Logarithm. Entering a number and then pressing this key calculates and displays the logarithm to base 10 of the displayed number. For example, to find the log of 532, enter 5 3 2 LOG. Pressing LOG displays the answer of 2.7259.
(Chapter 1 described logarithms.)
Antilogarithm. Entering a number and pressing this key (usually after pressing the calculator’s second function key) calculates and displays the antilogarithm of the displayed number. For example, enter 2 . 7 2 5 9 . Then, press
2nd to access the antilogarithm key; finally, press 10x. The calculator displays 531.9857, which rounds to 532. (Chapter 1 described antilogarithms.)
Natural Logarithm. Entering a number and pressing this key calculates and displays the natural logarithm of the displayed number. For example, enter 2 0 , press IN (or LN). The calculator displays 2.9957, the natural logarithm of 20. (Chapter 1 described natural logarithms.)
Natural Antilogarithm. Entering a number and pressing this key calculates and displays the natural antilogarithm of the displayed number. (Usually, you press the second function key first to access this key.) Just as an antilogarithm is the reverse of a logarithm to the base 10, a natural antilogarithm is the reverse of a natural logarithm. To find the natural antilogarithm of a number, enter the number and press the exponential function key (ex). For example, enter 2 .
9 9 5 7 , press 2nd , and press ex . The answer is 19.9993 (rounds to 20), the natural antilogarithm of 2.9957.
Power. After entering a number, pressing this key raises the entered number (y) to a power determined by the second number entered (x). (Chapter 1 discussed raising a number to powers). For example, press 2 yx 4 and = . The answer is 16 because two to the fourth power (24) is 16. Also, you can carry out numbers multiplied by 10 to a given power. For example, to carry out 9.3 × 107, press 9 . 3 × 10 yx 7 = 93,000,000. The first number entered must be posi-tive; the second one, however, may be negative.
xth Root. After entering the desired number, and, usually the second function key, pressing this key calculates the desired (x) root of the first number entered (y) provided the number is not negative. For example, press 16 2nd X √y
4 and = . The answer is 2 because the fourth root of 16 is 2.
Roll Stack. Pressing one or the other of these keys allows the user of an RPN calculator to review the contents of the working registers or to reposition numbers.
Fraction Key. A calculator with this key allows you to perform operations with proper and improper fractions. For example, to multiply ¾ × 3, press 3 ab/c 4
× 3 = . The answer is 2¼. Some calculators display fractions with a symbol between the nominator and denominator. So, the calculator displays ¾ as 34, and 2¼ as 2_14.
OPERATING A CALCULATOR
When properly operated, a calculator conveniently and accurately solves problems.
Most calculators are easy to operate if the instructions furnished by the manu-facturer are followed. Many operations may be obvious; however, some valuable features may be available but not immediately evident by simply looking at the keyboard. So, as mentioned many times, the first step in operating any calculator is to read the manufacturer’s instructions.
LOG
50 THE CALCULATOR
Most booklets or instruction sheets provide example calculations. These examples can help you understand functions. Working the examples in the instructions is often a good way to learn the function of the many keys on the calculator.
As pointed out earlier, a calculator functions according to its notation, or operating system. The simplest is the arithmetic operating system. More involved calculations require the algebraic or RPN operating systems.
Solving Problems with an Arithmetic Calculator
The arithmetic calculator (fig. 2.6) adds, subtracts, multiplies, and divides two numbers in logical order with the operation key pressed between the entry of the numbers. For example, to add two numbers, press the keys in this sequence:
6 + 3 = 9.
The display shows 9 as soon as the equal key = is pressed. Likewise, to subtract, multiply, or divide, press the keys in the order of the operation:
6 – 3 = 3.
6 × 3 = 18.
6 ÷ 3 = 2.
In chain calculations (a calculation that is followed by another calculation that is followed by another calculation, and so on, the sequence of operations is the same as entered. For example, in the chain calculation
6 – 4 × 5 = 10.
the calculator subtracts 4 from 6 first, then multiplies the resulting 2 by 5 to give the answer 10. An arithmetic calculator treats a chain calculation where multiplication or division precedes addition or subtraction in the sequence the operation is entered. For example, in the problem
4 × 5 + 9 = 29.
the calculator multiplies 4 by 5 to give 20, then adds 9 to 20 to give the answer 29.
When using an arithmetic calculator to calculate a chain of operations that requires algebraic notation, it is easier to make the calculations if the arithmetic calculator has a memory. All but the least expensive calculators have a memory.
However, even with models that feature memory, it may be only one level, so the calculator can remember only one piece of information. In this case, the memory solves one part of a problem, holds it, solves another part of the problem, retrieves the results of the first problem, and then performs a final operation using the results of both parts.
For example, when using an arithmetic calculator to work the problem (12 + 5) × (23 – 4)
follow this entry sequence:
12 + 5 = M+ 23 – 4 × MR = 323.
Figure 2.7 shows the entries and displays in graphic form.
In this sequence, the calculator stores the sum of the first two numbers in Figure 2.6 Arithmetic calculator
Figure 2.7 Entries and displays on an arithmetic calculator using memory key
Operating a Calculator 51
its memory, performs the subtraction in the next set, then multiplies this differ-ence by the sum in its memory to get the answer 323. Note that the equal key
= is not pressed to complete the subtraction; the times key × activates the operation. Also, most calculators give an indication when a number other than zero is in its memory. Usually, an m appears in the display. Be sure that the cal-culator’s memory is cleared (erased) before you begin using it. Most calculators store numbers in their memory until you take steps to remove it. For example, turning off the calculator may not clear the memory. So, to clear a calculator’s memory, check the instructions.
Memory is also useful when it is necessary to make several calculations when one of the numbers does not change. For example, suppose you wish to convert several measurements in feet to metres. Because the conversion factor is 0.3048, enter this number into the calculator’s memory. Then, simply recall this stored factor and multiply it by the number of feet. You do not have to enter 0.3048 each time because the calculator’s memory stores the number.
Example Problem: Convert 3.5, 8.25, 10, and 22.75 feet to metres.
Solution: Make sure the calculator’s memory is clear and then enter the feet-to-metres conversion factor into the calculator’s memory. Next, multiply the quantity of feet to be converted to metres.
The sequence is:
.3048 M+ × 3.5 = 1.0668
RM × 8.25 = 2.5146
RM × 10 = 3.0480
RM × 22.75 = 6.9342
Pressing M+ stores .3048 in memory. (On some calculators, this key is labeled
STO.) After making the first conversion, press RM to recall .3048 in the calculator’s memory. (On some calculators, the memory-recall key is labeled RCL.) Next, press
× and enter the second number. Finally, press = to obtain the answer. Repeat the sequence for the remaining conversions.
A calculator’s percent key % is useful for dividing numbers by 100 and for calculating markup and markdown percentages. Dividing a number by 100 merely moves the decimal point of that number two places to the left. Most calculators require the percentage number to be entered last. For example, to find 60% of 200, the keys are pressed in the following order:
2 0 0 × 6 0 % 120.
In effect, the calculator multiplies 200 by 0.6 to give 120.
Example Problem: A business borrows $5,000 at a simple interest rate of 7.75%.
How much interest will it pay at 7.75%?
Solution: Enter the figures as follows:
5 0 0 0 × 7 . 7 5 % 387.5.
The business will pay $387.50 in simple interest.
For a markup or add-on problem, the + key is used instead of the × key.
52 THE CALCULATOR
Example Problem: You have bought a carload of tires for $34.50 each. How much would you have to sell each for to make a 35% profit?
Solution: To work this problem without a % key, multiply $34.50 times 0.35 and add the product to $34.50 to get $46.58. On a calculator with a % key, enter:
3 4 . 5 0 + 3 5 % 46.575 or $46.58.
In any case, you would have to sell each tire for $46.58 to make a 35%
profit.
For a markdown, or discount, problem, the – key is used instead of the + key.
Example Problem: You normally sell an item for $398.99, but you discount the item by 16%. What is the new selling price?
Solution:
3 9 8 . 9 9 – 1 6 % 335.15.
The new selling price is $335.15.
Using the percent key greatly simplifies an operation.
Example Problem: Find the tax (8.5%) and the tip (15%) on an $8.93 luncheon;
$2.00 of the luncheon is for a glass of wine that is not taxable, but you want to pay the tip on the total. How much money will you have to pay?
Solution: First, find the cost of the lunch without the wine:
8.93 – 2.00 = 6.93.
Then, find the tax on 6.93 and add it to the total:
6.93 × 0.085 = 0.5891 + 6.93 = 7.52.
Next, add 7.52 to 2.00 and add 15% to all for the total amount:
7.52 + 2.00 = 9.52 9.52 × 0.15 = 1.43 1.43 + 9.52 = 10.95.
A calculator with a percent key can handle all these calculations by enter-ing the followenter-ing:
8 . 9 3 – 2 . 0 0 = 6.93 + 8.5 % 7.519 + 2 . 0 0 = 9.519 + 1 5 % 10.946.
So, the total cost of the luncheon is $10.95.
Many arithmetic logic calculators can also find the square root of a number.
Simply enter the number into the display, then press the key with the radical sign (√). The calculator displays the answer. For example, to find the square root of 84, press the keys in the following order:
8 4 √ 9.16515139.
Note that you do not press the = key to obtain the result.
Operating a Calculator 53
Using an Algebraic Calculator
In an algebraic calculator (fig. 2.8), two-number operations are performed in the same sequence as in the arithmetic calculator—that is, numbers and operations are entered as they appear. Chain calculations are also entered as they appear;
however, the sequence of operations actually taking place in the calculator fol-lows algebraic logic. So, the calculator computes powers first, multiplication and division second, and addition and subtraction last.
For example, the problem
2 + 4 × 32 = is entered as:
2 + 4 × 3 x2 2 = .
When the = key is pressed, the calculator begins to work, but it does not per-form the operations in the order given. Instead, it first calculates 3 to the 2nd power to produce 9; next, it multiplies the 9 by 4 to give 36; finally, it adds 36 to the 2 to give the answer, 38.
Most algebraic calculators have parenthesis or bracket keys, which allow you to properly group the expressions. For example, the problem
(2 + 3) × (4 + 5) = 45
if entered without parentheses in the order shown, is interpreted by the calcula-tor as
2 + (3 × 4) + 5 = 19.
Unfortunately, 19 is the wrong answer. So, you must also enter the paren-theses as well as the numbers and operational signs in their proper order for the calculator to come up with the right answer of 45.
An algebraic calculator may have a single-level memory system or one that allows storage of more than one piece of information by pressing location keys after the memory key. To use this type of memory, calculators may have the following keys:
STO , RCL , SUM , and PRD .
If the calculator’s memory has many levels (storage places), you must tell the calculator at what level to store the information. If a two-digit number designates the level, you must enter two digits (01, 12, etc.). To bring back a number stored in the memory, press RCL and the keys for the two-digit location where the number is stored. To add a displayed number to a number in the memory, press the SUM key and the two-digit memory location keys. Likewise, to multiply a displayed number by a number in the memory, press the PRD key and the two digits. With single-level memory, the same keys are used with the location keys, or some calculators have simply the STO and RCL keys.
Also keep in mind that to access some of these keys, you may have to press the 2nd or 3rd function keys. For example, on some calculators, the RCL and
SUM key may be shared. In this case, to use the RCL key, you simply press it;
however, to use the SUM key, you press 2nd first and then the SUM key. Press-ing 2nd gives the calculator access to the SUM key rather than the RCL key.
Some problems require both parentheses and memory functions to reach the correct answer. In this type of problem, it is especially important to clear the memory and the display before starting calculations. To clear memory in your
Figure 2.8 Algebraic logic scientific calculator
54 THE CALCULATOR
calculator, refer to the instruction booklet. Different calculators use different keys to clear memory; so, how you clear your calculator’s memory depends on the specific calculator.
Example Problem:
32.64 + 18 44.9 + 16.33 –––––—–– × –————– = ?
5.32 81
Solution: Treat the problem in two parts. First, write the first part for the cal-culator as:
(32.64 + 18) ÷ 5.32.
So, enter the figures in the following order after you have cleared the display and memory:
( 3 2 . 6 4 + 1 8 ) ‚ 5 . 3 2 = STO.
Pressing STO stores the answer to the first part of the problem in memory.
Now, do the second part of the problem in the same way. Write the second part as (44.9 + 16.33) ÷ 81 and enter
( 4 4 . 9 + 1 6 . 3 3 ) ‚ 8 1 = .
To multiply the two answers, press:
× RCL = .
The answer is 7.19550543, or rounded off to two places, 7.20. (Fig. 2.9 shows all the entries and displays.)
When solving complicated problems, keep in mind that a calculator with algebraic notation performs operations in a specific order. Since some calculators do not have parentheses keys, it is necessary to solve parts of an involved problem in a specific order—not necessarily in the order of appearance. The following problem is solved without parentheses.
Example Problem:
30 + √ 12 – (4 × 2) 8 + –––––––––––—–– = ?7 + (2 × 3)2
Solution: Solve the denominator part of the problem first and store it in the memory for later use. First, multiply 2 by 3 and square the answer. If 2 is not multiplied by 3 first, the calculator squares the 3 and gets a wrong answer. Once the problem in parentheses is solved, add 7 to it and then store the answer, 43, in the memory. To solve for the denominator, press the keys in this order:
2 × 3 = x2 + STO 1.
If the calculator has parentheses, you can solve the denominator as:
7 + ( 2 × 3 ) x2 = STO 1.
Now solve the numerator part of the problem. The parentheses are not a problem here because the calculator automatically does multiplication first. Solve the part of the problem under the radical sign, then add the answer to 30:
Figure 2.9 Entries and displays on an algebraic logic calculator
Operating a Calculator 55
The answer, 32, is not stored in the memory because this numerator is going to be divided by the denominator, 43, which is recalled from the memory:
‚ RCL 1 = Finally, add 8 for the answer:
+ 8 = 8.744186047, or rounded off to two places, 8.74.
Using an RPN Calculator
To solve a problem with an RPN calculator (fig. 2.10), you must enter the problem in a specific order. After the calculator stores parts of the problem in its stack, press the operation keys. Operations can be grouped into two categories. An operation with only one variable (one number to be dealt with, such as x, x2, 1/x) not only acts on the display contents, but also the answer shows in the display. An operation with two or more variables (addition, subtraction, multiplication, and division where more than one number is involved) acts on the display contents and on the contents of the next higher memory level in the stack. The answer shows in the display, but all stack contents move down one level. The numbers that are keyed in first are worked on last, and operation keys are entered in re-verse order of their use. Numbers are automatically stored in the memory and worked on in reverse order. After each number to be put into the memory stack is pressed, the ENTER key must be pressed to move the number into the stack.
Example Problem:
2 + √ 33 – 4 1 + –––––––––– = ?1 + (7 × 3)2
Solution: Enter the parts of the problem in the calculator as follows:
1 ENTER 2 ENTER 3 3 ENTER 4 – √ + ENTER 1 ENTER 7 ENTER 3 × x2 + ÷ + 1.1067.
To understand what happens in an RPN calculator, follow the display and memory of the problem as each entry is made (fig. 2.11). The number 1 was en-tered into the memory stack first and left there to be worked on later. Next, the numerator of the fraction was solved and stored for later use. The numerator was solved by displaying and entering the numbers from left to right. The last number, 4, was displayed but not entered so that it could be subtracted from 33 to get an answer of 29. Working backwards, the square root of 29 was found and added to the 2. The answer to the numerator, 7.38, was entered in the memory stack.
Next, the denominator of the fraction was solved in the same way, entering the numbers from left to right and then working backwards with the operations between these numbers. Because the 3 and 7 are in parentheses, they are one unit, and they had to be multiplied together first to get an answer of 21. Then the square of 21 was found and added to the 1. The result was 442, the answer to the denominator. This answer was not entered into the memory stack, as the next step was to divide the numerator by the denominator (7.38 ÷ 442). Finally, the result of the division was added to the whole number 1 to obtain the solution, 1.0167.
Figure 2.10 An RPN calculator
Figure 2.11 Stack memory of an
56 THE CALCULATOR
Practice Problems
The following problems can be solved by any calculator. Experiment with your calculator, read the instructions, and discover how it works. Then solve these problems, rounding off the answer to two decimal places.
1. 32.4 + 68.3 + 17.1 = _________________________________________
_________________________________________________________
2. 981.503 + 576 + 834.334 – 1,000 = _____________________________
_________________________________________________________
3. 5,621 – 5.86 – 7.90 – 3 – 569 = ________________________________
_________________________________________________________
4. 53.85 × 28 + 40.75 = _________________________________________
_________________________________________________________
5. (91 – 52) × 48 = ____________________________________________
_________________________________________________________
6. 8.9 × 0.87 × 0.065 = _________________________________________
_________________________________________________________
7. (575 × 286) ÷ 300 = _________________________________________
_________________________________________________________
8. 2,500 × (25 ÷ 50)2 = _________________________________________
_________________________________________________________
9. 350 + √ 3,368 – 4 = _________________________________________
_________________________________________________________
10. 46% of 3,500,152 = __________________________________________
_________________________________________________________
11. 72 – [(530 – 468) ÷ (95 × 0.052)] = _____________________________
_________________________________________________________
12. How far can a car be expected to travel without a refill if it gets 22 miles per gallon and its tank holds 14 gallons?
_________________________________________________________
13. If 9.8 pounds of peaches cost $12.65, what is the selling price per pound?
_________________________________________________________
14. A bag of nails regularly sells for $3.50. It is now advertised at 20% off. If you bought 5 bags at the sale price and paid a 5% sales tax, what is your total bill?
_________________________________________________________
15. 14.9 × (13.5 – 12.2)
–———–––––—–– × 1,450 = ________________________________35.5 – 13.5