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As a follow up question with respect to : Who wrote down minus times minus is equal to plus? and to : Historically, how did people define multiplication for negative numbers?, it can be interesting to trace the first "modern" justification for the rule :

"minus multiplied by a minus makes a plus".

If it is quite intuitive that "if you have a debt of $3$ by $3$ people, then you have a debt of $9$" (we can justify it reducing multiplication to repeated sum) it is not so easy to imagine a debt of $3$ by "minus" $3$ people.


For a (negative) reference, see :

See page 3 :

Quantities are either Affirmative, or greater than nothing; or Negative, or less than nothing. Thus in humane Affairs, Possessions or Stock may be calld affirmative Goods, and Debts negative ones. [...] A negative Quantity is denoted by the Sign $-$ ; the Sign $+$ is prefix'd to an affirmative one [...].

In an Aggregate of Quantities the Note $+$ signifies, that the Quantity it is prefix'd to, is to be added, and the Note $-$, that it is to be subtracted. And we usually express these Notes by the Words Plus (or more) and Minus (or less). Thus $2+3$, or $2$ more $3$, denotes the Sum of the Numbers $2$ and $3$, that is $5$. And $5-3$ or $5$ less $3$, denotes the Difference which arises by subducting $3$ from $5$, that is $2$. [Note the clear distinction of the two usages of the signs $+$ and $-$.]

Then see page 16 :

Simple Algebraick Terms are multiply'd by multiplying the Numbers into the Numbers, and the Species into the Species, and by making the Product Affirmative, if both the Factors are Affirmative, or both Negative : and Negative if otherwise. Thus $2a$ into $3b$, or $- 2a$ into $- 3b$ make $6ab$, or $6ba$; for it is no Matter in what Order they are plac'd. Thus also $2a$ by $- 3b$, or $- 2a$ by $3b$ make $- 6ab$.


Useful "post-Newtonian" references :

if the Signs of Multiplicand and Multiplicator are familiar (or the same) the Product will be Affirmative, but Negative if they are different.

The explanation of the minus times minus case is in terms of "symmetry", followed by an intuitive example :

There is taken away in this Case a negative Quantity, by which the Negation vanishes. So to take away a Debt is to pay it [emphasis added].


It is useful to compare with the fully "algebraic" explanation by one of the most brilliant Newton's followers:

By the definitions, $+a-a=0$; therefore, if we multiply $+a-a$ by $n$, the product must vanish or be $0$ because the factor $a-a$ is $0$. [...] Therefore $-a$ multiplied by $+n$ gives $-na$.

In like manner, if we multiply $+a-a$ by $-n$, the first term of the product being $-na$, the latter term of the product must be $+na$, because the two together must destroy each other, or they amount be $0$, since one of the factors (viz.$a-a$) is $0$. Therefore $-a$ multiplied by $-n$ must give $+na$.

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    $\begingroup$ "it is not so easy to imagine a debt of 3 by "minus" 3 people." What about a debt of 3, from 3 people less? (You have 3 people fewer to pay the debt of 3) That makes a savings of 9, doesn't it? $\endgroup$ – Aritra Das Nov 12 '15 at 18:07
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    $\begingroup$ @AritraDas - good point ! And this seems to be Wallis' approach below: if I have a debt of $-3$ from $3$ people, then I have a total debt of $-3 \times 3 = -9$; but if I remove the debt of $1$ person, the total left is $-9-1\times -3 = -6$. $\endgroup$ – Mauro ALLEGRANZA Nov 12 '15 at 20:14
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Possible source :

If a positive term of one quantity is multiplied by a positive term of another quantity, the product will be positive and if by a negative the result will be negative. The consequence of this rule is that multiplying a negative by a negative produces a positive, as when $A - B$ is multiplied by $D - G$. The product of $+ A$ and $- G$ is negative, but this takes away or subtracts too much since $A$ is not the exact magnitude to be multiplied. Similarly the product of $- B$ and $+ D$ is negative, which takes away too much since $D$ is not the exact magnitude to be multiplied. The positive product when $- B$ is multiplied by $- G$ makes up for this.

The geometrical interpretation can be traced back to ancient Greek geometrical algebra; see Euclid's Elements : II.7 and II.4.


For a "pure" algebraic exposition, 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 $+$.)

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  • $\begingroup$ Viete seems to justify only auxiliary use of negatives. Mumford credits Wallis's treatise as "the first place in Western literature in which the rule of signs is not merely stated but explained so clearly" for negatives as such. In his translation the last passage is:"But to multiply –A by −2 is twice to take away a defect or negative. Now to take away a defect is the same as to supply it; and twice to take away the defect of A is the same as twice to add A or to put 2A" dam.brown.edu/people/mumford/beyond/papers/… (p.137). $\endgroup$ – Conifold Nov 11 '15 at 19:26
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"Historical roots of the justification for the rule for multiplication of negative numbers"

This was an on going evolution process, it's impossible to pinpoint someone and give him all the credit. For hundreds of years (maybe thousands) merchants and accountants relied a lot on such rules for their bookkeeping, even without a formal description from the mathematicians about what negative numbers was.

Mentioned this, probably the first one to state such rules was the Indian mathematician Brahmagupta in the 7th century:

The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.

At that time there was no concept of number line (John Wallis at 17th century) or vectors (20th century). His proof should have been based only on simpler concepts. The most simple proof is that all the properties of the elementary arithmetic operations must be the same whether the number is positive or negative. In other words, the distributive property must be valid also for negative numbers:

(-1) . 0 = 0
(-1) . (1 + (-1)) = 0
(-1).1 + (-1).(-1) = 0
(-1) + (-1).(-1) = 0
(-1).(-1) = 1

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