CALCULATOR MATHEMATICS

A.2. CALCULATOR MATHEMATICS

There is such a variety of models of electronic calculators available that it is impossible to describe them all. This subsection describes many simple calculators. Read your owner’s manual to determine the exact steps to use to solve the various types of problems presented here.

A chemistry student needs to have and know how to operate a scientific calculator capable of handling exponential numbers. A huge variety of features are available on calculators, but any calculator with exponential capability should be sufficient for this and other introductory chemistry courses. These calculators generally have the other function keys necessary for these courses. Once you have obtained such a calculator, practice doing calculations with it so that you will not have to think about how to use the calculator while you should be thinking about how to solve the chemistry problems.

Some calculators have more than twice as many functions as function keys. Each key stands for two different operations, one typically labeled on the key and the other above the key. To use the second function (above the key), you press a special key first and then the function key. This special key is labeled INV (for inverse) on some calculators, 2ndF (for second function) on others, and something else on still others. On a few calculators, the MODE key gives several keys a third function. Most scientific calculators have a memory in which you can store one or more values and later recall the value.

Some of the examples in this section involve very simple calculations. The idea is to learn to use the calculator for operations that you can easily do in your head; if you make a mistake with the calculator, you will recognize it immediately. After you practice with simple calculations, you will have the confidence to do more complicated types.

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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)

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 = Ans.

(b)

(a) 17 (b) 32

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The rules of precedence followed by the calculator are exactly the same as those for algebra and arithmetic, except that in algebra and arithmetic, multiplication is done before division (multiplication has a higher precedence than division). Algebraically, ab/cd means that the product of a and b is divided by the product of c and d. On the calculator, since multiplication and division have equal precedence, if the keys are pressed in the order

a×b÷c×d=

the quotient ab/c is multiplied by d. To get the correct algebraic result for ab/cd, the division key should be pressed before the value of d is entered, or parentheses should be used around the denominator.

EXAMPLE A.12. What result will be obtained from pressing the following keys?

Ans. The result is 8; first, 36 is divided by 9 and then the result (4) is multiplied by 2. If you want 36 divided by both 9 and 2, use either of the following keystroke sequences:

or

EXAMPLE A.13. Solve:

Ans. (a) The answer is 3.1. The keystrokes are

(b) The answer is 0.775. The keystrokes used in part a do not carry out the calculations required for this part. The correct keystroke sequence is

0 ) = Either divide by each number in the denominator, or use parentheses around the two numbers so that their product

or

will be divided into the numerator.

Division

In algebra, division is represented in any of the following ways, all of which mean the same thing:

a/b

b a÷b

Note that any operation in the numerator or in the denominator of a built-up fraction (a fraction written on two lines), no matter what its precedence, is done before the division indicated by the fraction bar. For example, to simplify the expression

a+b c−d

the sum a + b is divided by the difference c − d. The addition and subtraction, despite being lower in precedence, are done before the division. In the other forms of representation, this expression is written as

( a + b)/(c − d)

or

( a + b) ÷ (c − d)

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where the parentheses are required to signify that the addition and subtraction are to be done first. A calculator has only one operation key for division, ÷ . When you use your calculator, be careful to indicate what is divided

by what when more than two variables are involved. EXAMPLE A.14. Write the sequence of keystrokes required to perform each of the following calculations, where a, b,and

c represent numbers to be entered: a+b

(a) (b) a + b/c + d

(c) a + b/(c + d)

c+d

Ans. (a) ( a+b) ÷ ( c+d ) = (c) a+b÷ ( c+d ) = (b) a+b÷c+d=

The Change-Sign Key

If we want to enter a negative number, the number is entered first, and then its sign is changed with the change-sign +/− key, not the subtraction − key. The change-sign key changes the value on the display from positive to negative or vice versa.

EXAMPLE A.15. Write the sequence of keystrokes required to calculate the product of 6 and −3. Ans.

The change-sign key converts 3 to −3 before the two numbers are multiplied. EXAMPLE A.16. What result will be displayed after the following sequence of keystrokes?

Ans. 7 (The first +/− changes the value to −7, and then the second +/− changes it back to +7.) EXAMPLE A.17. (a) What result will be displayed after the following sequence of keystrokes?

3 + /− x 2 = (b) What keystrokes are needed to calculate the value of −3 2 ?

Ans. (a) The +/− key changes 3 to −3 (minus 3). The squaring key x 2 squares the quantity in the display (−3), yielding +9.

(b) For the algebraic quantity −3 2 , the operations are done right to left; that is, the squaring is done first, and the final answer is −9. To make the operations on the calculator follow the algebraic rules, enter 3 press x 2 , and then press +/− .

Exponential Numbers

When displaying a number in exponential notation, most calculators show the coefficient followed by a space or a minus sign and then two digits giving the value of the exponent. For example, the following numbers

represent 2.27 × 10 3 and 2.27 × 10 − 3 , respectively:

Note that the base (10) is not shown explicitly on most calculators. If it is shown, interpreting the values of numbers in exponential notation is slightly easier.

To enter a number in exponential form, press the keys corresponding to the coefficient, then press either the EE key or the EXP key (you will have one or the other on your calculator), and finally press the keys

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corresponding to the exponent. For example, to enter 4 × 10 5 into the calculator, press

4 EE EE 5 or

4 EXP 5

Do not press the times × key or the

1 and 0 keys when entering an exponential number! The EE or EXP key stands for “times 10 to the power.” For simplicity, we will use EXP to mean either EXP or EE from this point on. If the coefficient is negative, press the +/− key before the EXP key. If the exponent is negative, press the + /− key after the EXP key.

EXAMPLE A.18. What keys should be pressed to enter the number −4.44 × 10 − 5 ? Ans. + 4 . 4 4 /− EXP 5 + /−

or

4 4 + /− EXP + /− 5

EXAMPLE A.19. What value will be displayed if you enter the following sequence of keystrokes?

4 4 × 1 0 EXP 5 =

Ans. These keystrokes perform the calculation

The resulting value will be 4.44 × 10 6 (which might be displayed in floating-point format as 4 440 000). These keystrokes instruct the calculator to multiply 4.44 by 10 × 10 5 , which yields a value 10 times larger than was

intended if you wanted to enter 4.44 × 10 5 .

EXAMPLE A.20.

A student presses the following sequence of keys: to get the value of the quotient

4 × 1 0 EXP 7 ÷ 4 × 1 0 EXP 7 =

What value is displayed on the calculator as a result? Ans.

The result is 1 × 10 16 . The calculator divides 4 × 10 8 by 4, then multiplies that answer by 10 times 10 7 . (See the precedence rules in Table A-1.) This answer is wrong because any number divided by itself should give an answer of 1. (You should always check to see if your answer is reasonable.)

EXAMPLE A.21. What keystrokes should the student have used to get the correct result in Example A.20? Ans.

4 EXP 7 ÷ 4 EXP 7 =

The precedence rules are not invoked for this sequence of keystrokes, since only one operation, division, is done. Some calculators display answers in decimal notation unless they are programmed to display them in

scientific notation. If a number is too large to fit on the display, such a calculator will use scientific notation automatically. To get a display in scientific notation for a reasonably sized decimal number, press the SCI key or an equivalent key, if available. (See your instruction booklet.) If automatic conversion is not available on your

calculator, you can multiply the decimal value by 1 × 10 10 (if the number is greater than 1) or 1 × 10 − 10 (if the number is less than 1), which forces the display automatically into scientific notation. Then you mentally subtract or add 10 to the resulting exponent.

The Reciprocal Key

The reciprocal of a number is 1 divided by the number. It has the same number of significant digits as the number itself. For example, the reciprocal of 5.00 is 0.200. A number times its reciprocal is equal to 1.

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On the calculator, the reciprocal 1/x key takes the reciprocal of whatever value is in the display. To get the reciprocal of a/b, enter the value of a, press the division ÷ key, enter the value of b, press the equals = key, and finally press the reciprocal 1/x key. Alternatively, enter the value of b, press the division key, enter the

value of a, and press the equals key. The reciprocal of a/b is b/a. The reciprocal key is especially useful if you have a calculated value in the display that you want to use as

a denominator. For example, if you want to calculate a/(b + c) and you already have the value of b + c in the display, you divide by a, press the equals key, and then press the reciprocal key to get the answer. Alternatively, with the value of b + c in the display, you can press the reciprocal key and then multiply that value by a.

b+c

b+c

EXAMPLE A.22. The value equal to 1.77 + 1.52 = 3.29 is in the display of your calculator. What keystrokes should you use to calculate the value of the following expression?

Ans.

0 8 = 1/x The display should read 0.93617 · · ·.

1/x × 3 .

or

Logarithms and Antilogarithms

Determining the logarithm of a number or what number a certain value is the logarithm of (the antilogarithm of the number) is sometimes necessary. The logarithm log key takes the common logarithm of the value in the display . The 10 x key, or the INV key or the 2ndF key followed by the log key, takes the antilogarithm of the value in the display . That is, this sequence gives the number whose logarithm was in the display.

The ln key takes the natural logarithm of the value in the display. The e x key or the INV key or the 2ndF key followed by the ln key, yields the natural antilogarithm of the value in the display.

EXAMPLE A.23. What sequence of keystrokes is required to determine (a) the logarithm of 1.15 and (b) the antilogarithm of 1.15? Reminder: Some calculators require log to be entered before the number. Check the owner’s manual.

Ans. (a) 1 .

1 5 2ndF log

or

1 5 INV log

EXAMPLE A.24. What sequence of keystrokes is required to determine (a) the natural logarithm of 5.1/5.43 and (b) the natural antilogarithm of 5.1/5.43?

Ans. (a) 5 .

4 3 = ln

(b) 5 .

4 3 = 2ndF ln

In part a, be sure to press = before ln so that you take the natural logarithm of the quotient, not that of the denominator. In part b, press = before 2ndF (or INV ).

EXAMPLE A.25. Write the sequence of keystrokes required to solve for x, and calculate the value of x in each case: (a) x = log(2.22/1.03)

(b) x = 2.22/log1.03

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Ans. (a) 2 .

2 2 ÷ 1 . 0 3 log = x= 0.334

Note the similarity of the keystroke sequences but the great difference in answers.

Significant Figures

An electronic calculator gives its answers with as many digits as are available on the display unless the last digits are zeros to the right of the decimal point. The calculator has no regard for the rules of significant figures (Sec. 2.5). You must apply the rules when reporting the answer. For example, the reciprocal of 9.00 is really 0.111, but the calculator displays something like 0.111 111 11. Similarly, dividing 8.08 by 2.02 should yield 4.00, but the calculator displays 4. You must report only the three significant 1s in the first example and must add the two significant 0s in the second example.

Summary

Note especially the following points that are important for your success:

1. Variables, constants, and units are represented with standard symbols in scientific mathematics (for example,

d for density). Be sure to learn and use the standard symbols.

2. Units may be treated as algebraic quantities in calculations.

3. Be sure to distinguish between similar symbols for variables and units. Variables are often printed in italics. For example, mass is symbolized by m, and meter is represented by m. Capitalization can be crucial: v represents velocity and V stands for volume.

4. Algebraic and calculator operations must be done in the proper order (according to the rules of precedence). (See Table A-1.) Note that multiplication and division are treated somewhat differently on the calculator than in algebra.

5. Operations of equal precedence are done left to right except for exponentiation and unary minus, which are done right to left.