Elements of Algebra

6 ELEMENTS SECT. I. thus 0, or nothing, will always be the value of $+a - a$; but if we wish to know the value of $+a - b$, two cases are to be considered. The first is when $a$ is greater than $b$; $b$ must then be subtracted from $a$, and the remainder (before which is placed, or understood to be placed, the sign $+$) shews the value sought. The second case is that in which $a$ is less than $b$: here $a$ is to be subtracted from $b$, and the remainder being made negative, by placing before it the sign $-$, will be the value sought. *** CHAP. III. *Of the Multiplication of Simple Quantities.* 23. When there are two or more equal numbers to be added together, the expression of their sum may be abridged: for example, $a + a$ is the same with $2 \times a$, $a + a + a$ - - - - - - $3 \times a$, $a + a + a + a$ - - - $4 \times a$, and so on; where $\times$ is the sign of multiplication. In this manner we may form an idea of multiplication; and it is to be observed that, $2 \times a$ signifies 2 times, or twice $a$ $3 \times a$ - - - - 3 times, or thrice $a$ $4 \times a$ - - - - 4 times $a$, &c. 24. If therefore a number expressed by a letter is to be multiplied by any other number, we simply put that number before the letter, thus; $a$ multiplied by 20 is expressed by $20a$, and $b$ multiplied by 30 is expressed by $30b$, &c. It is evident, also, that $c$ taken once, or $1c$, is the same as $c$. 25. Farther, it is extremely easy to multiply such products again by other numbers; for example: 2 times, or twice $3a$, makes $6a$ 3 times, or thrice $4b$, makes $12b$ 5 times $7x$ makes $35x$, and these products may be still multiplied by other numbers at pleasure. 26. When the number by which we are to multiply is also represented by a letter, we place it immediately before the other letter; thus, in multiplying $b$ by $a$, the product is