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Polynomials and equations in arabic algebra

Abstract

It is shown in this article that the two sides of an equation in the medieval Arabic algebra are aggregations of the algebraic “numbers” (powers) with no operations present. Unlike an expression such as our 3x + 4, the Arabic polynomial “three things and four dirhams” is merely a collection of seven objects of two different types. Ideally, the two sides of an equation were polynomials so the Arabic algebraists preferred to work out all operations of the enunciation to a problem before stating an equation. Some difficult problems which involve square roots and divisions cannot be handled nicely by this basic method, so we do find square roots of polynomials and expressions of the form “A divided by B” in some equations. But rather than initiate a reconsideration of the notion of equation, these developments were used only for particularly complex problems. Also, the algebraic notation practiced in the Maghreb in the later middle ages was developed with the “aggregations” interpretation in mind, so it had no noticeable impact on the concept of polynomial. Arabic algebraists continued to solve problems by working operations before setting up an equation to the end of the medieval period.

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Correspondence to Jeffrey A. Oaks.

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I thank Mahdi Abdeljaouad, who provided comments on an earlier version of this paper, and Haitham Alkhateeb, for his help with some of the translations.

Notes on references: When page numbers are separated by a “ / ”, the first number is to the Arabic text, and the second to the translation. Also, a semicolon separates page number from line number. Example: [Al-Khwārizmī, 1831, 31;6/43] refers to page 31 line 6 of the Arabic text, and page 43 of the translation.

Communicated by L. Berggren.

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Oaks, J.A. Polynomials and equations in arabic algebra. Arch. Hist. Exact Sci. 63, 169–203 (2009). https://doi.org/10.1007/s00407-008-0037-7

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Keywords

  • Unknown Quantity
  • Algebraic Solution
  • Arabic Text
  • Arabic Word
  • Algebraic Notation