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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.

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.


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:

Add: 
(2 + 4i) + (3 - 2i)
Subtract:
(6 - 3i) - (4 + 5i)
Multiply:
(3 + 2i) (4 - 2i)

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

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
84OS

Be careful to write your final answer correctly. 
Note the location of the i in the final answer:


NOTE: On OS2.53MP, 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:

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.
(E-13 is Scientific Notation meaning 10 raised to -13 power.)

 

 

"My Deer, BEWARE!!!

When raising i to a power on a graphing calculator, accuracy diminishes as the powers increase.

 

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):

1. conj(  returns the complex conjugate of a complex number.
    conj(2+5igives  2-5i

2. real(   returns the "a" value in an a+bi complex number.
    real(2+5igives  2

3. imag(   returns the "b" value in an a+bi complex number.
    imag(2+5igives  5

5.  abs(   returns the absolute value of the complex number.
     abs(5+12i)  gives  13

(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.)


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