Ted's Math World - Programming a 4-function Calculator

Calculator Magic #1:
Introduction

Programming a Four-Function Calculator

Powers, roots, compound interest, trigonometry, logarithms — all things are possible even on the simplest of units, if you know how. One could simply purchase a scientific calculator from Hewlett-Packard or Texas Instruments; or else you could enjoy mastering the power of the $4 solar unit that you received in your Christmas stocking or as an advertising promotion. The results will be perfectly useful, and you can amaze your friends and win bar-room bets in the process.

This discussion covers only features that apply to my related pages on specialized calculator application. Extensive use is made of the Memory and Constant features, but usage of the Percent function isn't covered at all.

ABOUT THE LOGIC CHIP

Important: There are two basic types of of calculator logic:

Arithmetic: these models emulate the older electric and manual units. A number is entered, then an operator. For example, 11 − 4. The entry is 11 + 4 −. For 17 ÷ 3, the entry is 17 + 3 ÷. These units sometimes have a paper-printing capability; but they are designed for account clerks, not for scientists such as ourselves.
Algebraic: this is the type you want. Most modern calculators, including solar units, support this logic. Calculations are entered just as you would write or say them. For 11 − 4, the entry is 11 − 4 =. For 17 ÷ 3, the entry is 17 ÷ 3 =.

Just as important: There are two types of algebraic logic: Casio, and all others. Many retailers such as Radio Shack market units with their own logos, but which are manufactured by other companies. If you are not sure of your brand, then perform the following test:

Enter 2 × 3 =. The display will show a {6}. Now, without clearing, enter 4 =. If the display shows a {4}, then you have a Casio-made unit; others will read {8} in the display.

Note: Some newer cheapo Casio units don't behave like their predecessors, choosing instead to emulate their inferior competition. This disappoints me greatly, but more importantly, you need to run the test even on Casio calculators.

USING THE CONSTANT (K)

Every pocket calculator has a built-in constant feature, meaning that you can establish a fixed multiplier or divisor to save steps in repetitive calculations. Although I am a strong detractor of substituting 'K' for 'C' in general (names such as "Kalico Kitchen" and "Krispy Kreme" turn my stomach), but it has a valuable usage for us programmers. Henceforth, this function will be designated as Konstant — or perhaps simply K, so as to differentiate from the algebraic meaning of 'constant'. In keeping with this protocol, Casio units display a 'K' when Konstant mode is initialized.

The following example sets a Konstant value of 2 in order to double a series of numbers:

Non-Casio:

K-Multiplier: 2 × 3 = {6} ; 4 = {8} ; 3.14 = {6.28}
2 × sets the Konstant. Following with 3 = completes a calculation, and the display shows a {6}.
Now, in order to multiply 2 × 4, the unit remembers the 2 ×, so you need only enter 4 = {8}. You may continue indefinitely: 5.3 = {10.6} ; .07 = {.14}, etc.
K-Divisor: 2 ÷ 5 = {.4} ; 18 = {3.6} ; 98.6 = {17.2}
It works the same as the multiplier.

Casio models: the Konstant is initialized by a double entry of the arithmetic operator:

K-Multiplier: 2 × × 3 = {6} ; 4 = {8} ; 3.14 = {6.28}
Think of (××) as "multiplied by."
K-Divisor — enter the divisor first: 5 ÷ ÷ 2 = {.4} ; 18 = {3.6} ; 98.6 = {17.2}
Think of (÷÷) as "divided into."

The Casio protocol may seem somewhat tedious, but it has certain advantages. For example, Konstants for addition (++) and subtraction (−−) can be set up in the same manner; other brands cannot do this at all:

K-Addend: 7 + + 5 = {12} ; 84 = {91} ; .55 = {7.55}.
Think of (++) as "added to."
K-Subtrahend: 4 − − 13 = {9} ; 32 = {28}.
Think of (−−) as "subtracted from."

When using Konstant mode, it can be useful to think of the mnemonic aids — or even mouth them — during a calculation; it may help you to keep track of what you are doing.

Important: In order to accommodate all types of calculators in program code, the multiplication constant will be denoted as a generic ×(×). Non-Casio users do not enter the second (×).

Equally important: Some Casios behave differently than others in Konstant mode, regarding their treatment of the M+ and M− keys. Run this test:

Enter 4 ×× M+ =

If the display shows a 64, then you have an "active" model, which allows the usage of M+ and M− without disrupting the Konstant series. 4² will be in memory, and 4³ is the active number. In other words, pressing M+ includes the function of an equal sign, squaring the number as it is copied to memory.

If the display show a 16, then you have a "passive" model, which allows M+ and M− to cancel Konstant mode. Pressing M+ placed 4² into memory all right, but the equal sign did nothing. Passive Casios and other brands need to use an extra equal sign in the code sequence: 4 ×(×) = M+ =

Note: "Active" units save one keystroke per loop of this type, and this savings is not optional. Remember that the memory key serves double duty; entering the extra equal sign would add an unwanted increment to the exponent in this particular calculation.

TAKING A RECIPROCAL

Reciprocals also are handled differently by the two types of logic. If the display shows {8.7} and you wish to take its reciprocal, 1/8.7:

Casio: {8.7} ÷ ÷ 1 = {.1149..} Or else, you can use: {8.7} ÷ ÷ = =.
Other: {8.7} ÷ = {.1149..} Or else, if the unit has a reciprocal [1/x] key, just press it.

Dividing the {display} into another number is slightly different. Example: divide {17} into 3:

Casio: {17} ÷ ÷ 3 = Normal division of one number into another.
Other: {17} ÷ 3 ÷ = Do an 'upside-down' division, then take the reciprocal.

ABOUT MEMORY

Your calculator might have only a 3-key memory setup; that is, it will have a dual-function MRC button. One keypress recalls the memory, and a second keypress clears it. A 4-key setup is preferable, as it can be useful to be able to clear the memory and re-use it during a calculation. We'll just have to work around that limitation with the following generic protocol:

MR recall the memory (one press of MRC)
MC clear the memory (two presses of MRC)

A final note: many online arithmetic calculators — even some on purported instructional sites — do not feature a Konstant mode. These are careless omissions by the programmers, as the utilities are thereby crippled. Those units are unsuitable for our purposes. If you are running Microsoft Windows, then you can use the good calculator utility supplied with the operating system. I have no familiarity with the offerings of other PC operating systems.