Algebra

1st millennium BC, Abelian group, Abstract algebra, Addition, Al-Karaji, Al-Khowarazmi, Algebra (disambiguation), Algebra (ring theory), Algebra over a field, Algebra over a set, Algebraic combinatorics, Algebraic geometry, Algebraic number theory, Algebraic structure, Algebraic structures, Analytic geometry, Arabic, Arithmetic, Arithmetica, Associativity, Axiomatization, Babylonian mathematics, Banach algebra, Banach space, Bhaskara II, Binary operation, Boolean algebra (structure), Brahmagupta, Brahmasphutasiddhanta, Calculus, Category theory, Chinese mathematics, Classification of finite simple groups, Closure (mathematics), Combinatorics, Commutative algebra, Commutative operation, Commutativity, Complex numbers, Constant (mathematics), Constructible number, Coordinate vector, Cubic equation, Cyclic group, Determinant, Digital object identifier, Diophantus, Distributivity, Division (mathematics), Dynamical systems theory, Edmund F. Robertson, Egyptian mathematics, Element (mathematics), Elementary algebra, Equation, Equation solving, Euclid's Elements, Examples of groups, Expository writing, Expression (mathematics), F-algebra, F-coalgebra, Factorization, Field (mathematics), Field theory (mathematics), Finite groups, Finite set, Function (mathematics), Functional analysis, Fundamental theorem of algebra, Gabriel Cramer, Galois theory, Game theory, Geometry, Glossary of field theory, Glossary of group theory, Glossary of linear algebra, Glossary of ring theory, Gottfried Leibniz, Greek mathematics, Gresham College, Group (mathematics), Group theory, Hellenistic civilization, Hero of Alexandria, Hilbert space, History of algebra, Identity element, Indeterminate equation, Indian mathematics, Information theory, Inner product, Inner product space, Integers, Integral domain, International Standard Book Number, Inverse elements, Japanese mathematics, John J. O'Connor (mathematician), K-theory, Kowa Seki, L. Euler, La Géométrie, Like terms, Linear Algebra, Linear algebra, Linear equation, List of abstract algebra topics, List of algebraic structures, List of group theory topics, List of linear algebra topics, List of mathematics articles, Logic, MacTutor History of Mathematics archive, Mahavira (mathematician), Mathematical analysis, Mathematical expression, Mathematical logic, Mathematics, Mathematics in medieval Islam, Matrix (mathematics), Matrix multiplication, Matrix theory, Metric (mathematics), Modular arithmetic, Monad (category theory), Monoid, Muhammad ibn Musa al-Khwarizmi, Multiplication, Natural numbers, Normed algebra, Normed linear space, Number, Number theory, Numerical analysis, Octonion multiplication, Omar Khayyam, Online Computer Library Center, Operation (mathematics), Optimization (mathematics), Order of operations, Order theory, Outline of algebra, Penguin Books, Polynomial, Polynomials, Probability theory, Problem, Pure mathematics, Quadratic Equation, Quadratic equation, Quartic equation, Quasigroup, Quintic equation, Rational number, Rational numbers, Real number, Real numbers, Reduction (mathematics), Relation (mathematics), Relational algebra, Rene Descartes, Rhind Mathematical Papyrus, Ring (mathematics), Ring theory, Root of a function, Secondary education, Semigroup, Set (mathematics), Set theory, Sharaf al-Dīn al-Tūsī, Sigma-algebra, Simple group, Springer Science+Business Media, Stanford Encyclopedia of Philosophy, Statistics, Subtraction, Term (mathematics), The Compendious Book on Calculation by Completion and Balancing, The Nine Chapters on the Mathematical Art, Theory of computation, Timeline of algebra, Topological algebra, Topological group, Topology, Trigonometry, Universal algebra, University of St Andrews, Variable (mathematics), Vector (geometric), Vector space, Zhu Shijie,

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics. The part of algebra called elementary algebra is often part of the curriculum in secondary education and introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalised and their precise definitions lead to structures such as groups, rings and fields.

Additional info

Abelian group

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel.

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings.

Addition

Addition is the mathematical process of combining quantities. It is signified by the plus sign (+). For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples. Therefore, 3 + 2 = 5. Besides counts of fruit, addition can also represent combining other physical and abstract quantities using different kinds of numbers: negative numbers, fractions, irrational numbers, vectors, and more.

Al-Karaji

Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī (or al-Karkhī) (c. 953 in Karaj or Karkh – c. 1029) was a 10th century Persian^[1] Muslim mathematician and engineer. His three major works are Al-Badi' fi'l-hisab (Wonderful on calculation), Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi fi'l-hisab (Sufficient on calculation).

Al-Khowarazmi

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī^[1] (c. 780, Khwārizm^[2]^[3]^[4] – c. 850) was a Persian^[5]^[2]^[6] mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad.

Algebra (ring theory)

In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R.

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it is an algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as an algebra, this multiplication must satisfy certain compatibility axioms with the given vector space structure, such as distributivity. In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field.^[1]

Algebra over a set

In mathematics a field of sets is a pair $\langle X, \mathcal{F} \rangle$ where

X

is a set and $\mathcal{F}$ is an algebra over $X$ i.e., a non-empty subset of the power set of

X

closed under the intersection and union of pairs of sets and under complements of individual sets. In other words $\mathcal{F}$ forms a subalgebra of the power set Boolean algebra of

X

. (Many authors refer to $\mathcal{F}$ itself as a field of sets.) Elements of

X

are called points and those of $\mathcal{F}$ are called complexes.

Algebraic combinatorics

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. It is one of the youngest combinatorial disciplines. Thus, in the preface to his Combinatorial Theory, published in 1979, Martin Aigner wrote about "growing consensus that combinatorics should be divided into three parts" (Enumeration, Order theory, Configurations), without even mentioning algebraic combinatorics by name.

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations, as to find some solution; this leads into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

Algebraic number theory

In mathematics, algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integers — such as unique factorization — need not hold. The virtue of the primary machinery employed — Galois theory, group cohomology, group representations, and L-functions — is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.

Algebraic structure

In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra.

Algebraic structures

Analytic geometry

Analytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. Analytic geometry is the foundation of most modern fields of geometry, including algebraic geometry, differential geometry, and discrete and computational geometry, and is widely used in physics and engineering.

Arabic

Egypt: Academy of the Arabic Language in Cairo
Iraq: Iraqi Academy of Sciences
Jordan: Jordan Academy of Arabic
Libya: Academy of the Arabic Language in Jamahiriya
Morocco: Academy of the Arabic Language in Rabat
Sudan: Academy of the Arabic Language in Khartum
Syria: Arab Academy of Damascus (the oldest)
Tunisia: Beit Al-Hikma Foundation

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. Professional mathematicians sometimes use the term (higher) arithmetic^[1] when referring to more advanced results related to number theory, but this should not be confused with elementary arithmetic.