Algebra

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[[File:Quadratic formula.svg|thumb|The quadratic formula expresses the solution of the equationwikipedia
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Mathematics

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Algebra (from الجبر, transliterated "al-jabr", literally meaning "reunion of broken parts" ) is one of the broad parts of mathematics, together with number theory, geometry and analysis.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis).

Areas of mathematics

broad partsfields of mathematicsareas
Algebra (from الجبر, transliterated "al-jabr", literally meaning "reunion of broken parts" ) is one of the broad parts of mathematics, together with number theory, geometry and analysis.

Abstract algebra

algebraalgebraicmodern algebra
The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra.
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

Elementary algebra

Algebraelementary formElementary
The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra.
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics.

Arithmetic

arithmetic operationsarithmeticsarithmetic operation
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.
Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis.

The Compendious Book on Calculation by Completion and Balancing

AlgebraAl-JabrAl-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala
The word algebra comes from the Arabic الجبر ( lit. "the reunion of broken parts") from the title of the book Ilm al-jabr wa'l-muḳābala by the Persian mathematician and astronomer al-Khwarizmi.
The Compendious Book on Calculation by Completion and Balancing (الْكِتَابْ الْمُخْتَصَرْ فِيْ حِسَابْ الْجَبْرْ وَالْمُقَابَلَة, Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala; Liber Algebræ et Almucabola), also known as Al-jabr, is an Arabic mathematical treatise on algebra written by Persian polymath Muḥammad ibn Mūsā al-Khwārizmī around 820 CE while he was in the Abbasid capital of Baghdad, modern-day Iraq.

Muhammad ibn Musa al-Khwarizmi

Al-Khwarizmial-KhwārizmīMuhammad ibn Mūsā al-Khwārizmī
The word algebra comes from the Arabic الجبر ( lit. "the reunion of broken parts") from the title of the book Ilm al-jabr wa'l-muḳābala by the Persian mathematician and astronomer al-Khwarizmi.
Because he was the first to treat algebra as an independent discipline and 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), he has been described as the father or founder of algebra.

Commutative algebra

commutativealgebrasP-Commutative Topological Algebras
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.
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.

Quadratic equation

quadratic equationsquadraticquadratic formula
For example, in the quadratic equation
In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form

Calculus

infinitesimal calculusdifferential and integral calculusclassical calculus
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.
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

Field (mathematics)

fieldfieldsfield theory
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. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields.
A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

Permutation

permutationscycle notationpermuted
These questions led extending algebra to non-numerical objects, such as permutations, vectors, matrices, and polynomials.
In algebra, and particularly in group theory, a permutation of a set

K-theory

K-theoriesK theoryalgebraic K_i
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.
In algebra and algebraic geometry, it is referred to as algebraic K-theory.

Chinese mathematics

Chinese mathematicianmathematicsChinese mathematical
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 Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral systems (base 2 and base 10), algebra, geometry, number theory and trigonometry.

History of algebra

algebraGreek geometric algebrasyncopation
The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them.
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects.

Babylonian mathematics

BabyloniansBabylonian mathematiciansBabylonian
The roots of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion.
The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem.

Mathematics in medieval Islam

mathematicianmathematicsIslamic mathematics
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.
Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for The Compendious Book on Calculation by Completion and Balancing by scholar Al-Khwarizmi), and advances in geometry and trigonometry.

Indian mathematics

Indian mathematicianmathematicianmathematics
The Hellenistic mathematicians Hero of Alexandria and Diophantus 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.
Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra.

Theory of equations

theory of algebraic equationstheory of polynomial equations
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 context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".
In algebra, the theory of equations is the study of algebraic equations (also called “polynomial equations”), which are equations defined by a polynomial.

Cubic equation

cubicCardano's formulaCardano's method
Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation.
In algebra, a cubic equation in one variable is an equation of the form

Diophantine equation

Diophantine equationsDiophantine analysisDiophantine
These texts deal with solving algebraic equations, and have led, in number theory to the modern notion of Diophantine equation.
The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra.

Quartic function

quartic equationquarticquartic polynomial
The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods.
In algebra, a quartic function is a function of the form

Quintic function

quintic equationquinticgeneral quintic equation
The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods.
In algebra, a quintic function is a function of the form

Arithmetica

Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica.
It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

Like terms

Combining like termsgrouping
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, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs, while treating algebra as an independent discipline in its own right.
In algebra, like terms are terms that have the same variables and powers.