algebra

Etymology

From Medieval Latin algebrāica, from Arabic word الْجَبْر (al-jabr, “reunion, resetting of broken parts”) in the title of al-Khwarizmi's influential work الْكِتَاب الْمُخْتَصَر فِي حِسَاب الْجَبْر وَالْمُقَابَلَة (al-kitāb al-muḵtaṣar fī ḥisāb al-jabr wa-l-muqābala, “The Compendious Book on Calculation by Completion and Balancing”).

noun

1. (uncountable, mathematics) A system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols.
   Also the rule of false position, with dyuers examples not only vulgar, but some appertaynyng to the rule of Algeber. 1551, James A.H. Murray, editor, A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society., volume 1, Oxford: Clarendon Press, published 1888, Part 1, page 217
   Let us conceive, then, of an Algebra in which the symbols x, y, z, &c. admit indifferently of the values 0 and 1, and of these values alone. 1854, George Boole, “Signs and their Laws”, in An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities, London: Walton and Maberly, page 37
2. (uncountable, medicine, historical, rare) The surgical treatment of a dislocated or fractured bone. Also (countable): a dislocation or fracture.
   Ne take noon hede to brynge togidere þe parties of þe boon þat is to-broken or dislocate, til viij. daies ben goon in þe wyntir, & v. in þe somer; for þanne it schal make quytture, and be sikir from swellynge; & þanne brynge togidere þe brynkis eiþer þe disiuncture after þe techynge þat schal be seid in þe chapitle of algebra. a1420, The British Museum Additional MS, 12,056, “Wounds complicated by the Dislocation of a Bone”, in Robert von Fleischhacker, editor, Lanfranc's "Science of cirurgie.", London: K. Paul, Trench, Trübner & Co, translation of original by Lanfranc of Milan, published 1894, page 63
   Algebra is used today by surgeons to mean bone-setting, i.e. the restoration of bones, and the idea of restoration is present in the mathematical context, too. 1987, John Newsome Crossley, “Latency”, in The emergence of number, Singapore: World Scientific, Al-Khwarizwi, page 65
3. (uncountable, mathematics) The study of algebraic structures.
4. (countable, mathematics) A universal algebra.
5. (countable, algebra) An algebraic structure consisting of a module over a commutative ring (or a vector space over a field) along with an additional binary operation that is bilinear over module (or vector) addition and scalar multiplication.
   In mathematics, a Lie algebra (pronounced /liː/ "Lee") is a vector space g together with a non-associative, alternating bilinear map g×g→g;(x,y)↦[x,y], called the Lie bracket, satisfying the Jacobi identity. 2018-03-23, Wikipedia contributors, “Lie algebra”, in English Wikipedia, Wikimedia Foundation, revision 831953572
6. (countable, set theory, mathematical analysis) A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and complements (and thereby also under intersections and differences).
7. (countable, mathematics) One of several other types of mathematical structure.
8. (figurative) A system or process, that is like algebra by substituting one thing for another, or in using signs, symbols, etc., to represent concepts or ideas.
   Fly ! Fly ! avaunt with that base cowardly gibbrish ; That Algebra of honour ; which had never Been nam'd, if all had equal courage—what? 1663, William Clark, edited by William Hugh Logan, Marciano; or, The discovery: A tragi-comedy, Edinburgh: Reprinted for Private Circulation, published 1871, page 13