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If we divide a positive number by another positive number, the result becomes greater as the divisor becomes smaller. If we continue this logic, division by a negative number should be greater than any division by a positive number.

So, I wonder whether the idea that the result of such division should be negative was ever controversial, or other ideas were ever proposed as alternatives (for instance, counting such result as infinitely large value or otherwise non-real number)?

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  • $\begingroup$ The main issue is about consistency of the manipulation rules. The basic property of division is that if $c = \dfrac a b$, then $a= b \times c$ $\endgroup$ Jan 29, 2021 at 7:43
  • $\begingroup$ Thus, the basic question is "why minus times minus is plus?" and "who wrote down minus times minus is equal to plus?" $\endgroup$ Jan 29, 2021 at 7:44
  • $\begingroup$ Not to mention that $\frac{a}{b}$ is the same as $a*\frac{1}{b}$ . $\endgroup$ Jan 29, 2021 at 12:21
  • $\begingroup$ @MauroALLEGRANZA A natural way to define division by negative numbers is by Laplace transform: $\frac1a=\int_0^\infty e^{-at}dt$ $\endgroup$
    – Anixx
    Mar 16, 2021 at 9:51

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Much has been written about various roadblocks to the acceptance of negative numbers, and I have a folder containing photocopies of a few such papers, but I don't have time now to look for that folder. However, I found the following citation and excerpt in a manuscript of mine. The paper is behind a paywall (which I don't have access to, but I have a photocopy from the original library journal volume in that folder), so I'm including the excerpt I have.

George Abram Miller (1863−1951), A crusade against the use of negative numbers, School Science and Mathematics 33 #9 (December 1933), 959−964.

(from pp. 959−960) It appears that the greatest opposition to the use of negative numbers in algebra developed just before the dawn of clear light on this subject near the beginning of the nineteenth century. Among the well known opponents to this use was Robert Simson who was professor of mathematics in the University of Glasgow for fifty years beginning with 1711 and edited a well known edition of Euclidís Elements. One of the serious obstacles in the way towards accepting the negative numbers as real numbers is that this acceptance implies that we cannot thereafter continue to support the view that the ratio of a smaller number to a larger number must differ from the ratio of a larger number to a smaller since it is obvious, for instance that $\frac{2}{3} = -2/-3$ and that $2$ is smaller than $3$ while $-2$ is larger than $-3.$ The example of greatest historical interest in this connection is $1/-1 = -1/1.$ In dealing only with positive numbers it is obvious that the ratio of a smaller number to a larger one is always less than unity while the ratio of a larger number to a smaller number is always greater than unity, and hence two such ratios cannot be equal to each other in this restricted number field. This difficulty is the more serious in view of the fact that the concepts of ratio and proportion are among the earliest mathematical concepts and that in Greek mathematics the latter largely replaced our modern concept of equation. In all these early uses of the concept of ratio, including the golden period of Greek mathematical developments, the ratio of a smaller number to a larger number was always assumed to be less than the ratio of a larger number to a smaller number. The former corresponds to a proper fraction while the latter corresponds to an improper fraction, but the division of fractions into these two classes was inaugurated after the middle ages although the Hindus calculated already with fractions whose numerators are larger than their denominators. I. Newton and others defined a number as the ratio between two line segments. These examples of the early use of the concept of ratio are cited here in order to exhibit more clearly an explanation of the reluctance with which mathematicians of the seventeenth and of the eighteenth centuries abandoned the idea that the ratio of a larger number to a smaller number is not always greater than the ratio of a smaller number to a larger one. In 1768 W. J. G. Karsten, who was then professor of logic and later professor of mathematics in a German university, published an article under the title “Von den Logarithmen verneinten Grösen” in which he argued that a negative number cannot be less than zero because if this were true it would follow from the correct proportion $1:-1=-1:1$ that the larger of two numbers may have the same ratio to the smaller as the smaller to the larger, which seemed to him to be a contradiction.

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