TC3 → Stan Brown → TI-83/84/89 → Complex Numbers (TI-83/84)
revised 26 Jan 2011

Complex Numbers on TI-83/84

Copyright © 2003–2014 by Stan Brown, Oak Road Systems

Summary: Your TI-83/84 can be set up to do all calculations with complex numbers in polar form or rectangular form. Here’s how.

See also: A separate TI-89 procedure is also available.


Selecting the Display Format

You can tell your TI-83/84 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 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-83/84 that you want results in rectangular mode. [MODE] [ 6 times] [] [ENTER] selects a+bi mode. Remember to press [ENTER].
MODE screen with a+bi setting
[2nd MODE makes QUIT] returns to home screen.

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, reiθ, 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]
mode screen with Degree and re^@i modes
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 DegreesPolar 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).

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 decimal point. Enter 8 [] 3 [2nd . 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

10+4i minus 7-3i yields 3+7i; 10+4i times 7-3i yields 82-2i; 10+4i over 7-3i yields 1+i; 10+4i squared yields 84+80i 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.

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 e^i) with angle in radians even if the calculator is in degree mode; the i is required.
Enter the absolute value or modulus, r. 4
Enter the separators between r and θ. Press [2nd LN makes ex]. 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]

4e^(2 pi i / 3) = -2+3.464i Here’s what you get if you enter the same number when the TI-83/84 is set for rectangular (a+bi) display.


Converting to Polar or Rectangular Form

Your TI-83/84 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. 2 e to the pi i/3 times 2.5 e to the pi i/6 equals 5i [MATH] [] [] [6]

To convert an answer to polar form:

Enter the number or expression, then ►Polar. [MATH] [] [] [7]
10+4i minus 5-8i equals 13 times e to the 1.176i 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.

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. [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. angle of -16+47i is 108.8 degrees [)] [ENTER]

Finding the Absolute Value r

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. absolute value of -16+47i is about 49.649 [)] [ENTER]

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