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.
