Números Complejos en sistemas de ecuaciones con calculadoras CASIO fx-570, fx-991 y fx-GC50 ClassWiz

This video shows how to solve systems of linear equations of complex coefficients with CASIO calculators fx-570 fx-991 and fx-CG50 ClassWiz using: a) The replacement method, b) The method of conversion to a system of real equations. c) The cramer rule (using the calculator's memories). The procedure with the previous calculator model (ES PLUS) is very similar, except that complex operations (conjugated ...) are not in the [OPTN] menu but pressing the [SHIFT] [2] keys and the RECALL command does not show all the memories at the same time on the screen. Some calculators like HP-Prime and its predecessors (HP-50, 49 and 48) respond directly to complex systems, but also to faces and prohibitions. The Casio FX 570, 991 and CG50 calculators have a tool to solve systems of linear equations, but unfortunately this only works with real numbers. The methods to solve systems of linear equations (substitution, equalization, elimination, Gaussian method, inverse matrix method, Cramer's rule, etc.) can also be applied to solve complex systems. When one faces a system of complex equations, it is convenient that it is resolved or reduced. If the system of equations has several null coefficients, I recommend solving it, perform the operations in the complex mode of the calculator. Let's see the following example. The first step is to put the calculator in complex mode. Optionally, I can adjust the angular unit, the complex default notation and delete the memories. When the calculator is in angular sexagesimal mode, the D symbol "Degrees" appears at the top of the screen. You can change the default format. The functions of complex numbers (module, argument, conjugate ...) are in the options menu [OPTN], which has several screens. Then we would eliminate the unknowns. I recommend storing the variables calculated in the memories to speed up the resolution and avoid errors. We can see the variables calculated using the [RECALL] command. To exit this screen, we can press [ON] or the corresponding key to a memory. DEVELOPMENT IN CONVENTIONAL EQUATIONS. Another option is, however, the equation in two reais, a true part of equality. (a + j b) (x + j y) + (c + j d) (z + j t) = e + j f Equality must be met for both the real part and the complex part, for what is desired in two equal. The real part is obtained by distributing the imaginary coefficients with alternative signs: a x - b and + c z - d t = e In the imaginary part, the signs are maintained, but the positions of the real and imaginary components alternate or interchange: b x + ay + d z + c t = f To remember this, a mnemonic rule is to expand a partially real and imaginary product: (a + j b) (x + j y) = (a x - b y) + j (b x + a y) If you use this technique, I recommend that you write the paper equations in advance and pay close attention to avoid making mistakes when entering them into the calculator. This technique requires a practice and a correct conversion because it is easy to make mistakes when transcribing the system. CRAMER RULE In the case that we have a very coupled system that we can not reduce its size easily, I recommend applying the Cramer rule because in 2x2 systems it is quite fast and memories are used. Next, I will solve for Cramer a system of two equations with two unknowns, where the imaginary unit is j. As the complex numbers that appear in the engineering usually have many digits, it is recommended to use the variables or the memories of the calculator to reduce the number of keys pressed and to avoid errors in the data entry. Therefore, I will rewrite the system using the coefficients from A to F: The matrix expanded [M | n] system of linear equations helps to calculate the determinants of Cramer's rule. Then, the calculation of the matrix M, which in a 2 x 2 matrix is ​​the product of the main diagonal minus the secondary diagonal. I keep it in the variable M. We verify that the system has a unique solution. We calculate the unknown x = (CE-BF) / | M |, storing the result in the variable x. The options menu [OPTN] has, in the second screen, an option to display in polar format. With the key [S = D] we see the result in two lines. Finally we calculate the unknown y = (AF-CD) / | M |, storing the result in the variable y. Thank you for your attention and I hope this video has been useful.