Complex Numbers
The graphing calculator can be a very
useful tool for checking your work with complex numbers.
Keep in mind, when working with a graphing
calculator,
that there may be more than one way
to arrive at an answer. 
Note: Complex numbers can be
accessed from Real Mode (without placing calculator in a + bi Mode).
Real mode, however, does not display complex results unless complex
numbers are entered as input. For example, if the calculator is NOT
in a + bi Mode, will create an error. 
Now, let's look
at the arithmetic of complex
numbers:
Using the calculator to investigate powers of i:
Investigate
the powers of i. 

These values
will appear when you are in either
Real Mode or
in
a + bi mode. 
You can look at many powers at once by using a list

.. .
use right arrow to scroll to the right to see all of the answers

What kind of number is
 3E  13  i???
This number is really just  i.
 3E  13 is so small, it is considered to be
zero.
(E13 is Scientific Notation meaning 10 raised to 13
power.)
The older the calculator, the "sooner" you will start to see these scientific notation answers. 
The TI84+CE is better with finding powers of i, but it, as well, will resort to the slightly less accurate scientific notation listings as the powers get larger. >
TI84+CE is OK through a power of 100. 


"My Deer, BEWARE!!!
When raising i to a power on the graphing calculator, accuracy diminishes as the powers increase.



We know mathematically
that i^{178} is i^{2} which is 1.
On all of the TI84+ family calculators,
i^{178} = 1+2.2E12i.
But it is actually just 1.

Special Calculator Functions for Complex Numbers:
There are also special functions on the
graphing calculator to deal with complex numbers
(but you probably won't
need a calculator for many of these functions):
Hit MATH key and arrow
to the right to CPX:

1. conj( returns the complex conjugate of a complex number. conj(2+5i)
gives 25i
2. real( returns the "a"
value in an a+bi complex number.
real(2+5i)
gives 2
3. imag( returns the "b"
value in an a+bi complex number.
imag(2+5i)
gives 5 
4. angle( returns the angle, or argument, of the complex number  the angle formed by the positive xaxis (the positive real axis) and the segment from the origin to the complex number point on an Argand diagram.
angle(2+5i) gives 68.199º (with calculator set in degree mode)
5. abs( returns the absolute value (magnitude) of the complex number.
abs(5+12i) gives
13
abs (2 + 5i) gives 5.385164807
(Note: The absolute
value of a complex number may also be called its magnitude.
It you plot a complex number as a single point, the absolute value
represents the distance from the origin to that point. If you
plot a complex number as a vector, the absolute value represents the length of the vector.)

