Calculators are available in a wide variety of capabilities and prices. The least expensive models not only perform addition, subtraction, multiplication, and division, but also percentages and square roots. Most also have a memory. The user can store a set of numbers in the calculator’s memory and retrieve it later without having to reenter the numbers.
For example, suppose you need to convert several measurements in feet to metres. To convert feet to metres, you multiply the number of feet by 0.3048.
By entering 0.3048 into the calculator’s memory, you may recall this conversion factor by pressing a single key rather than entering 0.3048 each time you wish to make the conversion. Also, when solving complicated equations, you can store part of the solved equation in the calculator’s memory. Then, you can retrieve the partial solution later when it is required to solve the entire problem. Later, this chapter covers both these uses of a calculator’s memory.
More expensive models offer trigonometric and logarithmic functions, among other things. Also, advanced models are equipped with more than one level of memory. That is, you can store parts of a calculation in more than one place. Although this chapter does not cover it because programming is beyond the scope of this manual, some calculators are programmable: you can load special mathematical operations into the calculator, which allow it to perform advanced functions.
Calculator Features
When choosing a calculator, it is important to match your calculation needs with the calculator’s capabilities. If you anticipate solving involved calculations, then consider buying a sophisticated model. On the other hand, if most calculations are little more than solving addition, subtraction, multiplication, and division problems, then a simple, inexpensive model is adequate.
Most of today’s handheld calculators are battery operated or have a power cell that operates the calculator when it is struck by light. Manufacturers often call such calculators solar powered, but sunlight is not needed to operate them.
Ordinary indoor or outdoor light is adequate. If batteries operate the calculator, it may come with an AC adapter-charger, which not only powers the calculator, but also charges the batteries when the calculator is plugged in. Desktop calcula-tors may plug into a normal electrical outlet.
Keyboards have the usual operational symbols of +, –, ×, ÷, and =. Advanced models contain other symbols, such as %, √, x2, yx, log, and sin. (This chapter discusses these keys shortly.) The keyboard also has a period key (.), which is a decimal point. The number of symbols and characters vary with the complexity and price of the calculator. The manufacturer may print the numbers and sym-bols directly on the keys, on the case near the keys, or on both. In this chapter,
Choosing a Calculator 41
reference to a key is by its symbol regardless of its printed location. Figure 2.1 shows typical calculator keyboards.
Small calculators display numbers and other entries in a window usually located at the top of the calculator. Light emitting diodes (LEDs) or, more often in today’s models, liquid crystal displays (LCDs) show the numbers and symbols in the window. (Some desktop calculators also print out characters on a paper tape.) The display should be easy to read—that is, the size, color, intensity, and visibility of the symbols and numbers should be readable in sunlight as well as in artificial light.
Figure 2.1 Typical calculator keyboards
The calculator you use with this text should be able to show eight or more digits in the display and should have—
1. number keys, which include the ten digits of
7 8 9 4 5 6
1 2 3 0
2. operation keys for the four basic arithmetic operations, which are + – · ‚
3. a decimal . key
4. an equals = key or, with calculators that use reverse Polish notation, an enter ENT or execute EXE key. (Reverse Polish notation is discussed shortly.)
Most calculators also offer square root √ and percent % keys, which are useful not only for solving many common problems, but also for the exercises in this text. If you choose a scientific calculator, it should have at least the following keys:
) xy SIN x2 (
yx COS TAN +/– or CHS LOG LN or IN 1/x
You will learn about these keys shortly.
42 THE CALCULATOR
Calculator Notation
A major consideration in selecting a calculator is the notation, or logic, the cal-culator uses to perform its work. Notation is a system of characters, symbols, and expressions used to represent mathematical operations. Notation is sometimes called logic because logic, in this sense of the term, refers to the way in which the calculator operates to perform its functions.
Generally, calculators use three types of notation, or logic: arithmetic, alge-braic, and reverse Polish (pronounced poe-lish). Reverse Polish notation is so called because it evolved from work the Polish mathematician Jan Lukasiewicz did in the 1920s. His notation scheme became known as Polish notation. Then, an Australian computer scientist, Charles Hamblin, refined Lukasiewicz’s work in the late 1950s and, because Hamblin reversed the order in which Polish notation was written, he called it reverse Polish notation and abbreviated it as RPN. Polish and reverse Polish notation is particularly useful in calculators and computers because of the manner in which these devices perform their mathematical operations.
Arithmetic Notation
The arithmetic notation calculator has a simple keyboard and is easy to use.
Arithmetic calculators are among the least expensive and are often given away as promotional gifts. An arithmetic calculator does addition, subtraction, multipli-cation, and division in the same order in which the keys are pressed. (This book shows the pressed keys in boxes; the final answers are not boxed.) So, pressing the keys in the order of 4 + 3 × 5 = results in the answer 35 appearing on the display and is shown as
4 + 3 × 5 = 35.
Figure 2.2 shows the steps in graphic form.
First, enter 4 by pressing the 4 key on the calculator. The number 4 appears in the display. Then, press the plus (+) key; the number 4 remains in the display until you press the 3 key, at which time 3 shows on the display. Next, press the times key (×) and 7 is displayed. Finally, press 5 and the equals (=) key to display the answer of 35. As previously mentioned, besides the four basic operations, arith-metic calculators may also have percentage, square root, and memory functions.
Algebraic Notation
A calculator with algebraic notation operates the same as an arithmetic notation calculator except that it performs operations in a specific order. An algebraic calculator does powers first, which are usually entered in the calculator by press-ing a key labeled xy. Then, such calculators perform multiplication (¥), division (÷), and finally addition (+) and subtraction (–). As you will learn in Chapter 5, algebra problems are also computed in this order. For example, to compute
2 + 4 × 32 press the keys in the order of:
2 + 4 × 3 x2 = 38.
When the = key is pressed, the calculator does its work, but it does not perform the operations in the order entered. Instead, the calculator first computes 3 to the second power, producing 9. Next, it multiplies the 9 times 4 and then adds the resulting 36 to the 2 to get the answer, 38. Figure 2.3 shows the entries and displays in graphic form.
Figure 2.2 Entries and displays on an arithmetic logic calculator
Figure 2.3 Entries and displays on an algebraic notation calculator
Choosing a Calculator 43
Calculators with algebraic notation may be inexpensive, medium-priced, or expensive, depending on their capabilities. In addition to the four basic operations, percentage, and square root, many medium-priced and more expensive models can square a displayed number, calculate the inverse of the displayed number, perform operations within parentheses first so that the contents are treated as one unit, and change the sign of a displayed number.
Some algebraic calculators have a memory system that can store several pieces of information rather than just one. To store several entries, the opera-tor usually presses location keys after the memory sopera-torage key. Information can be stored and recalled at any time. Calculators with several advanced keys and memory are often called scientific calculators.
Reverse Polish Notation
An enter, or execute, key characterizes a reverse Polish calculator—that is, it has a key labeled ENTER ENT EXECUTE or EXE . Although using an RPN cal-culator takes practice, once it is mastered, it is versatile, dependable, and reduces the number of key strokes required for many calculations.
RPN calculators work with a memory stack, which can be thought of as a ladder with several rungs. The display represents the bottom rung of the ladder, and each rung above the display is a memory level capable of holding one piece of information. When the ENTER key is pressed, the displayed number climbs one rung up the ladder and also pushes whatever else is on the ladder up one rung. With RPN calculators, the numbers that are keyed in first are worked on last. When a number is entered, the memory stack stores it for later use. For example, to solve the problem
5 + (2 × 4) = ?
press the following calculator keys:
5 ENTER 2 ENTER 4 × + 13.
Figure 2.4 shows the entries, displays, and the content that is in the memory of a reverse Polish (RPN) calculator for the preceding problem.
RPN-Algebraic Combination
Some calculators combine RPN and algebraic notation. Many of these combi-nation calculators also feature a large display screen on which users can display graphs. Thus, they are usually called graphing calculators. With a graphing calculator, to solve the problem
5 + (2 × 4) = ? press the keys in the following order:
5 + ( 2 × 4 ) ENTER .
When enter is pressed, the answer of 13 is displayed. Figure 2.5 shows the sequence. Notice that the display not only shows the number pressed, but also graphically shows the problem as the numbers and symbols are put into the calculator. This graphic display allows users to check the entries and ensure that they are correct. If an entry is wrong, cursor keys on the calculator allow users to enter the display, remove the wrong entry, and key in the correction.
If the problem is stated
Figure 2.4 Entries, displays, and memory in an RPN calculator
Figure 2.5 Entries and displays in a graphing calculator
44 THE CALCULATOR
Notation Differences
To illustrate the differences in the arithmetic, algebraic, and reverse Polish nota-tion, review the following problem and consider the calculator’s interpretation of the keystrokes when entered in this order:
2 + 3 × 4 =
The arithmetic logic calculator interprets the keystrokes as
(2 + 3) × 4 = 20.
The algebraic logic calculator performs the multiplication first and works the problem as
2 + (3 × 4) = 14.
The RPN calculator does not work with this sequence of keystrokes. For the arithmetic interpretation, the problem is entered as
2 ENTER 3 + 4 ×
and for the algebraic interpretation on an RPN calculator, the problem is entered as
2 ENTER 3 ENTER 4 × + .
Computer programmers were among the first to recognize the advantages of reverse Polish notation. The system scans an expression from left to right and therefore carries out the calculation immediately as it is encountered. This action means that less storage space is required on a calculator’s memory chip.
Also, fewer keystrokes are needed. Unlike algebraic logic, where the operation signs are sandwiched between the numbers, in reverse Polish they are generally entered last and are not stored. Not having to store the operation signs means less storage space is needed.
Many companies manufacturing small, nonscientific calculators use arith-metic logic with single-memory storage. Hewlett-Packard was the first manu-facturer to use reverse Polish notation in scientific calculators and continues to produce them, as do Texas Instruments, Casio, and others. Texas Instruments, Casio, Sharp, and other manufacturers also make graphing calculators as well as algebraic calculators.
Writing Problems
The way a calculator interprets a problem brings up an important point in mathematics: the problem must be written properly to obtain the desired results.
Consider the problem 2 + 3
×
4. Parentheses must be used before the problem can be correctly solved—that is, the correct result cannot be obtained unless the parentheses are placed in the correct position. The correct answer is 20 if the problem is stated as (2 + 3)×
4; however, the correct answer is 14 if the problem is stated as 2 + (3×
4).Note, too, that the way you enter the problem depends on the type of calculator in use. For example, if the problem is stated as (2 + 3)
×
4 and you are using an algebraic calculator, you must enter the problem as( 2 + 3 ) × 4 = 20.