Casio fx-991CW: Editing Variables

Casio fx-991CW: Editing Variables

Introduction

The newer set of Casio scientific calculators, better known as the Classwiz series, revised the way variables are dealt with. For the fx-991CW, storing and editing are dealt with through the Variable button, [ ⇄x ]. We have two options for recalling values of variables, one by the aforementioned variable button or by pressing [ SHIFT ] and the key that holds the variable that is needed.

In the Variables screen, use the key arrows to select a variable then press either [ OK ] or [ EXE ]. Depending on the calculator’s state and mode, we are given one of three options: Store, Edit, and/or Recall.

While editing variables, we are not restricted to just entering numbers. We can enter expressions that can be evaluated. This give us the ability to store several values in one screen.

Apps on the fx-991CW: Ability to Edit Variables in the Variable Screen

App	Access	Notes
Calculate	Yes
Statistics	Yes
Distribution	Yes
Spreadsheet	Yes
Table	Yes
Equation	Yes
Inequality	Yes
Complex	Yes	Catalog-Complex menu is limited to i, ∠
Base	No
Matrix	Yes
Vector	Yes
Ratio	Yes
Math Box	Only during input and Relative Freq (frequency) results screens

Notes:

* In the Variable Edit screen, the Catalog-Func Analysis menu contains only the functions logab, log, and ln.

* The [ ↓FORMAT↑ ] button is inactive in the Variable Edit screen.

* The [ f(x) ] (FUNCTION) button is inactive in the Variable Edit screen.

* Expressions can include other variables including its own.

* Executing the expression by pressing the [ EXE ] also stores the result to the variable.

Examples

Example 1: Use the Variable Edit screen to store the following:

A = 2^2, B = 2^4, and C = 2^3

Keystrokes:

[ ⇄x ] (variable key) select A, select Edit, 2 [ x² ] [ OK ]**

Select B, select Edit, 2 [☐^☐] 3 [ OK ]

Select C, select Edit, 2 [☐^☐] 4 [ OK ]

Result:

A = 4, B = 8, C = 16

Example 2: Use the Variable Edit screen to store the following:

E = 11/5, F = 11/(5 + E), x = 11 * F/E

Keystrokes:

[ ⇄x ], select E, select Edit, 11 [ ÷ ] 5 [ OK ]

Select F, select Edit, 11 [ ÷ ] [ ( ] 5 [ + ] [ SHIFT ] (E) [ ) ] [ OK ]

Select x, select Edit, 11 [ × ] [ SHIFT ] ( F ) [ ÷ ] [ SHIFT ] (E) [ OK ]

Result:

E = 2.2, F = 1.527777777, x = 7.638888888

** or [ EXE ]

Side note: it’s fun to find and use Unicode symbols, like ⇄ (21C4) and ☐ (2610).

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 author does not use AI engines and never will.

fx-991 CW: Finding the Eigenvalues of a 2 x 2 Matrix

fx-991 CW: Finding the Eigenvalues of a 2 x 2 Matrix

Finding the Eigenvalues of a 2 x 2 Matrix

To find the eigenvalues of a matrix M, the characteristic polynomial must be solved in terms of λ:

det( M – λ * I) = 0

where:

M = a square matrix of dimension n x n

I = an identity matrix of size n x n.

The identity matrix is a square matrix in which all elements have a value 0 except the diagonal elements, which have the value 1.

A 2 x 2 identity matrix: [ [ 1, 0 ] [ 0, 1 ] ]

A 3 x 3 identity matrix: [ [ 1, 0, 0 ] [ 0, 1, 0 ] [ 0, 0, 1 ] ]

For a 2 x 2 matrix:

M = [ [ a, b ] [ c , d ] ]

The characteristic polynomial used to find the eigenvalues are:

det( M – λ * I) = 0

det( [ [ a, b ] [ c , d ] ] - λ * [ [ 1, 0 ] [ 0, 1 ] ] ) = 0

det( [ [ a, b ] [ c , d ] ] - [ [ λ, 0 ] [ 0, λ ] ] ) = 0

det( [ [ a - λ, b ] [ c , d - λ ] ] ) = 0

(a – λ) * (d – λ) – b * c = 0

λ^2 – (a + d) * λ + (a * d – b * c) = 0

Note:

trace(M) = a + d

det(M) = a * d – b * c

The characteristic equation to be solved is:

λ^2 – (a + d) * λ + (a * d – b * c) = 0

λ^2 – trace(M) * λ + det(M) = 0

Casio fx-991CW Algorithm

The algorithm will involve two apps: Matrix and Equation. Here is a way of finding the eigenvalues of a 2 x 2 matrices using only the fx-991 CW calculator without the need for writing anything down.

Note: The variables I use in the procedure is just for illustrative purposes. You can use any variables you want to designate the corner elements, the trace, and the determinant. The point is to be organized.

Variables used in this procedure:

A = upper-left element

B = lower-right element

C = trace = A + B

D = determinant

Settings: It is assumed that the MathI/MathO Input/Output mode and a+bi is selected for Complex result.

The screen shots are generated using the ClassPad Math (classpad.net) emulator for the fx-991CW and illustrates finding the eigenvalues of the matrix:

MatA = [ [ 4, 2 ] [ 5, 4 ] ]

Matrix App

Step 1: Press the [ HOME ] key, select the Matrix app.

Step 2: Use the TOOLS key to define a Matrix of dimension 2 x 2.

Step 3: Enter the matrix’s elements. Register each element is registered by using [ OK ] or [ EXE ]. Be careful not to press [ EXE ] or [ OK ] without entering a value first, as we need to keep the matrix editing screen up.

Step 4: Go to element (1,1) (upper-left hand corner), press the [ VARIABLE] key, go to variable A, press [ OK ], and select Store. Then go to element (2,2) (lower-right hand corner), press the [ VARIABLE] key, go to variable B, press [ OK ], and select Store.**

Step 5: Without entering or editing an element, press either [ EXE ] or [ OK ] to leave the matrix editor. When the message “Press [TOOLS] to define Matrix.” appears, we are in the calculation mode of the Matrix app.

Note: If you leave the matrix edit mode before storing the corner elements, you can go back into matrix edit mode by pressing [ TOOLS ], selecting your matrix, and then selecting Edit. You can check to see if corner values are stored by pressing [ VARIABLE ].

Step 6: Press [ SHIFT ] [ 4 ] (A) + [ SHIFT ] [ 7 ] ( B ) [ EXE ]. This calculates the matrix’s trace. Then press [ VARIABLE ], choose C, press [ OK ], choose Store.

Step 7: Press [ CATALOG ], select the Matrix sub-menu, then the Matrix Calc sub-menu, and select Determinant. Then use the catalog to grab the matrix and press [ EXE ] to calculate the matrix. Use the variable list to store the value in D (similar procedure in Step 6).

When transferring between apps, the values stored in the memory registers A-F, x, y, and z are retained, even when the fx-991CW is turned off.

Equations App

Step 8: Press [ HOME ] and select the Equation app and press [ OK ]. Select Polynomial, ax²+bx+c (2^nd order polynomial, quadratic equation).

Step 9: Enter the following coefficients: 1 x² – C x + D (note the minus sign on C).

Step 10: Press [ OK ]. The first eigenvalue is displayed. Press the down arrow ([↓]) to get the other eigenvalue. 

 An optional step is to use the [ VARIABLE ] key to store the results (like in E or F, for example). 

 Another optional step is to press [ FORMAT ], select Decimal to see the decimal approximation.

The results are:

C = trace = 8

D = determinant = 6

Eigenvalues:

λ1 = 4 + √10

λ2 = 4 - √10

Other Examples

Find the eigenvalues of Mat B = [ [ -8, 1 ] [ 16, 7 ] ]

Results:

C = trace = -1

D = determinant = -72

Eigenvalues:

λ1 = 8

λ2 = -9

Find the eigenvalues of Mat C = [ [ -5, -7 ] [ 3, - 2] ]

Results:

C = trace = -7

D = determinant = 31

Eigenvalues:

λ1 = (-7 + 5 * √3 * i) / 2

λ2 = (-7 - 5 * √3 * i) / 2

I hope you find this useful and beneficial. 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.

Casio fx-991CW: Recurrence Sequences

Casio fx-991CW: Recurrence Sequences

Introduction

Today’s edition we will work with recurring sequences, which can take the form of:

u_n = f(u_n-1), with initial condition u0

u_n = f(u_n-1, u_n-2) with initial conditions u0, u1

In general:

u_n = f(u_n-1, u_n-2, … u_n-k) with initial conditions u0, u1, u2, …, u_n-(k-1)

There is no app on the Casio fx-991CW that is dedicated to be recurrence sequences.

Recurring Sequences the Quick and Easy Way with the Calculate App

A quick way to tackle recurrence sequences and involves the use of the last answers key ( [Ans] ). This works on recurrence sequences with only one initial condition, u0.

Start by entering u0 and press [ EXE ]. Then enter the recurrence function, using Ans as u_(n-1).

Example:

u_n = 0.75 * u_(n-1)^2 + 1.03 * u_(n-1) with u0 = 1

Settings: MathI/DecimalO (Input/Output), Number Format (Norm 1)

1 [ EXE ]

0.75×Ans²+1.03×Ans [ EXE ] (Disp: 1.78)

[ EXE ] (Disp: 4.2097)

[ EXE ] (Disp: 17.62717157)

[ EXE ] (Disp: 251.1938698)

[ EXE ] (Disp: 47582.49986)

…

Recurrence Sequences with the Spreadsheet App

A way to generate recurrence sequences is to use the Spreadsheet app. The app contains a spreadsheet of size 5 columns (A, B, C, D, E) by 45 rows. Depending on the number of the initial conditions, up to 44 terms can be generated.

For the purposes of this blog, we’ll use column A, but we can use any column, as long as there is memory in the Spreadsheet.

u_n = f(u_n-1), with initial condition u0

Set A1 as the value of u0.

At cell A2:

1. Press [ TOOLS ].

2. Select Fill Formula and press [ OK ].

3. Enter the recurrence function, using A1 as u_(n-1).

4. At the range, set as A2:A#, to create a list of #-1 terms. For example, the range A2:A11 to create a list of 10 terms.

5. Select Confirm and press [ OK ].

Example:

Under Show Cell, Value (using the [TOOLS] menu)

u_n = (u_(n-1) + 0.5) * u_(n-1) with u0 = 0.65 for 7 terms.

Cell A1 = 0.65

Cell A2:

Fill Formula:

Form = (A1+0.5)×A1

Range: A2:A8

	A (display)
1	0.65
2	0.7475
3	0.93250625
4	1.335821031
5	2.452328343
6	7.240078475
7	56.03877556
8	3168.363754

u_n = f(u_n-1, u_n-2), with initial condition u0, u1

Set A1 as the value of u0.

Set A2 as the value of u1.

At cell A3:

1. Press [ TOOLS ].

2. Select Fill Formula and press [ OK ].

3. Enter the recurrence function, using A1 as u_(n-2) and A2 as u_(n-1).

4. At the range, set as A3:A#, to create a list of #-2 terms. For example, the range A2:A11 to create a list of 9 terms.

5. Select Confirm and press [ OK ].

Example:

u_n = u_(n-1) + u_(n-2), u0 =1 , u1 = 1 for 7 terms.

Yes, this is ripe for the Fibonacci sequence.

Cell A1 = 1

Cell A2 = 1

Cell A3:

Fill Formula:

Form = A1+A2

Range: A3:A9

	A (display)
1	1
2	1
3	2
4	3
5	5
6	8
7	13
8	21
9	34

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.

Casio fx-991CW: The Inequalities Mode

Casio fx-991CW: The Inequalities Mode

Solving Inequalities

The Casio fx-991CW, along with other high-end scientific calculators from Casio has an Inequalities mode.

The inequalities mode solves quadratic, cubic, and quartic polynomials. The mode works similar to Equation-Polynomial module. Below is an example (all screenshots were created in Casio’s classpad.net website):

The Inequalities mode can assist in calculus and geometry problems.

Circle Problem

Task: Find valid points (x, y) that fit in the circle:

x^2 / 3 + y^2 / 6 = 1

Step 1: We can use the Inequalities mode to find the allowed x values.

Start by solving for y:

x^2 / 3 + y^2 / 6 = 1

6 × (x^2 / 3 + y^2 / 6) = 1 × 6

2 × x^2 + y^2 = 6

y^2 = 6 – 2 × x^2

y = √(6 – 2 × x^2)

For the equation to be “valid”, or to not return complex numbers, find all x where 6 – 2 × x^2 is not negative.

6 – 2 × x^2 ≥ 0

We find that the allow range is -√3 ≤ x ≤ √3.

Step 2: Store the Equations

We can do define f(x) in any mode. To do this, press [ f(x) ], select Define f(x). Press [ OK ] to store the function. The fx-991CW allows for two functions, f(x) and g(x).

In this example, we defined f(x) and g(x) as follows:

f(x) = √(6 – 2 × x^2)

g(x) = -f(x)

Step 3: Use the Table mode to find values in the appropriate range.

Set the Table Type as f(x)/g(x). Set the Table Range as Start: -√(3), End: √(3), Step: 2×√(3)÷5

Table:

	x ≈	f(x) ≈	g(x) ≈
1	-1.732	0	0
2	-1.0392	1.9595	-1.9595
3	-0.3464	2.4	-2.4
4	0.3464	2.4	-2.4
5	1.0392	1.9595	-1.9595
6	1.732	0	0

The Area Between f(x) and x ≥ 0

Find the area between the curves:

f(x) = -x^4 + 2 × x^3 + 25 × x^2 – 26 × x – 120 and x ≥ 0.

Step 1: Find the boundaries.

With the Inequalities mode, we can find the boundaries and intersection points where f(x) ≥ 0. We find that: -4 ≤ x ≤ -2 and 3 ≤ x ≤ 5. Write down these results. The area will be in the intervals [-4, -2] and [3, 5].

Confession: I ultimately chose this function because of its integer roots. Unfortunately we are not able to store coefficients or roots in variables directly on the fx-991CW to variables.

Step 2: Store the Equations

Store -x^4 + 2 × x^3 + 25 × x^2 – 26 × x – 120 to f(x). This will allow us to save keystrokes while calculating the area.

Step 3: Calculate the Area in the Calculate Mode

Use the integral function from the Catalog - Func Analysis menu.

∫( f(x), -4, -2) + ∫( f(x), 3, 5)

The area is: 1928/15 ≈ 128.5333333

Hint: To always get a decimal approximation (skipping the exact result), press [SHIFT] [ = ] ( ≈ ).

Another example:

Find the area between the curves:

f(x) = x^3 + 8 × x^2 – 3 and x ≥ 0.

The boundaries are: -7.952564217 ≤ x ≤ -0.63837179, 0.5909359176 ≤ x

Which means the intervals are: [ -7.952564217, -0.63837179] and [ 0.5909359176, ∞). We can see that the area given these boundaries, the area will approach infinity.  

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.