Casio fx-991 CW: Solving Linear Systems of Complex Numbers

Casio fx-991 CW: Solving Linear Systems of Complex Numbers

Introduction

The fx-991 CW is a capable calculator. It even handles complex number calculations and linear systems up to four variables. So we should be able to handle problems such as:

(A + Bi) * (x + yi) + (C + Di) * (z + ti) = E + Fi

(G +Hi) * (x + yi) + (J + Ki) * (z + ti) = L + Mi

where i=√-1 and A, B, C, D, E, F, G, H, J, K, L, and M are real and imaginary parts of complex numbers.

Well, yes. But the Equation mode does not allow for complex numbers, and the complex mode handles arithmetic and rectangular/polar conversions. True. However, all is not lost.

If we multiply the left side of both equations, we get:

(A * x – B * y + C * z – D * t) + (B * x + A * y + D * z + C * t)i = E + Fi

(G * x – H * y + J * z – K * t) + (H * x + G * y + K * z + J * t)i = L + Mi

Equating the real and imaginary parts, we get four equations since the real and imaginary parts are separated by addition.

A * x – B * y + C * z – D * t = E (real part)

B * x + A * y + D * z + C * t = F (imaginary part)

G * x – H * y + J * z – K * t = L (real part)

H * x + G * y + K * z + J * t = M (imaginary part)

In this form, we can use the linear system solver of the fx-991CW (as well as many other calculators).

In matrix form:

[ [ A, -B, C, D ] [ [ x ]   [ [ E ]

[ B, A, D, C ]   * [ y ]  = [ F ]

[ G, -H, J, -K ]   [ z ]    [ L ]

[ H, G, K, J ] ]   [ t ] ]  [ M ] ]

Casio fx-991CW: Algorithm

Step 1. Press the [ HOME ] key and select Equation.

Step 2. Select Simul Equation.

Step 3. Select 4 Unknowns.

Step 4. Set up the equations as follows:

A * x – B * y + C * z – D * t = E

B * x + A * y + D * z + C * t = F

G * x – H * y + J * z – K * t = L

H * x + G * y + K * z + J * t = M

Step 5. Press [ EXE ], or [ SHIFT ] [ EXE ] (≈) for all approximate solutions. Scroll to see the results x, y, z, and t. In the default setting, every answer, except for x is expressed in standard form. I’m not sure why this is. This is your solution to x + yi and z + ti.

Step 6. To do a new calculation, press [ EXE ] again.

Examples

Example 1:

(6 + 4i) * (x + yi) + (2 + 5i) * (z + ti) = 11 + 63i

(8 + 14i) * (x + yi) + (16 + 7i) * (z + ti) = 213 + 105i

Set up the four equations as:

6x – 4y + 2z – 5t = 11

4x + 6y + 5z + 2t = 63

8x -14y + 16z – 7t = 213

14x + 8y + 7z + 16t = 105

Results:

x ≈ -1.682508574

y = -2109 / 2041 ≈ -1.033317001

z = 2239 / 157 ≈ 14.2611465

t = 363 / 157 ≈ 2.312101911

x + yi ≈ -1.682508574 – 1.033317001i

z + ti ≈ 14.2611465 + 2.312101911i

Example 2:

(3 + 3i) * (x + yi) + (-9 + 7i) * (z + ti) = -4 + 2i

(-5 + 3i) * (x + yi) +(2 + 8i) * (z + ti) = 11 + 0i

Set up the four equations as:

3x – 3y - 9z – 7t = -4

3x + 3y + 7z - 9t = 2

-5x - 3y + 2z – 8t = 11

3x - 5y + 8z + 2t = 0

Results:

x ≈ -1.252444271

y = -1245 / 5114 ≈ -0.2434493547

z = 2131 / 5114 ≈ 0.4166992569

t = -2029 / 5114 ≈ -0.3967540086

x + yi ≈ -1.252444271 – 0.2434493547i

z + ti ≈ 0.4166992569 – 0.3967540086i

Until next time, take care,

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.

A Comparison of Algorithmique vs. Python Turtle (Casio fx-92 Collège vs. Numworks)

A Comparison of Algorithmique vs. Python Turtle (Casio fx-92 Collège vs. Numworks)

The Casio fx-92 Collège from France has an algorithm mode, which is essentially a program mode. The language is designed to be an easy-to-use program and can be viewed online as Scratch program code. I would love to see Casio sell this calculator world wide instead on just France, I think this calculator would give an excellent introduction to coding (at any age).

On today’s blog, I’m going to compare the Algorithmique programming language to Python’s Turtle module in drawing three simple shapes. The calculator I’m using for Python is Numworks. I’m forever grateful to Google Translate, as I am not fluent in French.

The Commands

Casio fx-92 Collège Algorithmique French	Casio fx-92 Collège Algorithmique French	Python Turtle Command (Numworks)	Python Built-In Command
Avancer de n	Move forward n	forward(n)
Tourner de ⟲ Θ	Turn from ⟲ Θ	Θ > 0: left(Θ) Θ < 0: right(Θ)
S’ orienter à Θ	Orient yourself to Θ	setheading(Θ)
Aller à x; y	Go to x; y	goto(x,y)
Stylo écrit	Pen writes	pendown()
Stylo relevè	Pen raised	penup()
Metire var á (Shown as expression → var)	Set var to (Shown as expression → var)		var = expression
Demander valeur (Shown as ? → var)	Ask for a value (Shown as ? → var)		var = input(“optional prompt string” ) [float, eval, int, default is a string]
Commenataire (shown as one of four comments)	Comment out of four per-programmed strings		print(“almost anything you want”)
Afficher résult var	Show result var		print(var)
Style	Style (Style of Cursor)	Turtle has a turtle character. Show or hide by showturtle()/hideturtle()
Attendre	Pause	N/A	N/A (though we can use an artificial for loop that does nothing)
Répéter n ( ↑ )	Repeat n ( ↑ )		For i in range(n):
Répéter jusqúa cond	Repeat until cond		While (not cond):
Si Alors (Fin)	If Then (End)		If: structure
Si Alors Sinon (Fin)	If Then Else (End)		If: Else: structure

The four per-programmed strings are:

“Oui” Yes

“Non” No

“Nombre?” Number?

“Résultat :” Result :

Using a per-programmed string pauses the algorithm and requires a press of either [ OK ] or [ EXE ] to continue.

The fx-92 Collège offers to styles of cursors: Arrows (Fléche) or Cross (Criox)

The pixel screen of the fx-92 Collège is 191 x 48 pixels, with the x coordinates ranging from -95 to 96 and the y coordinates raging from -24 to 48.

The pixel screen (Turtle) for the Numworks is 320 x 222 pixels, with the x coordinates ranging from -160 to 160 and the y coordinates ranging from -111 to 111.

Drawing a Square

fx-92 Collège:

? →A

Stylo relevé

Avancer de A÷2 pixels

Stylo écrit

Tourner de ⟲ 90 degrés

Avancer de A÷2 pixels

Répéter 3

  Tourner de ⟲ 90 degrés

  Avancer de A pixels

↑

Tourner de ⟲ 90 degrés

Avancer de A pixels

Numworks:

from math import *
from turtle import *

print("Draw a Square")
s=eval(input("side length? "))

penup()
forward(s/2)
pendown()
left(90)
forward(s/2)

for i in range(3):
  left(90)
  forward(s)

left(90)
forward(s/2)

Drawing a Triangle

fx-92 Collège:

? →A

? →B

Stylo relevé

Aller à x= －A÷2 ; y= －B÷2

Stylo écrit

Aller à x= A÷2 ; y= －B÷2

Aller à x= 0 ; y= B÷2

Aller à x= －A÷2 ; y= －B÷2

Numworks:

from math import *
from turtle import *

print("Draw a Triangle")
s=eval(input("base? "))
h=eval(input("height? "))

penup()
goto(-s/2,-h/2)

pendown()
goto(s/2,-h/2)
goto(0,h/2)
goto(-s/2,-h/2)

Drawing a Circle

fx-92 Collège:

? →A

Stylo relevé

Aller à x= A ; y= 0

Stylo écrit

0 →z

Répéter 120

  z＋3 →z

  Aller à x= A×cos(z°) ; y= A×sin(z°)

↑

Numworks:

from math import *

from turtle import *

print("Draw a Circle")
r=eval(input("radius? "))

penup()
right(90)
forward(r)
left(90)
pendown()
circle(r)


The Algorithmique (similar to Scratch) and Turtle language is comparable and these examples demonstrates similarities and differences of both languages.  

Source

Casio. “Algorithmque/Programmation” (French)

https://www.casio-education.fr/algorithmique-programmation/ Retrieved January 26, 2025

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.

HP Prime and fx-CG 50: Percentage of a Mortgage Paid

HP Prime and fx-CG 50: Percentage of a Mortgage Paid

Introduction

The program PERMORTGAGE for the HP Prime, and PERMORT for the Casio fx-CG 50 calculates the percentage of mortgage paid off any time during the mortgage’s term.

Inputs:

N = The length of the mortgage (or loan) in number of monthly payments. The payments are assumed to be made at the end of the month.

R = The annual rate of the mortgage. Assume that this loan is fixed.

L = The amount of the loan.

D = The number of payments already made.

Assume that there is no balloon payment.

The % of mortgage paid is calculated by the following steps:

Step 1: Let P by the monthly payment: P = PMT(N, R, L, 0, 12, 12)

(the last two arguments are payments per year and compounding periods per year, both set at 12)

Both calculators featured use the cash flow convention. That is, all cash inflows (receipts) are positive and cash outflows (payments) are negative. In this case, the monthly payment (P) and balance remaining (B) are negative.

Step 2: Let B be the approximate balance using the FV (future value) function:

B = FV(D, R, L, P, 12, 12)

Step 3: Calculate the % of mortgage paid:

T = (1 + B / L ) * 100%

HP Prime Program Code: PERMORTGAGE

Syntax: PERMORTGAGE( nterm, rate, loan, npaid )

nterm = number of monthly payments for the entire term. Example for a 30 year term, nterm = 360

rate = annual rate of a mortgage

loan = loan amount

npaid = number of payments made

Result: % of the principal paid

Code:

EXPORT PERMORTGAGE(nt,rate,loan,np)

BEGIN

// n term, rate, loan,,n paid

// Percent of a Mortgage Paid

// Monthly payments assumed, end mode assumed

// Assume no balloon payment

// EWS 2024-11-01

LOCAL pymt,prct,baln;

pymt:=Finance.TvmPMT(nt,rate,loan,0,12,12);

baln:=Finance.TvmFV(np,rate,loan,pymt,12,12);

// PMT and FV will be negative

prct:=(1+baln/loan)*100;

RETURN prct;

END;

Casio fx-CG 50 Program: PERMORT

The program asks for a single calculation or range of calculations which is stored in Matrix Mat A. (The first row is has two zeros, and is used as a “header”.) The first column is the number of payments made, the second is the percentage of mortgage (loan) paid.

Code:

PmtEnd

“N (TERM)”? → N

“RATE”? → R

“LOAN AMT”? → L

Menu “TYPE”,”SINGLE”,1,”RANGE”,2

Lbl 1

Cmpd_PMT(N,R,L,0,12,12) → P

Cmpd_FV(D,R,L,P,12,12) → B

(1+B÷L)×100 → T

“PERCENT PAID =”

T

Stop

Lbl 2

“N1”? → A

“N2”? → B

“STEP”? → C

[ [ 0 ] [ 0 ] ] → Mat A

For A → D To B Step C

Cmpd_PMT(N,R,L,0,12,12) → P

Cmpd_FV(D,R,L,P,12,12) → B

(1+B÷L)×100 → T

Augment(Mat A, [ [ D ] [ T ] ]) → Mat A

Next

Trn Mat A → Mat A

“IGNORE TOP ROW - “

“ [ N PER ]” ◢

Mat A

Stop

Example

$100,000 loan over 30 years (360 payments). Screen shots were taking with the HP Prime emulator.

Note that the % paid takes on a curve, the later we get into the term, the more principal is paid off. This program can be a demonstration of how amortization works.

Until next time,

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.

Spotlight: Casio fx-92 Collège

Spotlight: Casio fx-92 Collège

Quick Facts

Model: fx-92 Collège

Company: Casio

Timeline: 2023-present

Type: Scientific with Algorithm programming, 10 digits, Algebraic with Text Writing

Memory: 9 variables (A-F, x, y, z), 900 bytes for a script space

Power: 1 AAA battery

The fx-92 Collège is a scientific calculator designed for French school students. The calculator keys, functions, instructions, and commands are all in French. I purchased this calculator from a Swiss vendor online. The fx-92 Collège is sold in France, and I wish it was sold worldwide.

Modes

The fx-92 Collège has these following modes:

Calcul	Calculate Mode: This is the regular calculator mode. The input and output can be set to 2D input/output or linear. The 2D input/output shows mathematical calculations as they are written naturally. Also, fractions, terms of pi (π), and exact square roots (collectively known as QIRAC) are shown in 2D mode.
Stats	Statistics Mode: 1 and 2 statistics mode. The regression offered is linear regression.
Tableur	Spreadsheet: 45 rows x 5 columns. The spreadsheet has a capacity of 2,380 bytes.
Tabl fonct	Function Table: up to 2 functions, f(x) and g(x)
Équation	Equation Mode: Linear systems of orders 2, 3, or 4
Prod. croix	Ratio mode: Solve for X A / B = X / D A / B = C / X
Algo	Algorithmique: Algorithm Mode. This is the programming mode.
Math Box	Math Box: This features four applications: Lancer de dés: Dice Roll (up to 3 dice, up to 250 rolls of the dice) Pile ou face: Coin Toss (up to 3 coins, up to 250 coin tosses) Droite grad.: Graph line intervals, up to 3 (x<a, x≤a, x=a, x>a, x≥a, a<x<b, a≤x<b, a<x≤b, a≤x≤b) Cercle: Circle app (trigonometric circle, semi-circle, hourly clock (Horloge))

The fx-92 Collège is part of the Casio’s Classwiz series (think fx-991CW and the international fx-82CW). The calculator has a back button, four direction keys, an [ OK ] button, a button acts like a scroll button. The home button (ACCUEIL) calls up the modes while the configuration (CONFIG) shows the set up options.

The calculator uses a comma as a fraction indicator instead of a decimal point to align with how numbers are written in France.

In France, the approximation of √92 is written as 9,591663047.

In the United States, the approximation of √92 is written as 9.591663047.

Mathematical Functions

Some of the featured functions are:

Euclidean division: |-. [ SECONDE ] [ ÷ ]. This function returns the quotient and remainder.

Example: 2025 |- 47 returns Q = 43; R = 4

Last Answer: [ Rép ]. This is the returns the last answer processed by the previous calculation. This is often labeled elsewhere as Ans.

Fraction Simplification: Simp. [ SECONDE ] [ Rèp ]. Simplifies fractions and rational expressions or attempts to give a fraction representation of a decimal.

Example: .5757575757575757 >Simp returns 19/33.

Logarithm and Natural Logarithm: These are not on the keyboard, but instead found in the CATALOG – Analyse fonction menu.

Exponential Function (e^x): The Euler constant (e ≈ 2,718281…) is found by pressing [ CATALOG ], [ ↑ ], selecting Autre (Other) and selecting e.

Catalog-Probabilitè:

%

Factorielle (Factorial: n!)

Permutation (nPr)

Combinasion (Combinatio: nCr)

Nombre alèatoire (Random Number, Ran#)

Entier alèatoire (Random whole number, RandInt#(low; high))

Note arguments are separated by a colon. (;)

Catalog-Calcul numérique:

PGCD: Greatest common divisor (GCD)

PPCM: Least common multiple (LCM)

Valeur absolue: Absolute value (abs, |x|)

Tronc. À l’unité: Shown as Ent when called, this is the integer part function.

Arrondi: Rounds the number to the fix mode settings

Partie entière: The greatest integer less than x.

Arrondi(;): Rounds the number to any number of decimals.

Example: Arond(π; 4) returns 3927 / 1250 = 3,1416

Catalog-Angl/Coord/Sexag…:

This sub menu has angle designations (x°, x^r (radians), x^g (gradients), polar/rectangular conversions, markers for degrees/hours-minute-seconds calculations).

Catalog-Trigonométrique:

All the trigonometric (sin, cos, tan) functions and their inverses. These six functions are already on the keyboard.

Algorithmique Mode: Algorithm

This is the fx-92 Collège’s programming function. There is one program space, which allows up to 900 bytes of memory. Each command takes at least 4 bytes of memory. The language is similar to Scratch.

The output screen is a split screen. On the top, a graph and drawing screen. The dimensions of 191 pixel wide, 48 pixel height. The graph dimensions are x = [-95,96] and y=[-24,24]. Draws can leave the screen, up to ±999. At the bottom is one line where numerical results and per-programmed messages are shown. Below is a table of the commands that are available:

(Créer un algorithme: Create an algorithm)

French	English	Function
Avancer de n	Move forward n	Move the point n points/pixels
Tourner de ⟲ Θ	Turn from ⟲ Θ	Rotate the arrow Θ degrees
S’ orienter à Θ	Orient yourself to Θ	Turn the pointer to angle Θ
Aller à x; y	Go to x; y	Move pointer to point (x,y)
Stylo écrit	Pen writes	Put the pen down to write
Stylo relevè	Pen raised	Raises the pen to stop writing
Metire var á (Shown as expression → var)	Set var to (Shown as expression → var)	Makes a calculation and sets it to a variable A – F, or z.
Demander valeur (Shown as ? → var)	Ask for a value (Shown as ? → var)	Prompts for a value to be stored in A – F, or z
Commenataire (shown as one of four comments)	Comment out of four per-programmed strings	“Oui” Yes “Non” No “Nombre?” Number? “Résultat :” Result : Pauses execution
Afficher résult var	Show result var	Shows the contents of a variable and pauses execution
Style	Style (Style of Cursor)	Fléche: Arrow Croix: Cross
Attendre	Pause	Pause execution of script
Répéter n ( ↑ )	Repeat n ( ↑ )	Repeat loop, n time (up to 10,000 times). Loop ends at ↑.
Répéter jusqúa cond	Repeat until cond	Repeat a loop until condition cond is met. =, ≠, >, <, ≥, ≤ are present in the CATALOG-ALGO menu
Si Alors (Fin)	If Then (End)	If-Then structure. =, ≠, >, <, ≥, ≤ are present in the CATALOG-ALGO menu
Si Alors Sinon (Fin)	If Then Else (End)	If-Then-Else structure. =, ≠, >, <, ≥, ≤ are present in the CATALOG-ALGO menu

Algorithms start with the pointer at (0, 0) and the pen up. The pointer first starts at angle Θ = 0° (right along the x-axis).

The angle Θ is only accessed in this mode, [CATALOG], Algo, Θ (and only in certain prompts).

The variables x and y refer to the pixel coordinates. They cannot be stored or asked for in this mode.

With each command, you get a prompt screen to fill in the inputs.

When the script pauses, press [ OK ] to continue.  

Also with OUTILS (options):

French	English
Copier & Insérer	Copy & Insert (at Paste)	Press [ OK ] to insert the line.
Insérer linger	Insert Row	Press [ OK ] to insert the line.
Tout supprimer	Delete the Script	To delete a line, press [ ←x ] instead.

The fx-92 Collège is the second Casio calculator to have this type of algorithm mode. It succeeds the fx-92+ Spèicale Collège.

Next Sunday, February 16, we will explore the differences between the fx-92 Collège’s algorithm program and the Turtle module in Python.

Final Thoughts

The algorithm is erased when the calculator is turned off. I wish this wasn’t the case but I can see why the calculator does not retain programs. The fx-92 Collège is designed for teaching basic coding. This is the reason I would like to see this model available readily worldwide, as I think it would sell well. Even though the target audience are middle school students, I think this calculator can be enjoyed by everyone, at any age.

My only gripe: why aren’t the logarithm (log, ln) and exponential (e^x) functions on the keyboard? I get that middle school students probably don’t work with these functions, but the fx-92 Collège is a scientific calculation, and it is customary to have these functions on the keyboard. Not a deal breaker in any stretch of the imagination, just a minor thing for me. I’m glad they are there.

If you get a chance to get this calculator at a reasonable price, buy it. Now I can hope that the AAA battery is used is a universal AAA battery, I probably will have to invest in metric small screwdrivers when it comes time to replace the battery.

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.