Historical roots of the justification for the rule for multiplication of negative numbers

Question

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.

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For a (negative) reference, see :

• Isaac Newton, Universal Arithmetick : or, A Treatise of Arithmetical Composition and Resolution. To which is Added, Dr.Halley's Method of Finding the Roots of Equations Arithmetically (latin manuscript : Arithmetica Universalis (edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Cambridge, in 1707, based on Newton's lecture notes).

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 $- 6ab$.

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Useful "post-Newtonian" references :

• William Jacob s'Gravesande, The Elements of Universal Mathematics or Algebra (1728 - Latin original : 1727), page 9 :

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].

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

• Colin MacLaurin, A Treatise of Algebra (posthumous : 1748 - 3rd ed 1771), page 13 :

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 $+na$.

Answer 1

Possible source :

• Francois Viète (1540 – 1603), , The Analytic Art (In artem analyticam isagoge 1591), RULE III : To multiply one magnitude by another, page 19 :

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.

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For a "pure" algebraic exposition, see :

• John Wallis (23 November 1616 – 28 October 1703), A Treatise of Algebra (1685), page 78-79 :

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 $+$.)

Answer 2

"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