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Rafael Bombelli was the first European mathematician to write about the laws of arithmetic for negative numbers.

On Wikipedia I read that he wrote: “Minus 5 times minus 6 makes plus 30”.

I also read that Bombelli was interested in cubic equations and developed laws of arithmetic with imaginary numbers.

My question is that the laws of arithmetic he produced for imaginary numbers have commentaries and are generated by applying existing mathematical properties such as distribution over multiplication to equate whole number answers to cubic equations, which do not uphold to Cardano's general formula.

I was wondering if his laws for arithmetic with negatives arose in the same fashion.

If there is anywhere I can read about the subsequent progression of arithmetic with negative and imaginary numbers and how their acceptances were somewhat intertwined and “artificial” it would be appreciated.

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  • $\begingroup$ No, but his source, Cardano, did try, see Wagner, The natures of numbers in and around Bombelli’s L’algebra:"...Cardano’s attempt to explain why the product of negatives is negative in his De Regula Aliza. Bombelli himself, in his manuscript, claims that “meno times meno is più when it is accompanied by the più, but by itself alone is meno”, but the promised demonstration is lacking, a marginal note states that “this is not the case”, and the statement is omitted from the print edition." $\endgroup$
    – Conifold
    Feb 14 at 7:34
  • $\begingroup$ Bombelli was not "the first European mathematician who wrote about arithmetic of negative numbers". Much earlier was Diophantus who defined multiplication of negative numbers in his Arithmetic, Chap. I Def (IX). Technically it is unknown whether Diophantus worked in Africa (Alexandria) or Asia, or Europe, but he wrote in Greek, and certainly belongs to "European tradition". $\endgroup$ Feb 14 at 12:49
  • $\begingroup$ @AlexandreEremenko There are no chapters or definitions in Diophantus's Arithmetica, and there are no negative numbers there even where one would expect them. There is a sign rule in the preamble to Book I:"A deficit multiplied by a deficit makes an existence; a deficit by an existence makes a deficit; and the sign of the deficit is a truncated Ψ turned upside down, i.e., $\pitchfork$". But the "deficit" is his subtraction sign and the rule refers to expressions with it, not to negative numbers. $\endgroup$
    – Conifold
    Feb 15 at 6:32

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Leo Corry's recent text, A Brief History of Numbers, offers an authoritative account of these matters.

According to Corry, Bombelli's attitude to negative numbers was the same as Cardano's. Bombelli would not have written "Minus 5 times minus 6 makes plus 30”, as wikipedia suggests. Rather, Bombelli would have written "Subtract 5 times subtract 6 makes 30." Furthermore, Corry states that the law of signs for multiplication was established by the abacists and rejected by Cardano. What Corry gives Bombelli credit for is extending the law of signs to imaginary numbers.

What Bombelli did in his book [Algebra] was to extend to the case of square roots of negative numbers (with the help of some new terminology) the abacist rules for the arithmetic of binomials. ... Using this nomenclature, he wrote down correctly all the rules of sign multiplication with numbers of this kind.

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  • $\begingroup$ Does “a brief history of numbers” explore the acceptance of negative numbers and imaginary numbers, and the arithmetic with these entities, and how their acceptance and associated algebraic operations were progressed? $\endgroup$
    – Fraser
    Feb 12 at 20:52
  • $\begingroup$ @Fraser Yes, that is the subject of the text. It traces the gradual legitimation of broader classes of numbers from the Pythagorean and Euclidean concept of number as positive integer to the hypercomplex number systems and formalizations of the 19th and 20th century. Curiously, Corry makes no mention of p-adic numbers. $\endgroup$
    – nwr
    Feb 13 at 1:32
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    $\begingroup$ Re "extending the law of signs to imaginary numbers". This occurs in book I of L'algebra (1572). I have been unable to find a scan of the relevant page, and page numbers cited in the literature differ. He uses più di meno to refer to $+\sqrt{-1}$ and meno di meno to refer to $-\sqrt{-1}$ and proceeds to give eight rules from "più via più di meno, fa più di meno" (that is, $+(+i) = +i$) to "meno di meno via meno di meno, fa meno" (that is, $(-i)(-i)= -1$), where via means "times" and fa means "makes", and più is literally "plus" and meno literally "minus". $\endgroup$
    – njuffa
    Feb 13 at 9:34
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    $\begingroup$ A (rather poor) scan of L'algebra is available for download as a PDF document here. The eight rules appear on p. 169 of book 1. $\endgroup$
    – njuffa
    Feb 13 at 9:48
  • $\begingroup$ The author mentions that Cordano noticed negative appear quadratics of the form x^2 = px + c, and lists two formulas, but I can’t see how those are proved geometrically at the time and why there is two instead of one. $\endgroup$
    – Fraser
    Feb 14 at 5:04

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