Elements of Algebra

CHAP. III. OF ALGEBRA. 7 written $ab$; and $pq$ will be the product of the multiplication of the number $q$ by $p$. Also, if we multiply this $pq$ again by $a$, we shall obtain $apq$. 27. It may be farther remarked here, that the order in which the letters are joined together is indifferent; thus $ab$ is the same thing as $ba$; for $b$ multiplied by $a$ is the same as $a$ multiplied by $b$. To understand this, we have only to substitute, for $a$ and $b$, known numbers, as $3$ and $4$; and the truth will be self-evident; for $3$ times $4$ is the same as $4$ times $3$. 28. It will not be difficult to perceive, that when we substitute numbers for letters joined together, in the manner we have described, they cannot be written in the same way by putting them one after the other. For, if we were to write $34$ for $3$ times $4$, we should have $34$, and not $12$. When therefore it is required to multiply common numbers, we must separate them by the sign $\times$, or by a point: thus, $3 \times 4$, or $3.4$, signifies $3$ times $4$; that is, $12$. So, $1 \times 2$ is equal to $2$; and $1 \times 2 \times 3$ makes $6$. In like manner, $1 \times 2 \times 3 \times 4 \times 56$ makes $1344$; and $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$ is equal to $3628800$, &c. 29. In the same manner, we may discover the value of an expression of this form, $5.7.8.abcd$. It shews that $5$ must be multiplied by $7$, and that this product is to be again multiplied by $8$; that we are then to multiply this product of the three numbers by $a$, next by $b$, then by $c$, and lastly by $d$. It may be observed, also, that instead of $5.7.8$, we may write its value, $280$; for we obtain this number when we multiply $35$, (the product of $5$ by $7$) by $8$. 30. The results which arise from the multiplication of two or more numbers are called *products*; and the numbers, or individual letters, are called *factors*. 31. Hitherto we have considered only positive numbers; and there can be no doubt, but that the products which we have seen arise are positive also: viz. $+a$ by $+b$ must necessarily give $+ab$. But we must separately examine what the multiplication of $+a$ by $-b$, and of $-a$ by $-b$, will produce. 32. Let us begin by multiplying $-a$ by $3$ or $+3$. Now, since $-a$ may be considered as a debt, it is evident that if we take that debt three times, it must thus become three times greater, and consequently the required product is $-3a$. So if we multiply $-a$ by $+b$, we shall obtain $-ba$, or, which is the same thing, $-ab$. Hence we conclude, that if a positive quantity be multiplied by a negative quantity, the product will be negative; and it may be laid down