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
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