Al-Khwarizmi and the Loss of Division by Zero
DOI:
https://doi.org/10.36285/tm.131Abstract
We review two literal English translations, with pragmatic notes, of surviving accounts of Muhammad ibn Musa al-Khwarizmi's Arabic writings on the Hindu decimal number system and on algebra. Al-Khwarizmi, about 780-850 CE, showed no knowledge of division by zero as described in Sanskrit by Brahmegupta, 628 CE. Indeed, al-Khwarizmi's definition of division is inconsistent with division by zero. Al-Khwarizmi has lost Brahmegupta's: fractions as a syntactic object n/d, negative numbers, trichotomy, tetrachotomy, and division by zero, though he was close to obtaining general negative numbers from his sign convention for variables. We conclude that al-Khwarizmi's writings are an early cause of the mistaken European opinion that dividing by zero is impossible.
We hypothesise that Brahmegupta's division by zero did not diffuse from Sanskrit until Colebrooke's English translation of 1817 CE.
References
J.A.D.W. Anderson, “Review of Henry Thomas Colebrooke’s 1817 Translation of Sanskrit Works on Division by Zero”, Transmathematica, 2026.
J.A.D.W. Anderson, “Review of Fibonacci’s Liber Abaci by Sigler: The Loss of Division by Zero”, Transmathematica, 2026.
H.T. Colebrooke. “ALGEBRA WITH ARITHMETIC AND MENSURATION, FROM THE SANSCRIT OF BRAHMEGUPTA AND BHASCARA”, John Murray, Albermarle Street, London, 1817.
J.N. Crossley and A.S. Henry, “Thus Spake al-Khwsrizmi: A Translation of the Text of Cambridge University Library Ms. Ii.vi.5”, Historia Mathematica, 17 (2): 103–131, 1990.
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L.E. Sigler, “Fibonacci’s Liber Abaci: A translation into modern English of Leaonardo Pisano’s Book of Calculation”, Springer, 2002.
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Copyright (c) 2026 James Anderson, Jan Aldert Bergstra
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