I am not here to ask why "minus times minus is plus", this is a basic arithmetic fact. The related question most people ask is: why does $-\times-=+$. Of, course there may be several explanations for this fact. But I want to know who first wrote down that particular rule? Is there any historical evidence in some papers or books where it appeared for the first time?
- Thomas Heath, Diophantus of Alexandria : A Study in the History of Greek Algebra (1910), page 140 :
A minus multiplied by a minus makes a plus; a minus multipliedby a plus makes a minus ; and the sign of a minus is a truncated $\Psi$ turned upside down, thus $\Lambda$ [with a third central leg]. [Footnote. The literal rendering would be "A wanting multiplied by a wanting makes a forthcoming."]
See also :
- Thomas Heath, A History of Greek Mathematics. Volume II (1921), page 459 :
For subtraction alone is a sign used. The full term for wanting is λείψις as opposed to ύπαρξις, a forthcoming, which denotes a positive term. The symbol used to indicate wanting, corresponding to our sign for minus, is $\Lambda$ [with a third central leg]，which is described in the text as a ‘$\Psi$ turned downwards and truncated’.The description is evidently interpolated, and it is now certain that the sign has nothing to do with $\Psi$.Nor is it confined to Diophantus, for it appears in practically the same form in Heron's Metrica [...].
Clearly, also if the rule is "correctly described", there is no hint to its modern symbolic formulation for lack of the modern symbols, that dates to the late 16th- early 17th Centuries.
For the modern version, see :
Unde patet ratio tum hujus regulae, $+$ in $+$ facit $+$; tum hujus $-$ in $+$ facit $-$. [...] Indeque patet ratio tum hujus regulae, $+$ in $-$ facit $-$; tum hujus, $-$ in $-$ facit $+$.
The "justification" is that to multiply a given quantity by a positive factor is ponendi ("ubi $+2$ significat bis ponere") while to multiply it by a negeative factor is tollendi ("ibidem $-2$ est bis tollere, seu bis ponere contrarium").
Contra vero, $-A$ per $-2$ multiplicare, est bis tollere $-A$, seu defectum $-A$ bis supplere, quod est $+A$ bis ponere, facitque $+2A$, (adeoque $-$ in $-$ facit $+$.)
By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients.