ECE 220 users guide for HP graphing calculators (HP 48g thru 50g and g+ series)
Created by Christopher D. Peveler Sr. 4/14/2010
Section 1: Complex Numbers
The first step in using you HP to solve equations involving complex numbers is to enable complex mode. Often this is not the default setting. To verify your machine is setup to solve complex equations, make sure the complex box is checked in the CAS setup menu.
The next step in solving equations involving complex numbers is how to get them into the calculator. You may enter values in both polar and rectangular form. The calculator will handle mixed modes in a single equation. You do not need to convert to all one form first. All complex numbers, whether in polar or rectangular form, should be entered in to parenthesis (). As an example to enter the complex number 3.5-j1.2 you would press:
For entering in polar form you must make use of the special angle character, which is accessed using the key sequence So to enter the complex number 3.5-1.2 you would press:
That covers the basics of getting the numbers into your calculator. Now let’s look at getting your answers out of the calculator in the form that you need. You have two choices: polar or rectangular. You may switch between modes at any time. This will only affect the way the answers are shown, not how you enter them. To select which mode the answers are presented press the mode button then select either polar or rectangular from Coord System menu.
For conversion type problems, simply enter the number in the form it is given, press enter, then change the Coord System to the desired type. When you return to the display, the number will be shown in the proper form.
Let’s look at an example problem from HP’s Advanced user guide:
This is a very simple example, but demonstrates how to enter the numbers and interpret the answer given.
Nearly all math function that may be performed with real numbers on this calculator, may also be done using complex numbers. This includes matrix operations. In the next section we will examine how to solve a system of equations involving complex numbers using the matrix functions.
Section 2: Solving Systems of Equations
There are several methods for solving systems of equations simultaneously built into the calculator. We will examine the one that uses linear algebra and matrices to do this. By forming a matrix from the coefficients and another from the answers, it is possible to solve N equations for N number of unknowns at the same time. This method will also work for complex coefficients and answers. Here are some examples of the types of equations we are talking about:
EQN #1
2X+3Y-Z = 11
4X-3Y-2Z = 4
X-4Y+5Z = 0
Or something more complicated like:
EQN #2
(4+j37)X1 + (270-j3)X2 = (95-30)
(-52-j42)X1 + (56162)X2 = (67-j12)
4
0
2 / 3 / -1
4 / -3 / -2
1 / -4 / 5
While these systems look vastly different, they are solved by the same method. Strip away whatever is in front of the variables (real or complex) and create an NxN matrix from the coefficients. We will refer to this as the A matrix. Next we create an Nx1 matrix from the answers. We will call this matrix B. We can just forget the variable part or now. These are the answers the calculator will return to us in the X matrix. So for EQN #1 the A matrix is The B matrix is
(95-30)(67-j12)
(4+j37) / (270-j3)
(-52-j42) / (56162)
And for EQN #2 A would be And B would be
As mentioned before the method for getting the calculator to solve both of these is exactly the same. Start by accessing the solver menu by pressing this will bring up the numerical solver menu:
We want option 4 from this menu-- Solve Linear System. Press and the linear system entry screen will be displayed:
This is where we will enter our matrices. While the blank for A: is highlighted press and the MatrixWriter will be opened.
The MatrixWriter works just like a spreadsheet. Move around using the arrow keys and enter the elements from the A matrix we created earlier. **** Don’t forget if the elements are complex numbers you MUST use parenthesis () **** i.e. (4+j37). Once all the elements of A have been entered, click on an empty cell and you will be returned to the linear system screen.
Highlight B: and press to access the MatrixWriter once again and enter the B matrix elements. When finished press enter on an empty cell and you will be returned to the Linear system screen.
Finally highlight the X: blank. This time you want to press the softkey. Answers will be returned in the X matrix. You may need to press the followed by to reveal the desired number of significant digits.
Here is another complete example from HP’s Advanced user manual: