1. Imaginary and complex numbers on your calculator
Source

2. 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.

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

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

5. 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, −25 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.

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

7. Arithmetic of Complex Numbers
Add:  (2 + 4i) + (3 - 2i)
Subtract: (6 - 3i) - (4 + 5i)

8. Arithmetic of Complex Numbers
Multiply: (3 + 2i) (4 - 2i)

9. Note: 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.

10. Note:
TI-84+ with OS 2.53MP

11. Note:
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.

12. Powers of i
These values will appear when you are in either Real Mode or in a + bi mode.

13. Powers of i 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

14. Note: 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.)

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

16. Special Calculator Functions for Complex Numbers
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
5.  abs(returns the absolute value of the complex number.      abs(5+12i)  gives  1 3

17. 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