Algebra is a branch of mathematics connected with the study of structure, relation, and quantity.
Algebra (Bij Ganitam): The origin of Bij-ganitam is ancient India, where it was first developed as a systematic approach to solving mathematical problems. This rich mathematical tradition was characterized by innovative techniques and principles that laid the groundwork for algebraic concepts. Persians first learned it from here, assimilating its principles into their own educational frameworks, and subsequently, the Greeks adopted these ideas, further enriching the study of mathematics. The name “Algebra” itself is derived from the treatise written by the influential Persian mathematician Muhammad bin Mลซsฤ al-Khwฤrizmฤซ around 820 CE, which served as a cornerstone for mathematical literature. His work, titled (in Arabic ูุชุงุจ ุงูุฌุจุฑ ูุงูู ูุงุจูุฉ) Al-Kitab al-Jabr wa-l-Muqabala, meaning “The Compendious Book on Calculation by Completion and Balancing,” provided comprehensive methods for solving linear and quadratic equations. This foundational text not only transformed the landscape of mathematics in the regions of the Islamic Golden Age but also set the stage for algebra to become a vital component of modern mathematics around the globe.
Major fields of mathematics :
Logic โข Set theory โข Algebra (Abstract algebra – Linear algebra) โข Discrete mathematics โข Combinatorics โข Number theory โข Analysis โข Geometry โข Topology โข Applied mathematics โข Probability โข Statistics โข Mathematical physics โข Ring (mathematics) and Field (mathematics)
Elementary and Abstract Algebra
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. Although in arithmetic, only numbers and their arithmetical operations (such as +, โ, ร, รท) occur, in algebra, numbers are often denoted by symbols (such as a, x, y).
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.
The word algebra is also used for various algebraic structures:
Algebra over a field
Algebra over a set
Boolean algebra
F-algebra and F-coalgebra in category theory
Sigma-algebra
Fields of algebra
ยป Abstract algebra ยป Affine Hecke algebra ยป Algebra
ยป Algebra homomorphism ยป Algebra stubs ยป Algebraic element
ยป Algebraic extension ยป Algebraic function ยป Algebraic geometry
ยป Algebraic number ยป Algebraic solution ยป Algebraic topology
ยป Alternatization ยป Anyonic Lie algebra ยป Associative algebra
ยป Automorphism ยป Binary operation ยป Binomial
ยป Binomial theorem ยป Brahmagupta-Fibonacci identity ยป Canonical form
ยป Category theory ยป Closed-form expression ยป Coefficient
ยป Combinatorics ยป Completing the square ยป Computer algebra
ยป Congruence relation ยป Conjugate (algebra) ยป Consequence operator
ยป Cuntz algebra ยป Cycle (mathematics) ยป Degree (mathematics)
ยป Derivative (generalizations) ยป Determinant ยป Digital root
ยป Dimension ยป Distributive homomorphism ยป Distributive lattice
ยป Elementary algebra ยป Expression (mathematics) ยป Factorization
ยป Filtration (mathematics) ยป Galois theory ยป Generalized arithmetic progression
ยป Goursat’s lemma ยป Hall polynomial ยป Hecke algebra
ยป Homological algebra ยป Identity element ยป Immanant of a matrix
ยป Indeterminate (variable) ยป Intersection (set theory) ยป Inverse element
ยป Irreducible polynomial ยป Isomorphism ยป Iterated binary operation
ยป K-theory ยป K-theory (physics) ยป Kernel (algebra)
ยป Kernel (set theory) ยป Lattice (order) ยป Laws of Form
ยป List of basic algebra topics ยป Mathematical identities ยป Monomial
ยป Monomial basis ยป Multinomial theorem ยป Multiplicative inverse
ยป Nested radical ยป Operand ยป Operation theory
ยป Operator ยป Order of operations ยป Pairing
ยป Partial fraction ยป Partial fraction decomposition ยป Perfect square
ยป Permanent ยป Permutations ยป Plugging in (algebra)
ยป Polynomials ยป Power set ยป Ratio
ยป Rational root theorem ยป Recurrence relation ยป Relation algebra
ยป Ring theory ยป Square (algebra) ยป Sylvester’s determinant theorem
ยป Symbolic method ยป Symmetric difference ยป Symmetric functions
ยป System of linear equations ยป Temperley-Lieb algebra ยป Theory of equations
ยป Topological module ยป Transforming polynomials ยป Trinomial
ยป Union (set theory) ยป Unital ยป Universal algebra
ยป Variable ยป Variety (universal algebra)
More on Algebra
Derivative algebra (abstract algebra) Map algebra F-algebra
The Algebra of Ice C-algebra *-algebra
MV-algebra Algebra Pre-algebra
Max-plus algebra Rng (algebra) B-algebra
En (Lie algebra) Lie algebra Abstract algebra
Steenrod algebra Derivative algebra Hopf algebra
Multilinear algebra Group algebra Elements of Algebra
Hecke algebra Modular Lie algebra Graded algebra
De Morgan algebra Heyting algebra Plugging in (algebra)
Graded Lie algebra Elementary algebra Jordan algebra
Monster Lie algebra Linear algebra Operator algebra
Relation algebra Poisson algebra Free Lie algebra
Example of a non-associative algebra Characteristic (algebra) Magma (algebra)
Quaternion algebra Exterior algebra Cuntz algebra
Spectrum of a C*-algebra Strip algebra Topological algebra
Cube (algebra) Matrix algebra Enveloping algebra
Chronology of key algebraic developments :
Circa 2800 BC: Bodhayan Collected old Theories and compiled in Sulva Sutram [India]. It explained so-called Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax2 = c and ax2 + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations.
Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.
Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.
Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.
Circa 300 BC: In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.
Circa 300 BC: A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools.
Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations.
Circa 100 BC: The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.
Circa 150 AD: Hellenized Egyptian mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.
Circa 200: Hellenized Babylonian mathematician Diophantus, who lived in Egypt and is often considered the “father of algebra”, writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
499: Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation and gives integral solutions of simultaneous indeterminate linear equations.
Circa 625: Chinese mathematician Wang Xiaotong finds numerical solutions of cubic equations.
628: Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell’s equation, and gives rules for solving linear and quadratic equations.
820: The word algebra is derived from operations described in the treatise written by the Persian mathematician Muแธฅammad ibn Mลซsฤ al-แธดwฤrizmฤซ titled Al-Kitab al-Jabr wa-l-Muqabala (meaning “The Compendious Book on Calculation by Completion and Balancing”) on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered as the “father of algebra”, much of whose works on reduction was included in the book and added to many methods we have in algebra now.
Circa 850: Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.
Circa 850: Indian mathematician Mahavira solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations.
Circa 990: Persian Abu Bakr al-Karaji, in his treatise al-Fakhri, further develops algebra by extending Al-Khwarizmi’s methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, … and 1/x, 1/x2, 1/x3, … and gives rules for the products of any two of these.
Circa 1050: Chinese mathematician Jia Xian finds numerical solutions of polynomial equations.
1072: Persian mathematician Omar Khayyam develops algebraic geometry and, in the Treatise on Demonstration of Problems of Algebra, gives a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.
1114: Indian mathematician Bhaskara, in his Bijaganita (Algebra), recognizes that a positive number has both a positive and negative square root, and solves various cubic, quartic and higher-order polynomial equations, as well as the general quadratic indeterminant equation.
1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.
Circa 1300: Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations.
Circa 1400: Indian mathematician Madhava of Sangamagramma finds iterative methods for approximate solution of non-linear equations.
1535: Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation.
1545: Girolamo Cardano publishes Ars magna -The great art which gives Fontana’s solution to the general quartic equation.
1572: Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.
1591: Francois Viete develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge.
1631: Thomas Harriot in a posthumus publication uses exponential notation and is the first to use symbols to indicate “less than” and “greater than”.
1682: Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.
1680s: Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, and Bernoulli numbers.
1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer’s rule and studies algebraic curves, matrices and determinants.
1824: Niels Henrik Abel proved that the general quintic equation is insoluble by radicals.
1832: Galois theory is developed by รvariste Galois in his work on abstract algebra.
