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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.

Use the MODE key to place
the calculator in a+bi mode.


The complex i  is found above the
decimal point key or in the catalog.


  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, 2 will create an error.

You get an error message when you try to enter the square root of a negative number when in Real Mode.

You need to be in a + bi mode to evaluate the square root of a negative number.


Now, let's look at the arithmetic of complex numbers:

(2 + 4i) + (3 - 2i)
(6 - 3i) - (4 + 5i)
(3 + 2i) (4 - 2i)
7 8

Divide:  (2+3i)/(4-3i) (answer appears in a + bi form)   WOW!!
The calculator did the conjugate work for you. YEA!

For TI-83+ and for TI-84+ with OS prior to OS 2.53MP.
In this display, i is NOT in the denominator!  The calculator is simply listing order of operations.
TI-84+ with OS 2.53MP ...

TI-84+ CE
As a fraction, asnwer is

Be careful to write your final answer correctly. 
Note the location of the i in the final answer:
The i can also be written in the numerator with 18,
but it cannot be written in the denominator with 25.
NOTE: On OS2.53MP or the CE, if the decimals you are trying to "grab" to convert to fractions are really long, use MATH - CPX - real( and imag( to "grab" the full decimal value for converting "a" and "b" to fractions.


Using the calculator to investigate powers of i:

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 listccc
... 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.
(E-13 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 TI-84+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. ------>
TI-84+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 i178 is i2 which is -1.
On all of the TI-84+ family calculators,
i178 = -1+2.2E-12i.
But it is actually just -1.capture7

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  2-5i

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 x-axis (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.)