Complex Numbers on TI89
Copyright © 2003–2019 by Stan Brown
Copyright © 2003–2019 by Stan Brown
Summary: Your TI89 can be set up to do all calculations with complex numbers in polar form or rectangular form. Here’s how. (The same keystrokes should work with a TI92 or Voyage 200.)
See also: A separate TI83/84 procedure is also available.
You can tell your TI89 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 TI89 that you want results in rectangular mode.  [MODE ] [▼ 5 times ] [► ] brings up the choices
for complex format. Select [2 ] for Rectangular and press
[ENTER ].

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 ] [▼ 3 times ] [► ]. Then press
[1 ] for radian mode or [2 ] for degree
mode.

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. 
[▼ ] [▼ ] [► ] [3 ]
Then [ ENTER ] 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 

(r∠θ) with θ
in degrees. You may need to use green [ENTER ] for an
approximate answer. 
e^{θi}·r with θ in radians. Here again is 3−4i as an exact and an approximate answer. 
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 [CATALOG ]
key. Enter 8 [− ] 3 [2nd CATALOG 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.
(This screen shot was made on a TI84, but the TI89 produces identical results.)
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 ∠ θ) with angle in degrees or radians depending on calculator mode; the parentheses are required. 
Details:  
Enter the absolute value or modulus, r.  [( ] 4Caution: Parentheses are required, even if the complex number is not used in an expression. 
Enter the separator between r and θ.  [2nd EE makes ∠ ]

Enter the angle or argument, θ.  If the calculator is in degree mode, enter the angle in
degrees, 120. The degree sign is optional if the
calculator is in degree mode.
If the calculator is in radian mode, either enter the angle in radians, 2 [ 2nd ^ makes π ] [÷ ]
3, or enter the angle in degrees with a degree sign,
120 [2nd  makes ° ].

Enter the closing parenthesis.  [) ] [ENTER ]

Here’s what you get if you enter the same number when the TI89 is set for rectangular (a+bi) display. (The TI89 panel at right shows both exact and approximate answers.)
(You could also use re^(θi) for entry in polar form, but only if the calculator is in radian mode. Since the (r∠θ) form can be used in either degree or radian mode, I recommend you use it always.)
Your TI89 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 . 
You want the Math, Matrix, Vector ops menu.
Press [2nd 5 makes MATH ] [4 ] [▲ ] [► ] [5 ] [ENTER ].

To convert an answer to polar form:
Enter the number or expression, then
►Polar . 
You want the Math, Matrix, Vector ops menu.
Press [2nd 5 makes MATH ] [4 ] [▲ ] [► ] [4 ] [ENTER ].
The form of the answer depends on the calculator mode: Degree mode
Radian 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. 
[2nd 5 makes MATH ] [5 ] [4 ]

Enter the number.  [() ] 16 [+ ] 47
[2nd CATALOG makes i ] 
Enter the closing parenthesis and find the answer, about
108.8° or 1.8989 radians depending on your calculator mode.
(This screen shot was made on a TI84, but the TI89 produces identical results.) 
[) ] [◆ ] [ENTER ]
The illustration at right is shown in approximate mode. You could use exact mode by omitting the [ ◆ ], but that's not
useful for most number.

Let’s find r, the absolute value or modulus, of the number −16+47i.
Select the abs function. 
[2nd 5 makes MATH ] [1 ] [2 ] 
Enter the number.  [() ] 16 [+ ] 47
[2nd CATALOG makes i ]

Enter the closing parenthesis and find the answer, about
49.649.
(This screen shot was made on a TI84, but the TI89 produces identical results.) 
[) ] [◆ ] [ENTER ]

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