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Algebra in the Islamic World

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Episodes in the Mathematics of Medieval Islam

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Abstract

Many ancient mathematical works contain problems requiring the discovery of an unknown quantity.

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Notes

  1. 1.

    This is found in Chapter 4 of Section II of his al-Bāhir.

  2. 2.

    Our account relies on that given in Bajri et al.

  3. 3.

    He thinks of the product of two numbers, a and b, as the area of a rectangular surface with sides a and b.

  4. 4.

    Since Euclid has two different theories of proportion in his Elements, one for magnitudes in general in Book VI and another for numbers in particular it may be that al-Samaw’al had Elements VI, 1 in mind here.

  5. 5.

    The terminology ‘surface’ for ‘product’ is found in Euclid’s Elements, even though ‘the surface of a cube and another cube’ or ‘the surface of a surface’ makes no geometric sense.

  6. 6.

    He expresses this as saying that the number ab is divided at the point g. The practice of representing numbers as line segments is very much in the tradition of Euclid’s Elements. VII – IX. So a number written as a sum may be represented geometrically as a line segment, ab, divided at a point, g.

  7. 7.

    Although al-Samaw’al does not prove (pq)r = (pr)q it follows from Lemma 1 if we recall that r = r·1.

  8. 8.

    He refers to the fifth power with the algebraic terminology māl cube and the fourth as mâl mâl, although in the case of a simple square he uses the geometrical terminology for a square, murabba’.

  9. 9.

    Kitāb al-uṣūl wa’al-muqaddimāt fi’l-jabr wa’al-muqābbala. See Djebbar, A. 1990.

  10. 10.

    Chapter 4 of Sect. 2 of Part II.

  11. 11.

    According to M. Souissi, it was Ibn al-Bannā’ who introduced the word ‘uss, whose non-technical meaning is ‘foundation/basis,’ for ‘exponent.’’

  12. 12.

    For our information on this work we have relied on the edition of the Arabic text and French translation by Djebbar, 1990.

  13. 13.

    This same equation and the algebraic method of solving it is found in al-Karajī’s book, Al-Fakhrī, in his discussion of solving the three canonical equations of the second degree. (He refers to his algebraic method as being that of Diophantos.)

  14. 14.

    This method of abbreviating words used frequently in mathematics has been called syncopation, and can also be found in Byzantine manuscripts of Diophantos’s Arithmetica.

Bibliography

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada

    J. L. Berggren

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  1. J. L. Berggren
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Correspondence to J. L. Berggren .

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Berggren, J.L. (2016). Algebra in the Islamic World. In: Episodes in the Mathematics of Medieval Islam. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3780-6_4

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