Page:Elementary algebra (1896).djvu/32

$8+(13-5)$ means that to 8 we are to add the excess of 13 over 5; now if we add 13 to 8 we have added 5 too much, and must therefore take 5 from the result.

Thus

$8+(13-5)=8+13-5=10$.

Similarly $a+(b-c)$ means that to $a$ we are to add $b$, diminished by $c$.

Thus

$a+(b-c)=a+b-c$

(1).

In like manner,

$a+b-c+(d-e-f)=a+b-c-d-e-f{\mbox{.}}$

(2).

Conversely,

$a+b-c-d-e-f=a+b-c+(d-e-f){\mbox{.}}$

(3).

Again,

$a-b+c$	$=a+c-b,$	[Art. 24.]
	$=$ the sum of $a$ and $c-b$,
	$=$ the sum of $a$ and $-b+c$,	[Art. 24.]

therefore

$a-b+c=a+(-b+c)$

(4).

By considering the results (1), (2), (3), (4), we are led to the following rule:

Rule. When an expression within brackets is preceded by the sign $+$, the brackets can be removed without making any change in the expression.

Conversely: Any part of an expression may be enclosed within brackets and the sign $+$ prefixed, the sign of every term within the brackets remaining unaltered.

Thus the expression $a-b+c-d+e$ may be written in any of the following ways,

${\begin{aligned}&a+(-b+c-d+e)\\&a-b+(c-d+e),\\&a-b+c+(-d+e).\end{aligned}}$

26. The expression $a-(b+c)$ means that from $a$ we are to take the sum of $b$ and $c$. The result will be the same whether $b$ and $c$ are subtracted separately or in one sum.

Thus

$a-(b+c)=a-b-c$.