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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.

(Additional reference added nearly 3 years later)

George Abram Miller (1863−1951), Vague historical views relating to the negative numbers, The Mathematics Student 2 #1 (March 1934), pp. 1−6.

(first paragraph of the paper, on p. 1) In the history of negative numbers it is desirable to distinguish sharply between the following three points: their correct use, their correct theory and their introduction as a permanent element of mathematics. It is well-known that the Hindus used these numbers correctly and that this is the earliest such use thereof that has thus far been definitely established. On the other hand, the eighteenth century European mathematicians did not yet generally possess a correct theory of these numbers so that J. Tropfke could truly say in the third edition of his noted Geschichte der Elementar-Mathematik, Volume 2 (1933), page 101, that the eighteenth century suffered because a generally satisfactory introduction of these numbers was then lacking. The symbols $+$ and $-$ were employed in calculations and were then used at a bound to represent positive and negative numbers.

(first two complete paragraphs on p. 2) It should be emphasized in this connection that the use of negative numbers is not fully justified from their analogy to debt. The product of two debts is not a credit. When debts are multiplied, they are multiplied by positive numbers and not by negative numbers so as to become a credit. The history of negative numbers becomes clear only by noting that the analogy of negative numbers to debt relates only to some of the properties of these numbers while it is foreign to other properties thereof. Even such an eminent mathematician as H. Cardan tried to prove in the latter half of the sixteenth century that it is incorrect to say that $(-a).(-b)=+ab,$ and in the noted algebra of C. Clavius (1608) the author ascribes to the weakness of the human mind the impossibility of understanding this rule, but he did not doubt it correctness in view of the fact that it had been verified by many examples. The theory of negative numbers with respect to the operations of addition and subtraction is much more simple than their theory with respect to the operations of multiplication and division. In the former theory, the analogy of these numbers to debt when the corresponding positive numbers represent credit and to distances in the opposite direction from those represented by the corresponding positive numbers are illuminating. In trigonometry and in analytic geometry, negative numbers are especially useful with respect to the operations of addition and subtraction. Hence it is perhaps natural that some mathematical historians have unduly emphasized this analogy in presenting the extension of the number concept so as to include negative numbers. This extension, however, implied also a correct theory of these numbers with respect to the operations of multiplication and division, and the latter theory does not seem to have been satisfactorily presented before about the beginning of the nineteenth century and it seems to be entirely due to European mathematicians who, therefore, should have the credit of having completed the introduction of the negative numbers.

(last two paragraphs of the paper, on pp. 5−6) The three points to which we referred at the opening of this article can now be briefly answered as follows. The Hindus used negative numbers correctly at least as early as the times of Brahmagupta, who was born in 598; and they also possessed a correct theory of these numbers as regards addition and subtraction, but there is no definite evidence that a correct theory as regards the multiplication and division of these numbers was developed before about the beginning of the nineteenth century. The latter theory is a special case of the corresponding theory relating to complex numbers and it was developed simultaneously with this more general theory. The introduction of negative numbers as a permanent element of mathematics was inaugurated by the use of these numbers by the Hindus and was later inspired by the many simplifications which were effected by this use in various fields of mathematical developments. This introduction met with considerable opposition from time to time until a satisfactory theory of the negative numbers was developed around the beginning of the nineteenth century and thus completed their introduction more than a thousand years after this introduction was inaugurated. The introduction of the negative numbers belongs to the extensive field of mathematical developments which are due to a failure to state all the restrictions involved in certain early proofs. The history of these developments has never been written and involves many difficulties; but the general history of mathematics is much enriched by noting that many of the early proofs apply to wider fields than the authors of these proofs had in mind. When Diophantus first stated that the product of a subtracted number by a subtracted number is an added number, he naturally had no idea of the great generality of this rule, just as J. L. Lagrange had no idea of the generality of the so-called Lagrange theorem in group theory. Many mathematical reputations are largely due to the later growth of theorems and theories rather than to their observed generality at the time when they were first announced by those to whom they are now credited. Negative numbers entered mathematics as useful servants and later were accepted as staunch citizens after their real merits became known.

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