- #1

G037H3

- 280

- 2

could someone please explain this, I may be overly tired, but it's confusing me:

"A further illustration of this rule is generally given by algebraists as follows: First, we know that +a multiplied by +b gives the product +ab; and if +a be multiplied by a quantity less than b, as in b-c, the product must necessarily be less than ab; in short, from ab we must subtract the product of a, multiplied by c; hence a times (b-c) must be expressed by ab-ac; therefore it follows that a times -c gives the product -ac. If now we consider the product arising from the multiplication of the two quantities (a-b), and (c-d), we know that it is less than that of (a-b) times c, or of ac-bc; in short, from this product we must subtract that of (a-b) times d; but the product (a-b) times (c-d) becomes ac-bc-ad, together with the product of -b times -d annexed; and the question is only what sign we must employ for this purpose, whether + or -. Now we have seen from that the product ac-bc we must subtract a quantity less than ad, we have therefore subtracted already too much by the quantity bd; this product must therefore be added; that is, it must have the + sign prefixed, hence we see that -b times -d gives +bd for a product; or -minus multiplied by -minus gives +plus."