BrownMath.com → TI-83/84/89 → Complex Numbers (TI-89)
Updated 6 Nov 2020

# Complex Numbers on TI-89

Summary: Your TI-89 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 TI-92 or Voyage 200.)

See also: A separate TI-83/84 procedure is also available.

## Selecting the Display Format

You can tell your TI-89 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 Display Mode

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 TI-89 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 Display Mode

“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, 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 DegreesPolar Display in Radians (r∠θ) with θ in degrees. You may need to use green [`ENTER`] for an approximate answer. eθi·r with θ in radians. Here again is 3−4i as an exact and an approximate answer.

## Entering Numbers

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.

### Rectangular Form for Input

 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 negative-number key [`(-)`] and the subtract key [`−`]. Use the subtract key for numbers with interior minus like 7−3i and 2i−11; use the negative-number key for numbers with leading minus like −2i and −7+3i.

### Entering Expressions 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 TI-84, but the TI-89 produces identical results.)

Try these operations without parentheses and you’ll see that you get wrong answers.

### Polar Form for Input

Here’s how to enter the number 4∠120° or 4e120°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. [`(`] 4 Caution: 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 TI-89 is set for rectangular (a+bi) display. (The TI-89 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.)

## Conversions

### Converting to Polar or Rectangular Form

Your TI-89 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 ### Finding the Angle

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 TI-84, but the TI-89 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.

### Finding the Absolute Value r

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 TI-84, but the TI-89 produces identical results.) [`)`] [`◆`] [`ENTER`]