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around 820

Al-Khwārizmī's Al-Jabr

Landmark work that established and named algebra

Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa'l-muqābala (The Compendious Book on Calculation by Completion and Balancing), known as Al-Jabr for short, was a landmark work in the history of mathematics written by Persian polymath Muḥammad ibn Mūsā al-Khwārizmī around 820 CE.

Al-Khwārizmī's Al-Jabr

Al-Jabr established algebra as an independent discipline of study and is so central to the formulation of algebra that the term "algebra" itself is derived from the book's title. The work provided an exhaustive treatment of determining the positive roots of polynomial equations up to second degree with positive coefficients. The most famous page demonstrates the solution of the quadratic equation x² + bx = c by means of a diagram that determines the root by erecting rectangles on a central square and completing the square using smaller squares bordering those.

Artifact dimensions

6.3 in. × 9.5 in.

Artifact origin

Cairo, Egypt

Current artifact location

Rare Book & Manuscript Library, Butler Library, Columbia University in the City of New York

Catalog number

QA32 .K48 2009g


Algebra timeline Babylonian Metric Algebra Problems Tablet Berlin Pythagorean Theorem Papyrus Rhind Papyrus Al-Khwārizmī's Al-Jabr Khayyam's Al-jabr Cardano's Ars Magna Recorde's Whetstone of Witte Faulhaber's Academia Algebrae Wallis's Treatise of Algebra Emerson's Treatise of Algebra Hermes's Suitcase of Göttingen

Interactive Content

Computational Explanation

Additional Reading

  • Berggren, J. L. "Al-Khwārizmī's Compendium on Calculation by Completion and Reduction." In The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 542–548, 2007.
  • Gandz, S. "The Algebra of Inheritance." Osiris, Vol. 5, pp. 319–391, 1938.
  • Hughes, B. B. (Ed.). Robert of Chester's Latin Translation of Al-Khwarizmi's Al-Jabr: A New Critical Edition. Wiesbaden, Germany: F. Steiner Verlag, 1989.
  • Katz, V. J. and Parshall, K. H. Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton, NJ: Princeton University Press, 2020.
  • King, D. A. "Mathematics Applied to Aspects of Religious Ritual in Islam: The Arithmetic of Inheritance." Ch. 1.7, § 5 in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Vol. 1 (Ed. I. Grattan-Guinness). London: Routledge, p. 83, 1994.
  • Merzbach, U. C. and Boyer, C. B. "Al-Kharizmi." In A History of Mathematics, 3rd ed. New York: Wiley, pp. 206–212, 2011.
  • Rashed, R. The Development of Arabic Mathematics: Between Arithmetic and Algebra. Dordrecht, Netherlands: Kluwer, 1994.
  • Rosen, T. (Ed. and trans.). The Algebra of Mohammed ben Musa. London: J. L. Cox, 1831.
  • van der Waerden, B. L. "Three Muslim Authors: Al-Khwārizmī." Ch. 1, Part A in A History of Algebra: From al-Khwārizmī to Emmy Noether. Berlin: Springer-Verlag, pp. 3–15, 1985.