Algebra as a distinct method
The science of how to solve equations and transform them, regardless of what the equations describe. In al-Khwarizmi's treatise Kitab al-jabr wa'l-muqabala (Baghdad, c. 820) six canonical types of linear and quadratic equations are given, along with two reduction operations, al-jabr (restoration) and al-muqabala (balancing). Everything is set out in words, without symbols; even the numbers are written in words.
Rashed regards the book as the beginning of algebra as an independent discipline; Toomer and Crossley rate al-Khwarizmi's originality far lower, and the dispute is not closed. The word 'algebra' comes from the title of the book, and 'algorithm' from the Latinised name of its author.
The 12th-century Latin translation brought the method to Europe.
Why this date
approximate. The Kitab al-jabr wa'l-muqabala was written under Caliph al-Mamun (813-833); Britannica dates it "around 825"
What it grew from
-
The Graeco-Arabic translation movement in Baghdad
influenced · Confidence: medium
Al-Hajjaj, al-Khwarizmi's colleague in the Baghdad circle of translators, had translated Euclid's Elements into Arabic already under Harun al-Rashid; the geometrical figures with which al-Khwarizmi explains the solution of equations, in Toomer's view, argue for familiarity with Book II of the Elements, and Rashed holds that his treatment was very probably inspired by recent knowledge of the Elements.
Dispute: There are two disputes. First, about the 'House of Wisdom' itself: MacTutor and the common tradition place al-Khwarizmi and the translators there, while Gutas (1998) considers the bayt al-hikma a palace library with no direct connection to translation from Greek; the link here goes to the translation movement, not to the institution. Second, about the dependence itself: Toomer and Rashed see a trace of the Elements in al-Khwarizmi's geometrical proofs, while Gandz (according to MacTutor) writes that the Elements in spirit and letter are entirely unknown to al-Khwarizmi, who has no definitions, axioms or Euclidean demonstrations. MacTutor itself concludes that al-Khwarizmi in any case drew on other geometrical works, including Hebrew and Indian ones.
Link sources
- MacTutor History of Mathematics, «Abu Ja'far Muhammad ibn Musa Al-Khwarizmi» Verified 5 Oct 2026
In al-Rashid's reign, while al-Khwarizmi was still young, al-Hajjaj had translated Euclid's Elements into Arabic and al-Hajjaj was one of al-Khwarizmi's colleagues in the House of Wisdom.
- MacTutor History of Mathematics, «Abu Ja'far Muhammad ibn Musa Al-Khwarizmi» Verified 5 Oct 2026
in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid's "Elements".
- MacTutor History of Mathematics, «Abu Ja'far Muhammad ibn Musa Al-Khwarizmi» Verified 5 Oct 2026
treatment was very probably inspired by recent knowledge of the "Elements".
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- MacTutor History of Mathematics, «Abu Ja'far Muhammad ibn Musa Al-Khwarizmi» Verified 5 Oct 2026
What grew out of it
-
Decimal fractions in computation
influenced · Confidence: medium
In the Arabic tradition decimal fractions were developed in al-Karaji's school to approximate roots in algebraic problems (al-Samaw'al, 1172), and Stevin, who issued De Thiende together with L'Arithmetique, extended uniform rules of operation to all numbers, roots included.
Dispute: The European line of decimal fractions did not run through algebra: according to the DSB (Minnaert), before Stevin they "had previously found only occasional use in trigonometric tables", and Van Brummelen (2024) found decimal fractional numeration and a decimal point in Bianchini's tables of the 1440s. No direct dependence of Stevin on Arabic authors has been shown; the Muslim Heritage claim that Stevin "learnt of the Arab methods" is unsupported by a source. Stevin's systematic work on decimal expansions of polynomial roots came after De Thiende (1594, Fowler), so in Stevin's own case the fractions precede the algebraic application.
Link sources
- O'Connor J. J., Robertson E. F. «Al-Kashi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π
- Katz K. U., Katz M. G. «Stevin numbers and reality», Foundations of Science 17(2), 109-123 (2012) Verified 20 Sep 2026
The belief that all arithmetic operations, as well as extracting roots, etc., should follow the 'same' rules as the rationals, originates precisely with Stevin
- FSTC, «Decimal Arithmetic», Muslim Heritage (2007) Verified 20 Sep 2026
Al-Samawal (1172) considered decimal fractions in problems of division and extraction of roots.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- O'Connor J. J., Robertson E. F. «Al-Kashi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
-
Analytic geometry
prerequisite for · Confidence: high
Analytic geometry is the application of algebra to curves: Descartes and Fermat started from Viete's symbolic algebra and theory of equations, itself descended from al-Khwarizmi's algebra; without an algebraic language a curve cannot be written as an equation.
Dispute: Historians do not disagree here, but the immediate source for Descartes and Fermat is Viete's symbolic algebra (In artem analyticem isagoge, 1591), not al-Khwarizmi's rhetorical algebra of 820.
Link sources
- Knorr W. R. et al. «Mathematics: Analytic geometry», Encyclopaedia Britannica (2024) Verified 20 Sep 2026
After Viète's death the analytic art was applied to the study of curves by his countrymen Fermat and Descartes.
- Mahoney M. S. «Fermat, Pierre de», Dictionary of Scientific Biography (via Encyclopedia.com) (1971) Verified 20 Sep 2026
From Viète he took the new symbolic algebra and theory of equations that served as his basic research tools
- Domski M. «Descartes' Mathematics», Stanford Encyclopedia of Philosophy, §3.1 (2021) Verified 20 Sep 2026
his program for geometry begins where Viète's left off
- O'Connor J. J., Robertson E. F. «René Descartes», MacTutor History of Mathematics (2014) Verified 20 Sep 2026
La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Knorr W. R. et al. «Mathematics: Analytic geometry», Encyclopaedia Britannica (2024) Verified 20 Sep 2026
-
Liber abaci: Indian numerals and Arabic algebra in a Latin book
influenced · Confidence: high
The algebraic part of the Liber abaci is built on al-Khwarizmi's problems: the numerical examples are taken largely from his algebra and from al-Karaji.
Link sources
- Vogel K. «Fibonacci, Leonardo, or Leonardo of Pisa», Dictionary of Scientific Biography, vol. 4, New York: Scribner, 1971, encyclopedia.com (1971) Verified 4 Oct 2026
The numerical examples are taken largely from the algebra of al-Khw
- Karpinski L. C. «Robert of Chester's Latin Translation of the Algebra of al-Khowarizmi», New York: Macmillan, 1915, p. 34 (1915), 34 Verified 4 Oct 2026
Woepcke has shown that Leonard drew many of his problems from Al-Khowarizmi
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Vogel K. «Fibonacci, Leonardo, or Leonardo of Pisa», Dictionary of Scientific Biography, vol. 4, New York: Scribner, 1971, encyclopedia.com (1971) Verified 4 Oct 2026
Node sources
- O'Connor J. J., Robertson E. F. «Al-Khwarizmi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
The word algorithm derives from his name. His algebra treatise Hisab al-jabr w'al-muqabala gives us the word algebra and can be considered as the first book to be written on algebra.
- O'Connor J. J., Robertson E. F. «Al-Khwarizmi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
Al-Khwarizmi's mathematics is done entirely in words with no symbols being used. He first reduces an equation (linear or quadratic) to one of six standard forms: 1. Squares equal to roots. 2. Squares equal to numbers. 3. Roots equal to numbers. 4. Squares and roots equal to numbers; e.g. x2+10x=39. 5. Squares and numbers equal to roots; e.g. x2+21=10x. 6. Roots and numbers equal to squares
- O'Connor J. J., Robertson E. F. «Al-Khwarizmi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026, quotation assembled from fragments
The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation... The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation.
- O'Connor J. J., Robertson E. F. «Al-Khwarizmi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026, quotation assembled from fragments
Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials... From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions.
- Britannica, «Algebra», «Islamic contributions» (2024) Verified 24 Sep 2026
Islamic contributions to mathematics began around AD 825, when the Baghdad mathematician Muhammad ibn Musa al-Khwarizmi wrote his famous treatise al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala (translated into Latin in the 12th century as Algebra et Almucabal, from which the modern term algebra is derived).
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.
Further reading
Not evidence: these carry no quotation and support nothing we wrote. They are books and scans to read next.
- Boyer C. B. «A History of Mathematics», 2nd ed., «The Arabic Hegemony», 228-229 (1991)
- Rashed R. «Al-Khwarizmi: The Beginnings of Algebra», Saqi Books (2009)