Page 309 - Theory and Problems of BEGINNING CHEMISTRY

298 SCIENTIFIC CALCULATIONS [APP.

Precedence Rules
In algebra, when more than one operation is indicated in a calculation, the operations are done in a prescribed
order. The order in which they are performed is called the precedence,or priority, of the operations. The order
of the common algebraic operations is given in Table A-2. If operations having the same precedence are used,
they are performed as they appear from left to right (except for exponentiation and unary minus, which are done
right to left). For example, if a calculation involves a multiplication and an addition, the multiplication should
be done first, since it has a higher precedence. In each of the following calculations, the multiplication should
be done before the addition:
x = 2 + 4 × 6
x = 4 × 6 + 2

The answer in each case is 26. Try each of these calculations on your calculator to make sure that it does the
operations in the correct order automatically.

Table A-2 Order of Precedence of Common Operations

Calculator Algebra

Highest Parentheses Parentheses
Exponentiation (root) or unary minus* Exponentiation (root) or unary minus*
Multiplication or division Multiplication
Division
Lowest Addition or subtraction Addition or subtraction

*Unary minus makes a single value negative, such as in the number −2.

If parentheses are used in an equation, all calculations within the parentheses are to be done before the result
is used for the rest of the calculations. For example,

y = (2 + 4) × 6
means that the addition (within the parentheses) is to be done first, before the other operation (multiplication). The
parentheses override the normal precedence rules. We might say that parentheses have the highest precedence.
When you are using a calculator, some operation may be waiting for its turn to be done. For example,
when 2 + 4 × 6 is being entered, the addition will not be done when the multiplication key is pressed. It will
await the final equals key, when first the multiplication and then the addition will be carried out. If you want the
addition to be done first, you may press the equals key right after entering the four. If you want the calculator
to do operations in an order different from that determined by the precedence rules, you may insert parentheses
(if they are provided on your calculator) or press the equals = key to finalize all calculations so far before you
continue with others.

EXAMPLE A.10. What result will be shown on the calculator for each of the following sequences of keystrokes?

(a) 5 × 3 + 4 = (c) 5 × ( 3 + 4 ) =
(b) ( 5 × 3 ) + 4 = (d) ( 5 × 3 + 4 ) =

Ans. (a)19 (b)19 (c)35 (d)19

EXAMPLE A.11. What result will be shown on the calculator for each of the following sequences of keystrokes?

(a) 5 + 3 × 4 = (b) 5 + 3 =× 4 =

Ans. (a)17 (b)32