AOS Calculators: Duplicating a Value Without Retyping It
Note: The following applies to scientific classic calculators who operate under the algebraic operating system (AOS) (that is what Texas Instrument’s calls it). I tested this procedure with the following calculators: TI-30X ECO, HP 10bII+ (Algebraic mode), and Casio fx-260 Solar.
Introduction: Going Back to 1976
Imagine it is 1976 and you have have an SR-56 from Texas Instruments. Here is what an SR-56 looks like: http://www.datamath.org/Sci/WEDGE/ZOOM_SR-56.htm
You are tasked to calculate 1.401103287^1.401103287 and do not want to write the number twice. According to page 53 of the SR-56 manual, one approach is to key in:
1.401103287 [ y^x ] [ CE ] [ = ]
Result: 1.604057054
For that particular calculator, SR-56, pressing [ CE ] once stores the number in the display as the second operand allowing the value to duplicated without having to retype the number.
If we tried that on a modern TI-30Xa/TI-30 ECO RS, the display would clear to zero instead of showed the previous number.
However, there are a few tricks we can employ to achieve the similar result.
Trick 1: Pressing the Reciprocal Key Twice
As long as the number in the display is nonzero, pressing [ 1/x ] [ 1/x ] registers the number in the display for as a second operand. In calculators operating in AOS, executing one-argument functions only operate and effect the number in the display only.
Pressing [ 1/x ] takes the reciprocal of the number and registers the number in the display. Pressing [ 1/x ] again returns the number.
**The keystrokes omits any [ 2nd ] or [ SHIFT ] keys.
Example 1:
Expression: x * log x
Keystrokes: x [ × ] [ 1/x ] [ 1/x ] [ LOG ] [ = ]
5.8 * log 5.8
Keystrokes: 5.8 [ × ] [ 1/x ] [ 1/x ] [ LOG ] [ = ]
Result: 4.427882363
Example 2:
Expression: x^x
Keystrokes: x [ y^x ] [ 1/x ] [ 1/x ] [ = ]
3.088 ^ 3.088
Keystrokes: 3.088 [ y^x ] [ 1/x ] [ 1/x ] [ = ]
Result: 32.51797379
Example 3:
Expression: x * sin x
Keystrokes: x [ × ] [ 1/x ] [ 1/x ] [ SIN ] [ = ]
50° * sin 50°
Keystrokes: ([DRG] to DEG/[ MODE ] (DEG))
50 [ × ] [ 1/x ] [ 1/x ] [ SIN ] [ = ]
Result: 38.30222216
4^4 + 1 / (3^3)
Keystrokes:
4 [ y^x ] [ 1/x ] [ 1/x ] [ + ]
[ ( ] 3 [ y^x ] [ 1/x ] [ 1/x ] [ ) ] [ 1/x ] [ = ]
Result: 256.037037
If the calculator has a cube function (x^3), we can execute this keystroke:
4 [ y^x ] [ 1/x ] [ 1/x ] [ + ]
[ ( ] 3 [ x^3 ] [ ) ] [ 1/x ] [ = ]
Trick 2: Inverse Function Trick
This trick extends the reciprocal trick to include a function that acts on two (and theoretically more) “reversible” functions. This trick applies to the expressions with the following format:
f(x) OP g(x)
f(x) |
f^-1(x) |
f(x) |
f^-1(x) |
f(x) |
f^-1(x) |
SIN |
SIN^-1 |
e^x |
LN |
X^3 |
∛ |
COS |
COS^-1 |
LN |
e^x |
∛ |
X^3 |
TAN |
TAN^-1 |
10^x |
LOG |
Hyperbolic |
Inverse Hyperbolic |
SIN^-1 |
SIN |
LOG |
10^x |
Inverse Hyperbolic |
Hyperbolic |
COS^-1 |
COS |
X^2 |
√ |
|
|
TAN^-1 |
TAN |
√ |
X^2 |
|
|
OP covers the arithmetic operations: [ + ], [ - ], [ × ], [ ÷ ], [ y^x ], and [ y^(1/x) ]
The general keystroke sequence is: x [ f(x) ] [ OP ] [ f^-1(x) ] [ g(x) ] [ = ]
Let’s illustrate this with a few examples. Assume the calculator is in degrees mode.
Example 1:
sin 40° * cos 40°
f(x) = sin x, f^-1(x) = sin^-1 x, g(x) = cos x
Keystrokes: 40 [ SIN ] [ × ] [ SIN^-1 ] [ COS ] [ = ]
Result: 0.492403877
Example 2:
tan 32° * sin 32°
f(x) = tan x, f^-1(x) = tan^-1 x, g(x) = sin x
Keystrokes: 32 [ TAN ] [ × ] [ TAN^-1 ] [ SIN ] [ = ]
Result: 0.331130307
Example 3:
log 881 * ln 881
f(x) = log x, f^-1(x) = 10^x, g(x) = ln x
Keystrokes: 881 [ LOG ] [ × ] [ 10^x ] [ LN ] [ = ]
Result: 19.97005314
Example 4:
e^3.5 / √3.5
f(x) = e^x, f^-1(x) = ln x, g(x) = √x
Keystrokes: 3.5 [ e^x ] [ ÷ ] [ LN ] [ √ ] [ = ]
Result: 17.70095363
Example 5:
√4.555 + e^4.555
f(x) = √x, f^-1(x) = x^2, g(x) = e^x
Keystrokes: 4.555 [ √ ] [ + ] [ x^2 ] [ e^x ] [ = ]
Result: 97.24099983
The inverse function “recovers and registers” the original x. It’s kind of simulating the LAST x feature on RPN calculators.
Sources
Datamath. “Texas Instruments SR-56“ December 5, 2001. http://www.datamath.org/Sci/WEDGE/SR-56.htm
Texas Instruments. Programmable Slid-Rule Calculator SR-56: Owner’s Manual. Dallas, TX. 1976
Eddie
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