Complex Numbers on TI83/84
Copyright © 2003–2023 by Stan Brown, BrownMath.com
Copyright © 2003–2023 by Stan Brown, BrownMath.com
Summary: Your TI83/84 can be set up to do all calculations with complex numbers in polar form or rectangular form. Here’s how.
See also: A separate TI89 procedure is also available.
You can tell your TI83/84 to display results in rectangular or polar form by setting the mode (below). But however you set your calculator to display results, you can always enter expressions in rectangular form, polar form or a mixture.
Rectangular mode means you want answers in a+bi form, whether you use polar or rectangular form when entering your expressions.
Once only, you need to tell the TI83/84 that you want results in rectangular mode.  [MODE ] [▼ 6 times ] [► ] [ENTER ]
selects a+bi mode. Remember to press
[ENTER ].
[ 2nd MODE makes QUIT ] returns to home screen.

For complex numbers in rectangular form, the other mode settings don’t much matter.
“Polar form” means that the complex number is expressed as an absolute value or modulus r and an angle or argument θ. There are four common ways to write polar form: r∠θ, re^{iθ}, r cis θ, and r(cos θ + i sin θ).
Polar mode on your calculator means that you want answers in a polar form, even if you enter expressions in rectangular form. Here’s how to set polar mode for display:
Since polar mode involves an angle, select degree or radian mode.  [MODE] ] [▼ ] [▼ ]. Then cursor to
Radian or Degree and press [ENTER ].

Tell the calculator that you want results in polar
mode.
Caution: Degree mode is shown here by way of example. Make sure you select Radian mode if that’s what you want. 
[▼ 4 times ] [► ] [► ] [ENTER ]
Then [ 2nd MODE makes QUIT ] to return to the home screen.

Your calculator will display polar format differently, depending on whether you selected degree mode or radian mode:
Polar Display in Degrees  Polar Display in Radians 

re^(θi) with
θ in degrees. Example: 3−4i displays as
5e^(53.13010235i) . 
re^(θi) with
θ in radians. Example: 3−4i displays as
5e^(.927295218i) . 
You can enter numbers in rectangular form or polar form, regardless of how you have set the display mode. You can even mix the two forms in one expression.
Enter numbers just as you see them. For example,
here’s 8−3i.
Engineers, use i instead of j. 
Find i in yellow above the decimal point.
Enter 8 [− ] 3 [2nd . makes i ].

Remember to distinguish between the negativenumber key
[()
] and the subtract key [−
]. Use the
subtract key for numbers with interior minus like 7−3i
and 2i−11; use the
negativenumber key for numbers with leading minus like
−2i and −7+3i.
Even though a complex number is a single number, it is written as an addition or subtraction and therefore you need to put parentheses around it for practically any operation. The illustration shows correct methods for subtraction, multiplication, division, and squaring.
Try these operations without parentheses and you’ll see that you get wrong answers.
Here’s how to enter the number 4∠120° or 4e^{120°i} in your calculator. Note that 120° = 2π/3 radians.
Overview:  r e^ (θi) with angle in
radians even if the calculator is in degree mode; the i is required.

Details:  
Enter the absolute value or modulus, r.  4 
Enter the separators between r and θ.  Press [2nd LN makes e^{x} ]. The display shows
4e^( .

Enter the angle or argument, θ.  Since 120° = 2π/3 radians, you must
enter 2πi/3.
(Not 2π/3i: remember the order of operations.)
Press 2 [ 2nd ^ makes π ] [2nd . makes i ] [÷ ] 3.
Caution: The angle must be in radians, even if the calculator is in degree mode, and the imaginary symbol i is required. 
Enter the closing parenthesis.  [) ] [ENTER ]

Here’s what you get if you enter the same number when the TI83/84 is set for rectangular (a+bi) display.
Your TI83/84 will automatically convert all answers to polar or rectangular form, depending on how you set the display format. But you can convert a particular answer without changing the mode. The conversion command (to Rect or to Polar) comes at the end of the command line, never in the middle.
To convert an answer to rectangular form:
Enter the number or expression, then ►Rect . 
[MATH ] [► ] [► ] [6 ]

To convert an answer to polar form:
Enter the number or expression, then
►Polar . 
[MATH ] [► ] [► ] [7 ]
The calculator will display the angle (part of the exponent on e) in radians or degrees according to how you set the mode. Be careful to interpret the answer in the correct measure! This example shows the same calculation in radian mode and then in degree mode. 
You can find just the angle (or argument) for a complex number. The angle will be in radians or degrees, according to the calculator mode.
Example: What’s the angle for the complex number −16+47i? To begin with, since the number is in quadrant 2 (negative real part, positive imaginary part), the angle must be between 90° and 180° or between about 1.7 and 3.1 radians.
Select the angle function. 
[MATH ] [► ] [► ] [4 ]

Enter the number.  [() ] 16 [+ ] 47
[2nd . makes i ] 
Enter the closing parenthesis and find the answer, about 108.8° or 1.8989 radians depending on your calculator mode. 
[) ] [ENTER ]

Let’s find r, the absolute value or modulus, of the number −16+47i.
Select the abs function. 
[MATH ] [► ] [1 ]

Enter the number.  [() ] 16 [+ ] 47
[2nd . makes i ]

Enter the closing parenthesis and find the answer, about 49.649. 
[) ] [ENTER ]

Updates and new info: https://BrownMath.com/ti83/