Review of Fibonacci's Liber Abaci by Sigler

The Loss of Division by Zero

Abstract

We review "Fibonacci's Liber Abaci" by L. E. Sigler, 2002 CE. This is a literal English translation of a Latin text by Leonardo Pisano, better known as Fibonacci, dating back to 1202 CE. Fibonacci learned Hindu arithmetic and Hindu-Arabic algebra from Muslim North African Arabs and published a Latin treatise on it, incorporating his own mathematical developments. This text was highly influential in Europe. Fibonacci used a placeholder zero and had some knowledge of its algebraic structure, suggesting the re-emergence of an algebraic number zero. However, he seems to have had no direct contact with the works of Brahmegupta, 628 CE, who we credited with defining a total paraconsistent arithmetic of fractions that includes negative numbers and division by zero, nor of Bhascara, 1150 CE, who we credited with defining a total consistent version of Brahmegupta's fractional arithmetic. Fibonacci gave multiple definitions of division, but these are vague. He allows, but does not claim, a total paraconsistent division of non-negative fractions and a total inconsistent division of non-negative decimal integers. However, these inconsistencies would dissolve if Fibonacci did not regard zero as a number. The numberhood of zero is, therefore, critical to a proper understanding of Fibonacci's arithmetics.

Fibonacci gives a Euclidean proof of the correctness of the division of fractions that depends on their corresponding ratios, but he did not notice the corollary that division by zero exists because multiplication by zero exists. Thus, the Euclidean mathematics, of about 300 BCE, supported multiplication and division by zero as soon as zero became available in 628 CE! However, Fibonacci does not mention division by zero or negative numbers so, presumably, he did not learn Brahmegupta’s or Bhascara's fractional arithmetic, whence division by zero was lost until modern times.