Parentheses

If a compound expression is to be treated as a whole, it is enclosed in a parenthesis.

Thus, $2\times (10+5)$ means that we are to add $5$ to $10$ and multiply the result by $2$; if we were to omit the parenthesis and write $2\times 10+5$, the meaning would be that we were to multiply $10$ by $2$ and add $5$ to the result.

Like the parenthesis, we use with the same meaning any other sign of aggregation.

Thus, $(5+2)$, $[5+2]$, $\{5+2\}$, $\overline{5+2\phantom{(}}$, $$\begin{gathered} r5 \\ +2| \end{gathered}$$, all mean that the expression $5+2$ is to be treated as the single symbol $7$.

37. Parentheses preceded by $+$.

If a man has $10$ dollars and afterwards collects $3$ dollars and then $2$ dollars, it makes no difference whether he adds the $3$ dollars to his $10$ dollars, and then the $2$ dollars, or puts the $3$ and $2$ dollars together and adds their sum to his $10$ dollars.

The first process is represented by $10+3+2$.

The second process is represented by $10+(3+2)$.

If a man has $10$ dollars and afterwards collects $3$ dollars and then pays a bill of $2$ dollars, it makes no difference whether he adds the $3$ dollars collected to his $10$ dollars and pays out of this sum his bill of $2$ dollars, or pays the $2$ dollars from the $3$ dollars collected and adds the remainder to his $10$ dollars.

The first process is represented by $10+3-2$.

The second process is represented by $10+(3-2)$.

From (1) and (2) it follows that

If an expression within a parenthesis is preceded by the sign $+$, the parenthesis can be removed without making any change in the signs of the expression.

Any part of an expression can lie enclosed within a parenthesis and the sign $+$ prefixed, without making any change in the signs of the terms thus enclosed.

38. Parentheses preceded by $-$.

If a man has $10$ dollars and has to pay two bills, one of $3$ dollars and one of $2$ dollars, it makes no difference whether he takes $3$ dollars and $2$ dollars in succession, or takes the $3$ and $2$ dollars at one time, from his $10$ dollars.

The first process is represented by $10-3-2$.