html |website=Wolfram MathWorld |publisher=Wolfram |access-date=12 January 2024 |language=en |archive-date=10 January 2024 |archive-url=https://web.archive.org/web/20240110052934/https://mathworld.wolfram.com/RingofIntegers.html |url-status=live }}
* {{cite book |last1=Viterbo |first1=Emanuele |last2=Hong |first2=Yi |editor1-last=Hlawatsch |editor1-first=Franz |editor2-last=Matz |editor2-first=Gerald |title=Wireless Communications Over Rapidly Time-Varying Channels |date=2011 |publisher=Academic Press |isbn=978-0-08-092272-0 |url=https://books.google.com/books?id=d89QRR24jbMC&pg=PA127 |language=en |chapter=3.4 Algebraic Number Theory |access-date=2024-01-24 |archive-date=2024-01-24 |archive-url=https://web.archive.org/web/20240124094527/https://books.google.com/books?id=d89QRR24jbMC&pg=PA127 |url-status=live }}
* {{cite book |last1=Waerden |first1=Bartel L. van der |title=A History of Algebra: From al-Khwārizmī to Emmy Noether |date=2013 |publisher=Springer Science & Business Media |isbn=978-3-642-51599-6 |url=https://books.google.com/books?id=W6DwCAAAQBAJ |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126100104/https://books.google.com/books?id=W6DwCAAAQBAJ |url-status=live }}
* {{cite book |last1=Wagner |first1=Sigrid |last2=Kieran |first2=Carolyn |title=Research Issues in the Learning and Teaching of Algebra: The Research Agenda for Mathematics Education, Volume 4 |date=2018 |publisher=Routledge |isbn=978-1-135-43421-2 |url=https://books.google.com/books?id=uW4ECwAAQBAJ&pg=PT225 |language=en |access-date=2024-01-13 |archive-date=2024-01-12 |archive-url=https://web.archive.org/web/20240112102555/https://books.google.com/books?id=uW4ECwAAQBAJ&pg=PT225 |url-status=live }}
* {{cite book |last1=Walz |first1=Guido |title=Lexikon der Mathematik: Band 1: A bis Eif |date=2016 |publisher=Springer-Verlag |isbn=978-3-662-53498-4 |url=https://www.spektrum.de/lexikon/mathematik/algebra/12062 |language=de |chapter=Algebra |access-date=2024-01-13 |archive-date=2024-01-12 |archive-url=https://web.archive.org/web/20240112171819/https://www.spektrum.de/lexikon/mathematik/algebra/12062 |url-status=live }}
* {{cite book |last1=Weisstein |first1=Eric W. |title=CRC Concise Encyclopedia of Mathematics |date=2003 |publisher=Chapman & Hall/CRC |isbn=978-1-58488-347-0 |edition=2nd}}
* {{cite web |last1=Weisstein |first1=Eric W. |title=Field |url=https://mathworld.wolfram.com/Field.html |website=Wolfram MathWorld |publisher=Wolfram |access-date=12 January 2024 |language=en |date=2024a |archive-date=16 January 2024 |archive-url=https://web.archive.org/web/20240116223735/https://mathworld.wolfram.com/Field.html |url-status=live }}
* {{cite web |last1=Weisstein |first1=Eric W. |title=Field Axioms |url=https://mathworld.wolfram.com/FieldAxioms.html |website=Wolfram MathWorld |publisher=Wolfram |access-date=12 January 2024 |language=en |date=2024b |archive-date=24 December 2023 |archive-url=https://web.archive.org/web/20231224061232/https://mathworld.wolfram.com/FieldAxioms.html |url-status=live }}
* {{cite web |last1=Weisstein |first1=Eric W. |title=Ring |url=https://mathworld.wolfram.com/Ring.html |website=Wolfram MathWorld |publisher=Wolfram |access-date=12 January 2024 |language=en |date=2024c |archive-date=21 January 2024 |archive-url=https://web.archive.org/web/20240121142059/https://mathworld.wolfram.com/Ring.html |url-status=live }}
* {{cite book |last1=Williams |first1=Gareth |title=Linear Algebra with Applications |date=2007 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-5753-3 |url=https://books.google.com/books?id=HLQ9ocWuCzMC&pg=PA4 |language=en |access-date=2024-01-18 |archive-date=2024-01-17 |archive-url=https://web.archive.org/web/20240117143918/https://books.google.com/books?id=HLQ9ocWuCzMC&pg=PA4 |url-status=live }}
* {{cite book |last1=Young |first1=Cynthia Y. |title=Precalculus |date=2010 |publisher=John Wiley & Sons |isbn=978-0-471-75684-2 |url=https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA999 |language=en |access-date=2024-01-16 |archive-date=2024-01-13 |archive-url=https://web.archive.org/web/20240113140952/https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA999 |url-status=live }}
* {{cite book |last1=Young |first1=Cynthia Y. |title=Precalculus |date=2023 |publisher=John Wiley & Sons |isbn=978-1-119-86940-5 |url=https://books.google.com/books?id=pMSZEAAAQBAJ&pg=PA714 |language=en |access-date=2024-01-18 |archive-date=2024-01-17 |archive-url=https://web.archive.org/web/20240117144008/https://books.google.com/books?id=pMSZEAAAQBAJ&pg=PA714 |url-status=live }}
* {{cite book |last1=Zill |first1=Dennis |last2=Dewar |first2=Jacqueline |title=Algebra and Trigonometry |date=2011 |publisher=Jones & Bartlett Publishers |isbn=978-0-7637-5461-7 |url=https://books.google.com/books?id=MBAwrjc3gqMC&pg=PA529 |language=en |access-date=2024-01-16 |archive-date=2024-01-14 |archive-url=https://web.archive.org/web/20240114180535/https://books.google.com/books?id=MBAwrjc3gqMC&pg=PA529 |url-status=live }}
* {{cite book |last1=Zwillinger |first1=Daniel |title=CRC Standard Mathematical Tables and Formulae |date=2002 |publisher=CRC Press |isbn=978-1-4200-3534-6 |url=https://books.google.com/books?id=gE_MBQAAQBAJ&pg=PA812 |language=en |access-date=2024-01-27 |archive-date=2024-01-26 |archive-url=https://web.archive.org/web/20240126140400/https://books.google.com/books?id=gE_MBQAAQBAJ&pg=PA812 |url-status=live }}
{{Refend}}

== External links ==
{{EB1911 poster|Algebra}}

{{Algebra |expanded}}
{{Polynomials}}
{{Areas of mathematics |collapsed}}

[[Category:Algebra| ]]

{{Sourceattribution|Algebra}}