Algebra

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

Algebra (Arabic: الجبر Al-Jabr, transl. 'reunion of broken parts'^[1] or 'bone-setting';^[2] [ʔldʒbr] listen^ⓘ) is the study of variables and the rules for manipulating these variables in formulas.^[3] Originating in ancient Babylonian techniques of calculation, it is now a way of thinking that appears throughout almost all areas of mathematics.^[4]

The quadratic formula expresses the solution of the equation ax² + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.

Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. "Higher" or "abstract" algebra, which professional mathematicians typically just call "algebra", is the study of algebraic structures that generalize the operations familiar from ordinary arithmetic. For example, a group is a set with a binary operation, a rule for combining two members of that set to produce a third, which satisfies some of the same basic properties as addition of integers. Other algebraic structures include rings and fields. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory.

The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. Sometimes, the same phrase is used for a subarea and its main algebraic structures. For example, the subject known as Boolean algebra studies structures called Boolean algebras. A mathematician specialized in algebra is called an algebraist.

Etymology

 

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.^[5]

The word algebra comes from the Arabic: الجبر, romanized: al-jabr, lit. 'reunion of broken parts,^[1] bonesetting^[2]' from the title of the early 9th century book ʿIlm al-jabr wa l-muqābala "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi. In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala "balancing" referred to adding equal terms to both sides. Shortened to just algeber or algebra in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded (in English) in the 16th century.^[6]

Definition

Algebra is the branch of mathematics that studies algebraic operations^[a] and algebraic structures.^[8] An algebraic structure is a non-empty set of mathematical objects, such as the real numbers, together algebraic operations defined on that set, such addition and multiplication.^[9] Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it studies the use of variables in equations and how to manipulate these equations.^[10]

Algebra is often understood as a generalization of arithmetic.^[11] Arithmetic studies arithmetic operations, like addition, subtraction, multiplication, and division, in a specific domain of numbers, like the real numbers.^[12] 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.^[13] A higher level of abstraction is achieved in abstract algebra, which is not limited to a specific domain and studies different classes of algebraic structures, like groups and rings. These algebraic structures are not restricted to typical arithmetic operations and cover other binary operations besides them.^[14] 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.^[15]

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.^[16] When used as a countable noun, an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation.^[17] Depending on the context, "algebra" can also refer to other algebraic structures, like a Lie algebra or an associative algebra.^[18]

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.^[19] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

${\displaystyle ax^{2}+bx+c=0,}$ 

${\displaystyle a,b,c}$  can be any numbers whatsoever (except that ${\displaystyle a}$  cannot be ${\displaystyle 0}$ ), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity ${\displaystyle x}$  which satisfy the equation. That is to say, to find all the solutions of the equation.

Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as permutations, vectors, matrices, and polynomials. The structural properties of these non-numerical objects were then formalized into algebraic structures such as groups, rings, and fields.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of the 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown considerably and includes many branches of mathematics, as can be seen in the Mathematics Subject Classification^[20] where none of the first level areas (two digit entries) are called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

History

Main articles: History of algebra, Abstract algebra § History, and Timeline of algebra

The use of the word "algebra" for denoting a part of mathematics dates probably from the 16th century.^{[citation needed]} The word is derived from the Arabic word al-jabr that appears in the title of the treatise Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala (The Compendious Book on Calculation by Completion and Balancing), written in circa 820 by Al-Kwarizmi.

Al-jabr referred to a method for transforming equations by subtracting like terms from both sides, or passing one term from one side to the other, after changing its sign.

Therefore, algebra referred originally to the manipulation of equations, and, by extension, to the theory of equations. This is still what historians of mathematics generally mean by the term algebra.^{[citation needed]}

In mathematics, the meaning of algebra has evolved after the introduction by François Viète of symbols (variables) for denoting unknown or incompletely specified numbers, and the resulting use of the mathematical notation for equations and formulas. So, algebra became essentially the study of the action of operations on expressions involving variables. This includes but is not limited to the theory of equations.

At the beginning of the 20th century, algebra evolved further by considering operations that act not only on numbers but also on elements of so-called mathematical structures such as groups, fields and vector spaces. This new algebra was called modern algebra by van der Waerden in his eponymous treatise, whose name has been changed to Algebra in later editions.

Early history

 

A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala

The roots of algebra can be traced back to the ancient Babylonians,^[21] who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam.^[22]

By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them.^[19] Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations,^[23] and have led, in number theory, to the modern notion of Diophantine equation.

Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.^[24]

The Hellenistic mathematicians Hero of Alexandria and Diophantus^[25] as well as Indian mathematicians such as Brahmagupta, continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brāhmasphuṭasiddhānta are on a higher level.^{[26][better source needed]} For example, the first complete arithmetic solution written in words instead of symbols,^[27] including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta, published in 628 AD.^[28] Later, Persian and Arab mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.^[29]

In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".^{[30][31][32][33][34][35][36]} It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.^[37] Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to,^[38] and that he gave an exhaustive explanation of solving quadratic equations,^[39] supported by geometric proofs while treating algebra as an independent discipline in its own right.^[34] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".^[40]

According to Jeffrey Oaks and Jean Christianidis neither Diophantus nor Al-Khwarizmi should be called "father of algebra".^[41][42] Pre-modern algebra was developed and used by merchants and surveyors as part of what Jens Høyrup called "subscientific" tradition. Diophantus used this method of algebra in his book, in particular for indeterminate problems, while Al-Khwarizmi wrote one of the first books in arabic about this method.^[43]

Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe.^[44] Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.^[45] He also developed the concept of a function.^[46] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,^[47] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed Σn², Σn³ and used the method of successive approximation to determine square roots.^[48]

Modern history

 

Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.

François Viète's work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, René Descartes published La Géométrie, inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph-Louis Lagrange in his 1770 paper "Réflexions sur la résolution algébrique des équations" devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations.

Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues.^[49] George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).^[50]

Areas of mathematics with the word algebra in their name

 

Linear algebra lecture at the Aalto University

Some subareas of algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.

• Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics.
• Abstract algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
• Linear algebra, in which the specific properties of linear equations, vector spaces and matrices are studied.
• Boolean algebra, a branch of algebra abstracting the computation with the truth values false and true.
• Commutative algebra, the study of commutative rings.
• Computer algebra, the implementation of algebraic methods as algorithms and computer programs.
• Homological algebra, the study of algebraic structures that are fundamental to study topological spaces.
• Universal algebra, in which properties common to all algebraic structures are studied.
• Algebraic number theory, in which the properties of numbers are studied from an algebraic point of view.
• Algebraic geometry, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.
• Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.
• Relational algebra: a set of finitary relations that is closed under certain operators.

Many mathematical structures are called algebras:

• Algebra over a field or more generally algebra over a ring.
  Many classes of algebras over a field or over a ring have a specific name:
  • Associative algebra
  • Non-associative algebra
  • Lie algebra
  • Composition algebra
  • Hopf algebra
  • C*-algebra
  • Symmetric algebra
  • Exterior algebra
  • Tensor algebra
• In measure theory,
  • Sigma-algebra
  • Algebra over a set
• In category theory
  • F-algebra and F-coalgebra
  • T-algebra
• In logic,
  • Relation algebra, a residuated Boolean algebra expanded with an involution called converse.
  • Boolean algebra, a complemented distributive lattice.
  • Heyting algebra

Elementary algebra

Main article: Elementary algebra

 

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

Elementary algebra, also referred to as school algebra, college algebra, and classical algebra,^[51] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on the use of variables and examines how formulas may be transformed.^[52]

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using arithmetic operations like addition, subtraction, multiplication, and division. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in ${\displaystyle 2+5=7}$ .^[53]

Elementary algebra uses the same operations while allowing the use of variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true independent of which numbers are used. For example, the equation ${\displaystyle 2\times 3=3\times 2}$  belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combinations of numbers, as in the equation ${\displaystyle a\times b=b\times a}$ .^[54]

Elementary algebra is interested in algebraic expressions, which are formed by using arithmetic operations to combine variables and numbers. For example, the expression ${\displaystyle 5x+3}$  is an algebraic expression created by multiplying the number 5 with the variable x and adding the number 3 to the result. Other examples of algebraic equations are ${\displaystyle 32xyz}$  and ${\displaystyle 64x^{2}+7y-13}$ .^[55]

Algebraic expressions are used to construct statements that relate two expressions to one another. An equation is a statement formed by comparing two expressions using an equals sign (=), as in ${\displaystyle 5x^{2}+6x=3y+4}$ . Inequations are formed using symbols like the less-than sign (<) and the 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 ${\displaystyle x^{2}=4}$  is true if x is either 2 or -2 and false otherwise.^[56]

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 solving the equation for that variable. For example, the equation ${\displaystyle x-7=4}$  can be solved for x by adding 7 to both sides, which isolates x on the left side and results in the equation ${\displaystyle x=11}$ .^[57]

There are many other techniques used to solve equations. Simplification is used to replace a complicated expression with an equivalent simpler one. For example, the expression ${\displaystyle 7x-3x}$  can be replaced with the expression ${\displaystyle 4x}$ .^[58] Factorization is used to rewrite an expression as a product of several factors. This technique is common for polynomials to determine for which values the expression is zero. For example, the polynomial ${\displaystyle x^{2}-3x-10}$  can be factorized as ${\displaystyle (x+2)(x-5)}$ . The polynomial as a whole is zero if one of its factors is zero, i.e., if x is either -2 or 5.^[59] For statements using several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that ${\displaystyle y=3x}$  then one can simplify the expression ${\displaystyle 7xy}$  to arrive at ${\displaystyle 21x^{2}}$ .^[60] Other techniques include making use of the commutative, distributive, and associative properties.^[61]

 

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.^[62] An important application in the field of geometry concerns the use of algebraic equations to describe geometric figures in the form of a graph. To do so, the different variables in the equation are interpreted as coordinates and the values that solve the equation are interpreted as points of the graph. For example, if x is set to zero in the equation ${\displaystyle y=0.5x-1}$  then y has to be −1 for the equation to be true. This means that the x-y-pair (0, −1) is part of the graph of the equation. The x-y-pair (0, 7), by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of all x-y-pairs that solve the equation.^[63]

Education

See also: Mathematics education

It has been suggested that elementary algebra should be taught to students as young as eleven years old,^[64] though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States.^[65] However, in some US schools, algebra instruction starts in ninth grade.

Linear algebra

Main article: Linear algebra

Linear algebra employs the methods of elementary algebra to study systems of linear equations.^[66] An equation is linear if no variable is multiplied with another variable and no operations like exponentiation, extraction of roots, and logarithm are applied to variables. For example, the equations ${\displaystyle 0.25x-4=y}$  and ${\displaystyle x_{1}-7x_{2}+3x_{3}=0}$  are linear while the equations ${\displaystyle x^{2}=y}$  and ${\displaystyle 3x_{1}x_{2}+15=0}$  are non-linear. Several equations form a system of equations if they all rely on the same set of variables.^[67]

Systems of linear equations are often expressed through matrices^[b] and vectors^[c] to represent the whole system in a single equation. This can be done by moving the variables to the left side of each equation and moving the constant terms to the right side. The system is then expressed by formulating a matrix that contains all the coefficients of the equations and multiplying it with the vector made up of the variables.^[68] For example, the system of equations

1. ${\displaystyle 9x_{1}+3x_{2}-13x_{3}=0}$ 
2. ${\displaystyle 2.3x_{1}+7x_{3}=9}$ 
3. ${\displaystyle -5x_{1}-17x_{2}=-3}$ 

can be written as

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

 

Like elementary algebra, linear algebra is interested in manipulating and transforming equations to solve them. It goes beyond elementary algebra by dealing with several equations at once and looking for the values for which all equations are true at the same time. For example, if the system is made of the two equations ${\displaystyle 3x_{1}-x_{2}=0}$  and ${\displaystyle x_{1}+x_{2}=8}$  then using the values 1 and 3 for ${\displaystyle x_{1}}$  and ${\displaystyle x_{2}}$  does not solve the system of equations because it only solves the first but not the second equation.^[69]

Two central questions in linear algebra are whether a system of equations has any solutions and, if so, whether it has a unique solution. A system of equations that has solutions is called consistent. This is the case if the equations do not contradict each other. If two or more equations contradict each other, the system of equations is inconsistent and has no solutions. For example, the equations ${\displaystyle x_{1}-3x_{2}=0}$  and ${\displaystyle x_{1}-3x_{2}=7}$  contradict each other since no values of ${\displaystyle x_{1}}$  and ${\displaystyle x_{2}}$  exist that solve both equations at the same time.^[70]

Whether a consistent system of equations has a unique solution depends on the number of variables and the number of independent equations. Several equations are independent of each other if they do not provide the same information and cannot be derived from each other. A unique solution exists if the number of variables is the same as the number of independent equations. Underdetermined systems, by contrast, have more variables than equations and have an infinite number of solutions if they are consistent.^[71]

 

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.^[72] The elimination method creates a new equation by adding one equation to another equation. This way, it is possible to eliminate one variable that appears in both equations. For a system that contains the equations ${\displaystyle -x+7y=3}$  and ${\displaystyle 2x-7y=10}$ , it is possible to eliminate y by adding the first to the second equation, thereby revealing that x is 13.^[d][73] Many advanced techniques implement algorithms based on matrix calculations, such as Cramer's rule, the Gauss–Jordan elimination, and LU Decomposition.^[74]

On a geometric level, systems of equations can be interpreted as geometric figures. For systems that have two variables, each equation represents a line in two-dimensional space. The point where the two lines intersect is the solution. For inconsistent systems, the two lines run parallel, meaning that there is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to graphically look for solutions by plotting the equations and determining where they intersect.^[75] The same principles also apply to systems of equations with more variables, with the difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to planes in three-dimensional space and the points where all planes intersect solve the system of equations.^[76]

Abstract algebra

Main articles: Abstract algebra and Algebraic structure

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are the listed fundamental concepts in abstract algebra.

Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: collections of objects called elements. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax² + bx + c), the set of all two dimensional vectors of a plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.

Binary operations: The notion of addition (+) is generalized to the notion of binary operation (denoted here by ∗). The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a ∗ b is another element in the set; this condition is called closure. Addition (+), subtraction (−), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are generalized to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy a ∗ e = a and e ∗ a = a, and is necessarily unique, if it exists. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, ...) has no identity element for addition.

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written −a, and for multiplication the inverse is written a⁻¹. A general two-sided inverse element a⁻¹ satisfies the property that a ∗ a⁻¹ = e and a⁻¹ ∗ a = e, where e is the identity element.

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (a ∗ b) ∗ c = a ∗ (b ∗ c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes a ∗ b = b ∗ a. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.

Groups

Main article: Group (mathematics)

See also: Group theory and Examples of groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:

• An identity element e exists, such that for every member a of S, e ∗ a and a ∗ e are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member a⁻¹ such that a ∗ a⁻¹ and a⁻¹ ∗ a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c).

If a group is also commutative – that is, for any two members a and b of S, a ∗ b is identical to b ∗ a – then the group is said to be abelian.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

The non-zero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is 1/4, which is not an integer.

The theory of groups is studied in group theory. A major result of this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which separates the finite simple groups into roughly 30 basic types.

Semi-groups, quasi-groups, and monoids are algebraic structures similar to groups, but with less constraints on the operation. They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A semi-group has an associative binary operation but might not have an identity element. A monoid is a semi-group which does have an identity but might not have an inverse for every element. A quasi-group satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however, the binary operation might not be associative.

All groups are monoids, and all monoids are semi-groups.

Examples

Set	Natural numbers N		Integers Z		Rational numbers Q Real numbers R Complex numbers C				Integers modulo 3 Z/3Z = {0, 1, 2}
Operation	+	×	+	×	+	−	×	÷	+	×
Closed	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes
Identity	0	1	0	1	0	N/A	1	N/A	0	1
Inverse	N/A	N/A	−a	N/A	−a	N/A	1/a (a ≠ 0)	N/A	0, 2, 1, respectively	N/A, 1, 2, respectively
Associative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Commutative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Structure	monoid	monoid	abelian group	monoid	abelian group	quasi-group	monoid	quasi-group	abelian group	monoid

Rings and fields

Main articles: Ring (mathematics) and Field (mathematics)

See also: Ring theory, Glossary of ring theory, Field theory (mathematics), and Glossary of field theory

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings and fields.

A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have an identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

Distributivity generalises the distributive law for numbers. For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.

The integers are an example of a ring. The integers have additional properties which make it an integral domain.

A field is a ring with the additional property that all the elements excluding 0 form an abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a⁻¹.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

See also

•  Mathematics portal

• Algebra tile
• Outline of algebra
• Outline of linear algebra

References

Notes

1. When understood in the widest sense, an algebraic operation is mapping from a Cartesian power of a set into that set, expressed formally as ${\displaystyle \omega :A^{n}\to A}$ . Addition of real numbers is an example of an algebraic operations: it takes two numbers as input and produces one number as output. It has the form ${\displaystyle +:\mathbb {R} ^{2}\to \mathbb {R} }$ .^[7]
2. A matrix is a table of numbers, such as
   $${\displaystyle {\begin{bmatrix}3&-7&19\\0.3&7&-4\end{bmatrix}}}$$

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

    
4. In some cases, an equation has to be multiplied by a constant before adding it to another equation.

Citations

1. . Oxford Dictionaries UK English Dictionary. Oxford University Press. Archived from the original on 2013-11-20. . Archived from the original on 2013-12-31. Retrieved 2013-11-20..
2. Menini, Claudia; Oystaeyen, Freddy Van (2017). Abstract Algebra: A Comprehensive Treatment. CRC Press. ISBN 978-1-4822-5817-2. from the original on 2021-02-21. Retrieved 2020-10-15.
3. See Herstein 1964, page 1: "An algebraic system can be described as a set of objects together with some operations for combining them".
4. See Herstein 1964, page 1: "...it also serves as the unifying thread which interlaces almost all of mathematics".
5. Esposito, John L. (2000). The Oxford History of Islam. Oxford University Press. p. 188. ISBN 978-0-19-988041-6.
6. T. F. Hoad, ed. (2003). "Algebra". The Concise Oxford Dictionary of English Etymology. Oxford: Oxford University Press. doi:10.1093/acref/9780192830982.001.0001. ISBN 978-0-19-283098-2.
7. EoM Staff 2023, Lead Section
8. • EoM Staff 2020, Lead Section
   • Gilbert & Nicholson 2004, p. 4
9. • Fiche & Hebuterne 2013, p. 326
   • EoM Staff 2020, § The subject matter of algebra, its principal branches and its connection with other branches of mathematics.
   • Gilbert & Nicholson 2004, p. 4
10. • Pratt 2022, Lead Section, § 1. Elementary Algebra, § 2. Abstract Algebra, § 3. Universal Algebra
    • EoM Staff 2020, § The subject matter of algebra, its principal branches and its connection with other branches of mathematics.
11. • Maddocks 2008, p. 129
    • Burgin 2022, p. 45
12. • Romanowski 2008, pp. 302–303
    • HC Staff 2022
    • MW Staff 2023
    • EoM Staff 2020a
13. • Maddocks 2008, pp. 129–130
    • Pratt 2022, Lead Section, § 1. Elementary Algebra
    • Wagner & Kieran 2018, p. 225
14. • Maddocks 2008, pp. 131–132
    • Pratt 2022, Lead Section, § 2. Abstract Algebra
    • Wagner & Kieran 2018, p. 225
15. • Pratt 2022, § 3. Universal Algebra
    • Grillet 2007, p. 559
16. • Weisstein 2003, p. 46
    • Renze & Weisstein
    • Walz 2016, Algebra
17. • Weisstein 2003, p. 46
    • Renze & Weisstein
    • Golan 1995, pp. 219–227
18. EoM Staff 2017
19. See Boyer 1991, Europe in the Middle Ages, p. 258: "In the arithmetical theorems in Euclid's Elements VII–IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."
20. "2010 Mathematics Subject Classification". from the original on 2014-06-06. Retrieved 2014-10-05.
21. Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. ISBN 978-0-486-60255-4.
22. See Boyer 1991.
23. Cajori, Florian (2010). A History of Elementary Mathematics – With Hints on Methods of Teaching. Read Books Design. p. 34. ISBN 978-1-4460-2221-4. from the original on 2021-02-21. Retrieved 2020-10-15.
24. Roshdi Rashed (2009). Al Khwarizmi: The Beginnings of Algebra. Saqi Books. ISBN 978-0-86356-430-7.
25. . Archived from the original on 2013-07-27. Retrieved 2014-10-05.
26. "History of Algebra". from the original on 2014-11-11. Retrieved 2014-10-05.
27. Mackenzie, Dana (2012). The universe in zero words : the story of mathematics as told through equations. Princeton, N.J.: Princeton University Press. p. 61. ISBN 978-0-691-15282-0. OCLC 761851013.
28. Bradley, Michael J. (2006). The birth of mathematics : ancient times to 1300. New York: Chelsea House. p. 86. ISBN 978-0-7910-9723-6. OCLC 465077937.
29. Meri, Josef W. (2004). Medieval Islamic Civilization. Psychology Press. p. 31. ISBN 978-0-415-96690-0. from the original on 2013-06-02. Retrieved 2012-11-25.
30. Corona, Brezina (2006). Al-Khwarizmi: The Inventor Of Algebra. New York, United States: Rosen Pub Group. ISBN 978-1404205130.
31. See Boyer 1991, page 181: "If we think primarily of the matter of notations, Diophantus has good claim to be known as the 'father of algebra', but in terms of motivation and concept, the claim is less appropriate. The Arithmetica is not a systematic exposition of the algebraic operations, or of algebraic functions or of the solution of algebraic equations".
32. See Boyer 1991, page 230: "The six cases of equations given above exhaust all possibilities for linear and quadratic equations...In this sense, then, al-Khwarizmi is entitled to be known as 'the father of algebra'".
33. See Boyer 1991, page 228: "Diophantus sometimes is called the father of algebra, but this title more appropriately belongs to al-Khowarizmi".
34. See Gandz 1936, pp. 263–277: "In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
35. Christianidis, Jean (August 2007). "The way of Diophantus: Some clarifications on Diophantus' method of solution". Historia Mathematica. 34 (3): 289–305. doi:10.1016/j.hm.2006.10.003. It is true that if one starts from a conception of algebra that emphasizes the solution of equations, as was generally the case with the Arab mathematicians from al-Khwārizmī onward as well as with the Italian algebraists of the Renaissance, then the work of Diophantus appears indeed very different from the works of those algebraists
36. Cifoletti, G. C. (1995). "La question de l'algèbre: Mathématiques et rhétorique des homes de droit dans la France du 16e siècle". Annales de l'École des Hautes Études en Sciences Sociales, 50 (6): 1385–1416. Le travail des Arabes et de leurs successeurs a privilégié la solution des problèmes.Arithmetica de Diophantine ont privilégié la théorie des equations
37. See Boyer 1991, page 228.
38. See Boyer 1991, The Arabic Hegemony, p. 229: "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation".
39. See Boyer 1991, The Arabic Hegemony, p. 230: "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions".
40. Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 978-0-7923-2565-9. OCLC 29181926.
41. Oaks, Jeffrey (2014). The Oxford Encyclopedia of Islam and Philosophy, Science, and Technology. p. 458.
42. Christianidis, Jean (2007). "The way of Diophantus: Some clarifications on Diophantus' method of solution". Historia Mathematica. 34 (3): 303. doi:10.1016/j.hm.2006.10.003.
43. Oaks, Jeffrey; Christianidis, Jean (2013). "Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria". Historia Mathematica. 40 (2): 158–160. doi:10.1016/j.hm.2012.09.001.
44. Mathematical Masterpieces: Further Chronicles by the Explorers. p. 92.
45. O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics Archive, University of St Andrews
46. Victor J. Katz, Bill Barton; Barton, Bill (October 2007). "Stages in the History of Algebra with Implications for Teaching". Educational Studies in Mathematics. 66 (2): 185–201 [192]. doi:10.1007/s10649-006-9023-7. S2CID 120363574.
47. See Boyer 1991, The Arabic Hegemony, p. 239: "Abu'l Wefa was a capable algebraist as well as a trigonometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus – but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax²ⁿ + bxⁿ = c (only equations with positive roots were considered),"
48. "Al-Qalasadi biography". www-history.mcs.st-andrews.ac.uk. from the original on 2019-10-26. Retrieved 2017-10-17.
49. "The Origins of Abstract Algebra 2010-06-11 at the Wayback Machine". University of Hawaii Mathematics Department.
50. "The Collected Mathematical Papers". Cambridge University Press.
51. • Renze & Weisstein
    • Benson 2003, pp. 111–112
52. • Maddocks 2008, p. 129
    • Berggren 2015, Lead Section
    • Pratt 2022, § 1. Elementary algebra
    • EoM Staff 2020, § 1. Historical survey
53. • Romanowski 2008, pp. 302–303
    • HC Staff 2022
    • MW Staff 2023
    • EoM Staff 2020a
54. • Maddocks 2008, p. 129
    • Berggren 2015, Lead Section
    • Pratt 2022, § 1. Elementary algebra
    • EoM Staff 2020, § 1. Historical survey
55. • Maddocks 2008, pp. 129–130
    • Young 2010, p. 999
    • Majewski 2004, p. 347
    • Buthusiem & Toth 2020, pp. 24–28
    • Pratt 2022, § 1. Elementary algebra
56. • Maddocks 2008, pp. 129–130
    • Buthusiem & Toth 2020, pp. 24–28
57. • Maddocks 2008, p. 130
    • Buthusiem & Toth 2020, pp. 25–28
    • Pratt 2022, § 1. Elementary algebra
    • EoM Staff 2020, § 1. Historical survey
58. • Tan, Steeb & Hardy 2012, p. 306
    • Lamagna 2019, p. 150
59. • Buthusiem & Toth 2020, pp. 24–28
    • Berggren 2015, § Algebraic expressions, § Solving algebraic equations
60. • Zill & Dewar 2011, p. 529
    • Berggren 2015, § Solving systems of algebraic equations
    • McKeague 2014, p. 386
61. • Axler 2011, pp. 7–11
    • Berggren 2015, § Algebraic expressions, § Solving algebraic equations
62. • Maddocks 2008, pp. 130–131
    • Walz 2016, Algebra
63. • Maddocks 2008, pp. 130–131
    • Rohde et al. 2012, p. 89
    • Walz 2016, Algebra
64. "Hull's Algebra" (PDF). The New York Times. July 16, 1904. (PDF) from the original on 2021-02-21. Retrieved 2012-09-21.
65. Quaid, Libby (2008-09-22). "Kids misplaced in algebra" (Report). Associated Press. from the original on 2011-10-27. Retrieved 2012-09-23.
66. • Maddocks 2008, p. 131
    • Barrera-Mora 2023, p. ix, 1–2,
67. • Anton & Rorres 2013, pp. 2–3
    • Maddocks 2008, p. 131
    • EoM Staff 2011
68. • Barrera-Mora 2023, p. ix, 1, 12–13
    • Young 2010, pp. 726–727
    • Anton & Rorres 2013, pp. 32–34
69. • Maddocks 2008, p. 131
    • Andrilli & Hecker 2022, p. 57–58
70. • Anton & Rorres 2013, pp. 3–7
    • Mortensen 2013, p. 73–74
    • Williams 2007, pp. 4–5
    • Young 2023, pp. 714–715
71. • Maddocks 2008, p. 131
    • Harrison & Waldron 2011, p. 464
    • Anton 2013, p. 255
72. • Young 2010, pp. 697–698
    • Maddocks 2008, p. 131
    • Sullivan 2010, pp. 53–54
73. • Anton & Rorres 2013, pp. 7–8
    • Sullivan 2010, pp. 55–56
74. • Maddocks 2008, p. 131
    • Anton & Rorres 2013, pp. 7–8, 11, 491
75. • Anton & Rorres 2013, pp. 3–5
    • Young 2010, pp. 696–697
    • Williams 2007, pp. 4–5
76. • Anton & Rorres 2013, pp. 3–5
    • Young 2010, p. 713
    • Williams 2007, pp. 4–5

Works cited

• Boyer, Carl B. (1991). A History of Mathematics (2nd ed.). John Wiley & Sons. ISBN 978-0-471-54397-8.
• Gandz, S. (January 1936). "The Sources of Al-Khowārizmī's Algebra". Osiris. 1: 263–277. doi:10.1086/368426. JSTOR 301610. S2CID 60770737.
• Herstein, I. N. (1964). Topics in Algebra. Ginn and Company. ISBN 0-471-02371-X.
• Golan, Jonathan S. (1995). "Algebras Over A Field". Foundations of Linear Algebra. Springer Netherlands. ISBN 978-94-015-8502-6.
• Walz, Guido (1 December 2016). "Algebra". Lexikon der Mathematik: Band 1: A bis Eif (in German). Springer-Verlag. ISBN 978-3-662-53498-4.
• Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics (2nd ed.). Chapman & Hall/CRC. ISBN 1-58488-347-2.
• Renze, John; Weisstein, Eric W. "Algebra". Wolfram MathWorld. Wolfram. Retrieved 12 January 2024.
• Gilbert, William J.; Nicholson, W. Keith (30 January 2004). Modern Algebra with Applications. John Wiley & Sons. ISBN 978-0-471-46989-6.
• Grillet, Pierre Antoine (2007). "Universal Algebra". Abstract Algebra. Springer. ISBN 978-0-387-71568-1.
• EoM Staff (2020a). "Arithmetic". Encyclopedia of Mathematics. Springer. Retrieved 23 October 2023.
• MW Staff (2023). "Definition of Arithmetic". www.merriam-webster.com. Retrieved 19 October 2023.
• HC Staff (2022). "Arithmetic". www.ahdictionary.com. HarperCollins. Retrieved 19 October 2023.
• Romanowski, Perry (2008). "Arithmetic". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). The Gale Encyclopedia of Science (4th ed.). Thompson Gale. ISBN 978-1-4144-2877-2.
• Burgin, Mark (2022). Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective. World Scientific. ISBN 978-981-12-3685-3.
• Wagner, Sigrid; Kieran, Carolyn (7 December 2018). Research Issues in the Learning and Teaching of Algebra: the Research Agenda for Mathematics Education, Volume 4. Routledge. ISBN 978-1-135-43421-2.
• Maddocks, J. R. (2008). "Algebra". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). The Gale Encyclopedia of Science (4th ed.). Thompson Gale. ISBN 978-1-4144-2877-2.
• Fiche, Georges; Hebuterne, Gerard (1 March 2013). Mathematics for Engineers. John Wiley & Sons. ISBN 978-1-118-62333-6.
• Pratt, Vaughan (2022). "Algebra". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 11 January 2024.
• EoM Staff (2017). "Algebra". Encyclopedia of Mathematics. Springer. Retrieved 11 January 2023.
• EoM Staff (2020). "Algebra(2)". Encyclopedia of Mathematics. Springer. Retrieved 11 January 2023.
• EoM Staff (2023). "Algebraic operation". Encyclopedia of Mathematics. Springer. Retrieved 11 January 2023.
• Benson, Donald C. (2003). A Smoother Pebble: Mathematical Explorations. Oxford University Press. ISBN 978-0-19-514436-9.
• Rohde, Ulrich L.; Jain, G. C.; Poddar, Ajay K.; Ghosh, A. K. (12 January 2012). Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners. John Wiley & Sons. ISBN 978-1-118-13014-8.
• Axler, Sheldon (8 March 2011). Algebra and Trigonometry. John Wiley & Sons. ISBN 978-0-470-58579-5.
• McKeague, Charles P. (10 May 2014). Intermediate Algebra: A Text/Workbook. Academic Press. ISBN 978-1-4832-1417-7.
• Zill, Dennis; Dewar, Jacqueline (19 January 2011). Algebra and Trigonometry. Jones & Bartlett Publishers. ISBN 978-0-7637-5461-7.
• Lamagna, Edmund A. (15 January 2019). Computer Algebra: Concepts and Techniques. CRC Press. ISBN 978-1-351-60583-0.
• Berggren, John L. (2015). "Elementary algebra". Encyclopædia Britannica. Retrieved 14 January 2024.
• Buthusiem, Gregory; Toth, Gabor (2020). Precalculus. Linus Learning. ISBN 978-1-60797-798-8.
• Young, Cynthia Y. (19 January 2010). Precalculus. John Wiley & Sons. ISBN 978-0-471-75684-2.
• Majewski, Miroslaw (2004). MuPAD Pro computing essentials (2 ed.). Berlin Heidelberg New York: Springer. ISBN 978-3-540-21943-9.
• Tan, Kiat Shi; Steeb, Willi-Hans; Hardy, Yorick (6 December 2012). SymbolicC++:An Introduction to Computer Algebra using Object-Oriented Programming: An Introduction to Computer Algebra using Object-Oriented Programming. Springer Science & Business Media. ISBN 978-1-4471-0405-6.
• Sullivan, Michael (29 March 2010). Finite Mathematics: An Applied Approach. John Wiley & Sons. ISBN 978-0-470-87639-8.
• Harrison, Michael; Waldron, Patrick (31 March 2011). Mathematics for Economics and Finance. Routledge. ISBN 978-1-136-81921-6.
• Young, Cynthia Y. (16 May 2023). Precalculus. John Wiley & Sons. ISBN 978-1-119-86940-5.
• Williams, Gareth (17 August 2007). Linear Algebra with Applications. Jones & Bartlett Learning. ISBN 978-0-7637-5753-3.
• Mortensen, C. E. (14 March 2013). Inconsistent Mathematics. Springer Science & Business Media. ISBN 978-94-015-8453-1.
• Andrilli, Stephen; Hecker, David (5 April 2022). Elementary Linear Algebra. Academic Press. ISBN 978-0-323-98426-3.
• Anton, Howard; Rorres, Chris (4 November 2013). Elementary Linear Algebra: Applications Version. John Wiley & Sons. ISBN 978-1-118-47422-8.
• Barrera-Mora, Fernando (8 May 2023). Linear Algebra: A Minimal Polynomial Approach to Eigen Theory. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-113591-5.
• EoM Staff (2011). "Linear equation". Encyclopedia of Mathematics. Springer. Retrieved 10 January 2024.
• Anton, Howard (4 November 2013). Elementary Linear Algebra. John Wiley & Sons. ISBN 978-1-118-67730-8.

Further reading

• Allenby, R. B. J. T. (1991). Rings, Fields and Groups. ISBN 0-340-54440-6.
• Asimov, Isaac (1961). Realm of Algebra. Houghton Mifflin.
• Euler, Leonhard (2005). . ISBN 978-1-899618-73-6. Archived from the original on 2011-04-13.
• Herstein, I. N. (1975). Topics in Algebra. ISBN 0-471-02371-X.
• Hill, Donald R. (1994). Islamic Science and Engineering. Edinburgh University Press.
• Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books. ISBN 978-0140277784.
• O'Connor, John J.; Robertson, Edmund F. (2005). . MacTutor History of Mathematics archive. University of St Andrews. Archived from the original on 2016-03-03. Retrieved 2011-12-10.
• Sardar, Ziauddin; Ravetz, Jerry; Loon, Borin Van (1999). Introducing Mathematics. Totem Books.

External links

 

Wikiquote has quotations related to Algebra.

 

Look up algebra in Wiktionary, the free dictionary.

 

Wikibooks has a book on the topic of: Algebra

 

Wikisource has the text of the 1911 Encyclopædia Britannica article "Algebra".

• Khan Academy: Conceptual videos and worked examples
• Khan Academy: Origins of Algebra, free online micro lectures
• Algebrarules.com: An open source resource for learning the fundamentals of Algebra
• , lecture by Robin Wilson, at Gresham College, October 17, 2007 (available for MP3 and MP4 download, as well as a text file).
• Pratt, Vaughan. "Algebra". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.