6. Complex Numbers on the Calculator - Polar Form

Our first job is to get the calculator options set up for the task at hand. Key MODE DA DA and F2-CHOOS to see a drop down list with three choices of angle measure. Use DA or UA as needed to select Degrees from the list then press F6-OK. This puts the calculator in degree mode. Now key DA and F2-CHOOS to see three choices for coordinate systems. Again use DA or UA as needed to select Polar from this list then press F6-OK. For more information about angle modes and coordinate systems see page 1-22 of UG. To make sure that the constants in the calculator give numeric rather than symbolic results. Key F1-FLAGS then use DA and F3-CHK as needed to make sure that Flags 2 and 3 are checked and Flag 27 is clear, then F6-OK. For more information about the roll of Flag 2 see page 2-64 of UG. Finally press F3-CAS DA DA to highligt "_Approx." The other two items on that row are "_Numeric" on the left and "_Complex" on the right. Use LA, RA, and F3-CHK as needed to check all three of these items, then press F6-OK on this dialog box and on the next to return to the main screen.

Now look at the row of annunciators at the top of the screen. They should be "Degree," "HEX," "C~." and "'X'."

In the calculator, the complex number is expressed as One limitation of the calculator is that it will not accept variables for the magnitude and argument, only numbers. In particular, if it can't be expressed as it must be expressed as As a general rule, this limitation makes working with degrees much easier than with radians. We will, therefore, start with degrees, but we will see some tricks that will make working with radians not quite as bad as it looks. Actually, working in degrees has its own much more subtle problems. Since the calculator actually works with radians internally, there are two conversions in moving data from the keyboard to the display, and these sometimes cause annoying roundoff errors. To avoid this, set your display to Fix 2 for the time being.

NOTE: The angle mark, is created with the key stroke sequence AS RS 6.

Calculator Example 2.6.1

Let find PQ. With the calculator in degrees and polar coordinates, key in LS ( ) 2 (NOTE: At this point you may key in the comma which takes two key strokes or a space, SPC, and the calculator will convert it to a comma after you hit ENTER, but neither is necessary.) AS RS 6 43 ENTER LS ( ) 3 AS RS 6 29 You should see on the display. This clearly satisfies the definition of product from Section 2 of this chapter.

Now try it with The result you get is The magnitude is certainly correct, but we were expecting an argument of 199. What happens is that the calculator always normalizes the argument so that or in radians,

Calculator Example 2.6.2

Figure 14

Let and find P + Q. Key in the complex numbers as in the previous example, and press + to find the sum. The result is to two decimal places. We now use the definition of sum in Section 2 and geometry to verify that this answer is correct. See Figure 14. Construct SA and PB perpendicular to OU, and PC parallel to OU. Now is a 30o, 60o, 90o triangle with hypotenuse 4, so PB = 2 = CA, and Since is a 30o, 60o, 90o triangle with hypotenuse 2, so and SC = 1. OA = OB - AB = and AS = AC + SC = 3, so is a 30o, 60o, 90o triangle. Thus which agrees with our calculator answer at least to the level of accuracy possible on the calculator.

Calculator Example 2.6.3

An engineer wishes to build a straight railroad from town A to town B with a tunnel through the mountain from point C to point D. [See Figure 15] She cannot, of course see B from A, so she does not know what direction to go from A to find one end of the tunnel at C, nor what direction to go from B to find the other end of the tunnel at D. She can, however, see a tall tree at P from A, from P she can see a large rock at Q, and from Q she can see B. How should she proceed?

Figure 15

As mentioned above, the calculator's representation of a complex number requires r and to be expressed in decimal form. If one can compute the magnitude and argument to twelve significant figures on the calculator, then key them into the parentheses to get but this is a very tedious and error prone procedure. We will consider some tricks for entering such complex numbers more efficiently. For what follows, set the calculator numeric mode to standard to see the full 12 significant digits.

The easiest case is when r is "nice" and the argument is of the form with b a divisor of 180. Such an argument is an integer when expressed in degrees, hence easy to enter in that form. For example, if the argument is 45^o. To get this complex number into the calculator in radian form, first set the calculator to degree mode, enter the number in degree form, then switch the calculator to radian mode; the 45 will change to .785398163398.

We now return to the more general case, Although there are several ways to handle this, we will discuss only one; the equation writer. (See page 2-10 and Appendix E of UG.) With the calculator in radian mode, key in RS EQW 2 3 RA LS e^x LS 7 RA LS i ENTER EVAL. You should now see on the display. Note that we have used the equation writer to create the algebraic object then used the evaluation command, EVAL, to convert it to the complex number

The concepts of negative, subtraction, and conjugate are all easily handled on the calculator. The +/- key converts the complex number in level 1 of the stack to its negative. Enter your favorite complex number and try it! Note that pressing +/- twice returns the original number as it should. The - key will subtract the complex number in level 1 from the complex number in level 2. Try it with your choice of complex numbers A and B. After computing S = A - B, verify that S + B returns your original A. There is also a conjugate function, but it is in one of the menus. Key in LS MTH NXT F3-CMPLX NXT and you will find CONJ as the third item in the menu. Key in a complex number and try it. Again, pressing it twice gives back the original complex number as expected.

There are two other functions on this menu page; NEG and SIGN. NEG works the same as the +/- key and SIGN converts the complex number on level 1 into a complex number with the same argument but with absolute value 1. Press NXT and you see six more functions related to complex numbers. The first four will be discussed in the next section, but ABS returns the magnitude of the complex number on level 1, and ARG returns its argument. ABS and ARG are also on the keyboard in conjunction with the key.