Algebra is a major branch of mathematics. It studies quantities written as symbols and the rules for combining them. Elementary algebra extends arithmetic by using letters for unknown or general numbers, so that relationships can be stated and solved as equations. Abstract (or modern) algebra studies the structures themselves, such as groups, rings, fields and vector spaces, and leaves aside which objects are being combined. Algebra sits alongside geometry and analysis as a pillar of mathematics. It supplies the working language for most of science, engineering and computer science.
Etymology
The word comes from the Arabic al-jabr, "restoration" or "completion". It appears in the title of a treatise by the Persian scholar Muḥammad ibn Mūsā al-Khwārizmī, Al-Kitab al-Jabr wa-l-Muqabala ("The Compendious Book on Calculation by Completion and Balancing"), on the systematic solution of linear and quadratic equations, written around 820 CE. In ordinary Arabic, jabr referred to the surgical reduction of a fracture. The word passed into medieval Latin as algebra. Al-Khwārizmī's name also gave rise to the word algorithm.
Early history
Algebraic problem-solving is far older than the word. Scribes in Mesopotamia solved problems that are equivalent to quadratic equations. They did this nearly four thousand years ago, using step-by-step numerical recipes. Mathematical papyri from ancient Egypt contain linear problems about an unknown "quantity". The Chinese Nine Chapters on the Mathematical Art, compiled by the Han dynasty, solved systems of linear equations by a procedure that is essentially Gaussian elimination. It also handled negative numbers.
In ancient Greece, quantities were usually treated geometrically, as lengths and areas. Around the 3rd century CE, Diophantus of Alexandria wrote the Arithmetica. It introduced abbreviations for unknowns and their powers, and it looked for rational solutions to equations. This work later inspired number theory. Indian mathematicians, notably Brahmagupta in the 7th century, gave rules for working with negative quantities and zero, and gave general solutions to quadratic equations.
The Islamic Golden Age
Under the Abbasid Caliphate, al-Khwārizmī's book shows how to solve linear and quadratic equations, how to calculate the area and volume of certain geometric shapes, and how to reduce equations by "completion" and "balancing". His presentation was altogether rhetorical, i.e., devoid of all symbolism. The equations were written out in words and justified with geometric diagrams. Historians often credit him with founding algebra as a discipline because he taught it in an elementary form and for its own sake. Later scholars built on this work. Abu Kamil handled irrational coefficients, al-Karaji developed an algebra of polynomials, and Omar Khayyam classified cubic equations and solved them geometrically by intersecting conic sections.
Symbolic algebra in Europe
Arabic algebra reached Europe through Latin translations and through Fibonacci's Liber Abaci (1202). During the Renaissance, Italian mathematicians found general formulas for the cubic equation: Scipione del Ferro, Niccolò Tartaglia and Gerolamo Cardano, who published the method in Ars Magna (1545). Lodovico Ferrari solved the quartic. Working with these formulas pushed mathematicians toward accepting negative and complex numbers.
Before the late 16th century, algebra consisted largely of techniques for solving particular kinds of equations. François Viète changed this with his work of 1591. The idea of using letters to denote both unknowns and coefficients was introduced by François Viète (1540–1603), and this allowed whole families of equations to be studied at once. René Descartes gave notation close to the modern form in La Géométrie (1637). He also joined algebra to geometry through coordinates. This analytic geometry prepared the ground for calculus.
Theory of equations and the rise of abstract algebra
Two questions dominated the 18th and early 19th centuries. The first was whether every polynomial equation has roots. The fundamental theorem of algebra says every polynomial with real or complex coefficients has as many real or complex roots as its degree (counted with multiplicity). A widely recognized first proof was given by Carl Friedrich Gauss in his 1799 dissertation, though Gauss's proof had gaps by modern standards.
The second question was whether higher-degree equations could be solved by formulas that use only radicals (roots). The Abel–Ruffini theorem says that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Paolo Ruffini gave an incomplete proof in 1799, and Niels Henrik Abel gave a proof in 1824. Évariste Galois went further. He used group theory techniques to give a criterion for deciding if an equation is solvable using radicals. This was the origin of Galois theory and of group theory.
During the 19th century, the focus shifted from solving equations to studying the structures behind them. George Boole created an algebra of logic, now called Boolean algebra. Hamilton's quaternions showed that multiplication need not be commutative, and Cayley developed the algebra of matrices. Dedekind, Kronecker and Hilbert developed algebraic number theory and the theory of ideals. Abstract algebra is essentially a creation of the nineteenth century, but it became an independent and flourishing subject only in the early decades of the twentieth, largely through the pioneering work of Emmy Noether. Emmy Noether's axiomatic approach, spread by van der Waerden's textbook Moderne Algebra (1930–31), shaped how the subject is taught today.
Main branches
- Elementary algebra covers variables, expressions, and the solution of equations and inequalities. It is a core part of secondary education.
- Linear algebra studies vector spaces, linear maps, matrices and systems of linear equations.
- Abstract algebra studies algebraic structures defined by axioms. A group is a set with one associative operation, an identity element and inverses. Rings and fields have two operations, like addition and multiplication of integers or rational numbers.
- Commutative algebra and algebraic geometry study polynomial rings and the geometric shapes defined by polynomial equations.
- Universal algebra and category theory study what all algebraic structures have in common. They draw on set theory and the methods of mathematical proof.
Applications
Algebra is used throughout quantitative work. In physics, group theory describes symmetry. Through Noether's theorem, symmetries are linked to conservation laws, and group representations are used to classify elementary particles. Linear algebra is central to quantum mechanics, statistics, computer graphics and machine learning, which relies on large matrix computations. Finite fields and number-theoretic algebra are the basis of modern cryptography and error-correcting codes. Boolean algebra underlies the design of digital circuits in the computer. Computer algebra systems can manipulate symbolic expressions automatically, which carries forward the rule-based approach that al-Khwārizmī first set out.
References
- Timeline of algebraen.wikipedia.org
- Algebra: from Al-Khwarizmi to modern algebratangente-mag.com
- Muhammad Al-Khwarizmimathigon.org
- History of algebra - Wikiquoteen.wikiquote.org
- Al-Jabren.wikipedia.org
- Historylink.springer.com
- A History of Abstract Algebra - Jeremy Grayscribd.com
- graphsearch.epfl.chgraphsearch.epfl.ch
- Nasehpour - History of Algebranasehpour.com
- kleiner-history of abstract algebra_2007_Birkhauser.pdfmathscitech.org