3. Complex Numbers on the Calculator - Polar Form
Our first job is to get the calculator options set up for the task at
hand. Key RS MODES to get into the modes dialog box. There are three settings
which we must be sure are properly set: the angle units, the coordinate
system, and constant type. Key DA and m-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 m-OK. For more information about angle modes see
pages 4-3 to 4-4 of UG. Now key DA and m-CHOOS to see three
choices for coordinate systems. Again use DA or UA as needed to select
Polar from this list then press m-OK. For more information about coordinate
systems see pages 4-4 to 4-5 of UG. Finally, we must make
sure that the constants in the calculator give numeric rather than symbolic
results. Key m-FLAG DA to highlight Flag 2. Now press m-CHK as needed to
make sure that Flag 2 is checked then m-OK and m-OK. For more information
about the roll of Flag 2 see pages 4-7 and 11-5 of UG.
Now look at the second row of annunciators. The leftmost
should be blank indicating that the calculator is in degree mode, and the
next one to the right should be 
indicating that the polar coordinate system is in effect. For more information
about the annunciators see pages 1-2 to 1-3 of UG. Notice
that the top left key on the calculator, the MTH key, has RAD as its left
shifted command and POLAR as its right shifted command. If the setup steps
from the previous paragraph were completed correctly, LS RAD will now toggle
between radian mode and degree mode. When in radian mode the leftmost annunciator
will be RAD. Also, RS POLAR will toggle between polar coordinates and rectangular
coordinates. When in rectangular coordinates, the second annunciator is
blank.
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 which 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.
Calculator Example 2.3.1
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.) RS 
43
ENTER LS ( ) 3 RS
29 
You should
see 
on the
display. This clearly satisfies the definition of product from above.
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 2.3.2
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 above
and geometry to verify that this answer is correct. See Figure 4.
Figure 4 |
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 2.3.3
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 5] She cannot, of course see B from
A,
so she doesn't 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 5
Solution: She visualizes the map as the Argand Plane with the
origin at A and U one kilometer east of A. She finds
that from A to P is 10.13 kilometers in a direction 19.8o
measured counterclockwise from east, hence 
can be thought of as the complex number
From
P
to Q is 125.3o measured counterclockwise from east and
the distance is 3.17 kilometers, so 
can be thought of as the complex number 
Finally, she finds that from Q to B is 8.68 kilometers and
the direction is 177.4o clockwise from east, so 
is considered to be the complex number 
She adds these three complex numbers on her calculator and finds the sum
to be 
so
from A to B is a distance of 5.71 kilometers in the direction
99.8o counterclockwise from east. She now goes in that direction
from A until she comes to the mountain at a point she labels C,
then goes in the opposite direction from B until she comes to the
mountain at the point she labels D.
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 45o. 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 Chapter 7 of UG.) With the calculator
in radian mode, key in LS EQUATION 2 
3 RA LS ex LS 
7 RA AS LS I RA 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 