Elements of Algebra

CHAP. III. OF ALGEBRA. 9 34. The rules which we have explained are expressed more briefly as follows: Like signs, multiplied together, give $+$; unlike or contrary signs give $-$. Thus, when it is required to multiply the following numbers; $+a, -b, -c, +d$; we have first $+a$ multiplied by $-b$, which makes $-ab$; this by $-c$, gives $+abc$; and this by $+d$, gives $+abcd$. 35. The difficulties with respect to the signs being removed, we have only to shew how to multiply numbers that are themselves products. If we were, for instance, to multiply the number $ab$ by the number $cd$, the product would be $abcd$, and it is obtained by multiplying first $ab$ by $c$, and then the result of that multiplication by $d$. Or, if we had to multiply $36$ by $12$; since $12$ is equal to $3$ times $4$, we the nature of Addition; for $9 \times \frac{1}{2} = 4\frac{1}{2}$, $9 \times \frac{1}{9} = 1$, $9 \times \frac{1}{100} = \frac{9}{100}$, &c. The same will be found true with respect to algebraic quantities; $a \times b = ab$, $-9 \times 3 = -27$, that is, $9$ negative integers multiplied by $3$, or taken $3$ times, are equal to $-27$, because the measure of multiplication is $3$. In the same manner, by inverting the factors, $3$ positive integers multiplied by $-9$, or taken $9$ times negatively, must give the same result. Therefore a positive quantity taken negatively, or a negative quantity taken positively, gives a negative product. From these considerations, we may illustrate the present subject in a different way, and shew, that the product of two negative quantities must be positive. First, algebraic quantities may be considered as a series of numbers increasing in any ratio, on each side of nothing, to infinity. [See Art. 19.] Let us assume a small part only of such a series for the present purpose, in which the ratio is unity, and let us multiply every term of it by $-2$. $5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5,$ $-2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,$ -------------------------------------------------- $-10, -8, -6, -4, -2, 0, +2, +4, +6, +8, +10.$ Here, of course, we find the series inverted, and the ratio doubled. Farther, in order to illustrate the subject, we may consider the ratio of a series of fractions between $1$ and $0$, as indefinitely small, till the last term being multiplied by $-2$, the product would be equal to $0$. If, after this, the multiplier having passed the middle term, $0$, be multiplied into any negative term, however small, between $0$ and $-1$, on the other side of the series, the product, it is evident, must be positive, otherwise the series could not go on. Hence it appears, that the taking of a negative quantity negatively destroys the very property of negation, and is the conversion of negative into positive numbers. So that if $+\times-=-$, it necessarily follows that $-\times-$ must give a contrary product, that is, $+$. See Art. 176, 177.