AOS Calculators: Duplicating a Value Without Retyping It

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 [ 2^nd ] 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

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

All posts are 100% generated by human effort.  The author does not use AI engines and never will.

Casio fx-991CW: Generating Graphs

Casio fx-991CW: Generating Graphs

But Wait, the fx-991CW is Not a Graphing Calculator!

Technically, this is correct. We cannot generate graphs of the Casio fx-991CW calculator itself. But through the magic of the QR code, we can use that to generate graphs that appear on our smart phones.

What is needed:

(1) QR Reader. It does not have to be the official Casio EDU+ app, any QR reader app should do.

(2) A smart phone or device that can read QR codes.

(3) Your fx-991CW calculator. I believe that these instructions can apply to the fx-991EX as well, however, the instructions presented today are specific to the fx-991CW calculator.

The fx-991CW has storage for up to two functions: f(x) and g(x).

Steps

Step 1: Determine the number of functions you want to graph. You may either graph f(x) alone, g(x) alone, or both f(x) and g(x). To define functions, press the [FUNCTION] button.

Step 2: Press the [HOME] button, and select the Table app.

Step 3: Choose the number of functions to be evaluated. Press the [ TOOLS ] button, select Table Type, press [ → ], and select from f(x)/g(x), f(x), or g(x). Remember that contents of f(x) and g(x) are not retained when the calculator is turned off.

Step 4: Now we need to give the range. Press the [ TOOLS ] button, select Table Range, and then press [ → ]. Give a start value, an end value, and step. In terms of the graph, the value of step isn’t that important. Think of the start value as Xmin (minimum x value) and the end value as Xmax (maximum x value). Once you are satisfied, scroll down to Execute and either press [ EXE ] or [ OK ]. A table is generated.

Step 5: Next, while the table is shown, generate a QR code by pressing [ SHIFT ] [ x ] (QR).

Step 6: Grab your smart phone or device with your QR reader app open. Scan the QR code. A successful scan will give you a link to the classpad.net website. Some camera apps will read QR codes and give direct access to the link as well, neat!

Step 7: Click on the link and you will see the graph generated. At this point, you can clear the QR code off the calculator by pressing either [ EXE ] or the back key.

I took the following pictures with my smart phone. It’s a challenge to get pictures in a room in a house with two dogs and three cats…

Defining the function

Setting the table type.   Since I'm graphing just f(x), I'm selecting f(x) only for the table type.

Selecting Table Range

xmin =1, xmax = 10

A table is generated

QR code

The graph is generated!

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

All posts are 100% generated by human effort.  The author does not use AI engines and never will.

Spotlight: TI-32 Solar from 1988

Spotlight: TI-32 Solar from 1988

Today’s spotlight is on a standard scientific calculator from Texas Instruments… with a twist. First, let’s start with the quick facts.

Quick Facts

Model: TI-32 Solar

Company: Texas Instruments

Timeline: 1988 (possibly 1989?), this model was very short lived

Type: Scientific

Memory: 1 memory register

Power: Solar

Screen: 10 digits

The TI-32 Solar was a scientific calculator that was sold briefly by Texas Instruments in the last 1980s. The function set is a combination of the TI-30 Solar and the original one line TI-34:

* trigonometry, decimal/dms conversion, rectangular/polar conversions

* logarithms, anti-logarithms (10^x, e^x), powers

* one variable statistics (mean, deviation, sums)

* binary/decimal/hexadecimal conversion (and just conversions, strangely no octal)

* 3-decimal random number generator (RND)

* percent function

The Polar/Rectangular conversions use the [ a ] and [ b ] keys. [ a ] is for x and r, while [ b ] is for y and Θ.

The statistics mode is one-variable only. Though I am not a fan of the STAT mode being the alternative function of the [ C ] key, thankfully, the stat mode works.

What truly sets the TI-32 Solar apart are two things:

(1) The orange trim and keys on a dark gray keyboard. This is a very rare color scheme, especially orange until Texas Instruments started to release the TI-30XIIS and the TI-84 Plus CE in various colors.

(2) The screen tilts. The screen is a set on a hinge that allows the user to tilt the screen to different angles, which allows for more ergonomic operation. To put the calculator away, tilt the screen to be aligned with the keyboard and use the slide case. I think this is a really neat idea. The tilt mechanism seems to hold out well for a nearly 40 year old calculator.

Source

Woerner, Joerg. “Texas Instruments TI-32 SOLAR” Datamath Calculator Museum. September 20, 2002. Accessed July 7, 2025. http://www.datamath.org/Sci/Modern/TI-32Solar.htm

Eddie

 All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

All posts are 100% generated by human effort.  The author does not use AI engines and never will.

HP 71B and HP Prime Python: Gaussian Quadrature

HP 71B and HP Prime Python: Gaussian Quadrature

A Very Brief Introduction

This is a top-level overview of the Gaussian Quadrature method. For more details, please check out the Sources section.

The design of GQ is to calculate:

∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)

Where do xi and wi come from?

* The xi values come from find the roots of the polynomials P’(x)= 0

* Pn’(x) are the derivatives of the Legendre polynomials Pn(x)

* After getting the roots, xi, the weights, w are calculated by the formula:

wi = 2 / ((1 -xi)^2 * Pn’(xi)^2)

Weights and Points of orders 3 and 5:

Among 3 points:

Roots (x) come from the derivative of the Legendre polynomial of the 3^rd order.

Pn’(x) = 3/2 * (5*x^2 – 1)

Point xi	Weight wi
-√(3/5) ≈ -0.7745 9669 2 = -√0.6	5/9
0	8/9
√(3/5) ≈ 0.7745 9669 2 = √0.6	5/9

Among 5 points:

Roots (x) come from the derivative of the Legendre polynomial of the 5^th order.

Pn’(x) = 15/8 * (21*x^4 – 14*x^2 + 1)

Point xi	Weight wi
-0.90617 98459	0.23692 68851
-0.53846 93101	0.47862 86705
0	128/225 ≈ 0.56888 88889
0.53846 93101	0.47862 86705
0.90617 98459	0.23692 68851

We can fit any interval [a, b] by this conversion:

∫( f(x) dx, a, b) = (b – a) / 2 * ∫( f( (b – a) / 2* x + (b + a) / 2 dx, -1, 1)

= (b – a) / 2 * Σ( wi * f( (b – a) / 2 * xi + (b + a) / 2 ), i = 1, n)

However, the programs on this blog entry will focus on the basic version of the Gaussian Quadrature.

The Code

The following code works with the following integrals:

∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)

HP 71B Basic

HP 71B: GQ3 (Gaussian Quadrature – 3 Point: Integrals from -1 to 1)

10 DEF FNF(X) = insert function of X here

15 S = 0

20 RADIANS

25 FOR I = 1 TO 3

30 READ X, W

31 DATA -SQR(0.6),5/9

32 DATA 0,8/9

33 DATA SQR(0.6),5/9

40 S = S + W * FNF(X)

55 NEXT I

50 DISP “ INTEGRAL= ”; S

HP 71B: GQ5 (Gaussian Quadrature – 5 Point: Integrals from -1 to 1)

10 DEF FNF(X) = insert function of X here

15 S = 0

20 RADIANS

25 FOR I = 1 TO 5

30 READ X, W

31 DATA -.9061798459, .2369268851

32 DATA -.5384693101, .4786286705

33 DATA 0, 128/255

34 DATA .5384693101, .4786286705

35 DATA .9061798459, .2369268851

40 S = S + W * FNF(X)

45 NEXT I

50 DISP “ INTEGRAL= ”; S

Integral Test: ∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)

Function f(x)	Actual (approx)	Gaussian 3 point (GQ3)	Gaussian 5 point (GQ5)
(x-3)*(x+4)*(x-5)	352/3 ≈ 117.333333333	117.333333333	117.333333334
-x^2 + 9*x + 10	58/3 ≈ 19.33333333	19.3333333334	19.3333333333
1.5 * cos(x)	2.524412954	2.52450532159	2.52441295561
exp(-x)	2.350402387	2.35033692868	2.35040238646
exp(sin(x))	2.283194521	2.28303911433	2.28319665353
sin(x)	0	0	0
1/(x - 2)	-1.09861228867	-1.09803921569	-1.09860924181

HP Prime Python App

HP Prime Python: gq3.py

# Gaussian Quadrature 3 point

from math import *

def f(x):

  # insert Function here

  return 1/(x-2)

s=0

x=[-sqrt(0.6),0,sqrt(0.6)]

w=[5/9,8/9,5/9]

for i in range(3):

  s=s+w[i]*f(x[i])

print("Integral: ",s)

HP Prime Python: gq5.py

# Gaussian Quadrature 5 point

from math import *

def f(x):

  # insert Function here

  return sin(x)

s=0

x=[0,-.5384693101056831,.5384693101056831,-.9061798459386640, .9061798459386640]

w=[.5688888888889,.4786286704993665,.4786286704993665,.2369268850561891,.2369268850561891]

for i in range(5):

  s=s+w[i]*f(x[i])

print("Integral: ",s)

Function f(x)	Actual (approx)	Gaussian 3 point (GQ3)	Gaussian 5 point (GQ5)
(x-3)*(x+4)*(x-5)	352/3 ≈ 117.333333333	117.333333333	117.333333333
-x^2 + 9*x + 10	58/3 ≈ 19.33333333	19.3333333333	19.3333333334
1.5 * cos(x)	2.524412954	2.52450532159038	2.52441295561081
exp(-x)	2.350402387	2.35033692868001	2.35040238646284
exp(sin(x))	2.283194521	2.2830391143268	2.28319665352905
sin(x)	0	0.0	0.0
1/(x - 2)	-1.09861228867	-1.09803921568627	-1.09860924181248

Sources

Kamermans, Mike “Pomax”. “Gaussian Quadrature Weights and Abscissae” https://pomax.github.io/bezierinfo/legendre-gauss.html 2011. Retrieved January 19, 2025.

Wikipedia. “Gauss-Legendre quadrature” https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_quadrature Last edited January 19, 2025. Retrieved January 19, 2025.

Wikipedia. “Legendre Polynomials” https://en.wikipedia.org/wiki/Legendre_polynomials Last edited December 4, 2024. Retrieved January 19, 2025.

When it comes to evaluating integrals numerically, is Gaussian Quadrature better than Simpson’s Method?

Eddie

 All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

The content on this blog is 100% generated by humans. The author does not use AI engines and never will.