TI-89 Complex Numbers and Rectangular to Polar

Electrical engineering, AC circuits and precalculus all run on complex numbers, and the TI-89 Titanium handles them well once you know three keys and one mode setting. This guide covers TI-89 complex numbers from typing ⅈ to TI-89 rectangular to polar conversion and back.

Typing ⅈ: 2nd CATALOG, not the letter i

The imaginary unit is the 2nd function of CATALOG. It looks like an i in the display, but it is a different symbol. The letter i from the alpha layer is just a variable name, and 3+4i with that letter is an expression in a variable called i, not a complex number.

2ndCATALOG→ ⅈ

With ⅈ in place you can add, multiply, divide and raise complex numbers to powers like any other number. Put a multiplication sign or nothing between the coefficient and ⅈ: 4ⅈ and 4*ⅈ both work.

The Complex Format setting in MODE

MODE has a Complex Format field with three choices:

REAL shows only real results. A square root of a negative number gives a Non-real result error unless the input already contains ⅈ.
RECTANGULAR shows complex results as a + bⅈ.
POLAR shows complex results by magnitude and angle.

For a course that uses complex numbers, set RECTANGULAR once and leave it. The Angle field in the same dialog decides whether angles appear in radians or degrees, which matters for every polar result below.

MODEthen scroll to Complex Format→ RECTANGULAR

Real part, imaginary part and conjugate

real(, imag( and conj( take a complex number apart. On the handheld they are in the Complex submenu of MATH (2nd 5), or type the name with the alpha keys.

2nd5then the Complex submenu→ MATH Complex

Real part, imaginary part and conjugate
Input	Result
real(3+4ⅈ)	3
imag(3+4ⅈ)	4
conj(3+4ⅈ)	3−4·ⅈ

Magnitude and angle: abs( and angle(

For a complex number, abs( returns the magnitude r and angle( returns the argument θ, measured from the positive real axis. In RADIAN mode the angle comes out in radians, exact where possible.

angle( example
Input	Result
angle(1+ⅈ)	π/4

The result is π/4 because 1 + ⅈ lies on the diagonal of the first quadrant. In DEGREE mode the same input gives the angle in degrees. Press ◆ ENTER for a decimal when the exact form is not what you need.

Rectangular to polar with ▶Polar

▶Polar converts to polar form. Type the value, then ▶ with 2nd MODE, then Polar (from the MATH Complex submenu, the CATALOG or with the alpha keys). It works on a complex number and on a two element vector.

2ndMODE→ ▶

Rectangular to polar
Input	Result
[3,4]▶Polar	5∠tan⁻¹(4/3)

The vector [3,4] has length 5 and the angle tan⁻¹(4/3). The CAS keeps that angle exact; ◆ ENTER gives the decimal.

Polar to rectangular with ▶Rect

The way back is ▶Rect. To enter a value in polar form, use the angle symbol ∠, which is 2nd EE. Keep magnitude, ∠ and angle together in brackets or parentheses.

2ndEE→ ∠

Polar to rectangular
Input	Result
[5,∠(π/2)]▶Rect	05

Magnitude 5 at a right angle points straight up, so the rectangular form is [0, 5]. If you typed the angle in degrees while the mode is RADIAN, the result is wrong without any warning, so check the Angle setting first.

Coordinates: R▶Pr(, R▶Pθ(, P▶Rx( and P▶Ry(

For a point instead of a complex number, four functions return one coordinate each. R▶Pr( and R▶Pθ( go from x and y to r and θ; P▶Rx( and P▶Ry( go from r and θ to x and y. They are handy in physics when you need one component of a vector.

Coordinate conversions
Input	Result
R▶Pr(3,4)	5
R▶Pθ(1,1)	π/4
P▶Rx(2,π/3)	1
P▶Ry(2,π/6)	1

Solving with complex roots

solve( returns only real solutions. For complex ones use cSolve(, cZeros( and cFactor( from the Complex submenu of F2.

F2►►1→ cSolve(

Complex solutions and factors
Input	Result
cSolve(x^2=-4,x)	x=−2·ⅈ or x=2·ⅈ
cZeros(x^2+1,x)	{−ⅈ,ⅈ}
cFactor(x^2+1)	(x+ⅈ)·(x−ⅈ)

Common mistakes

alpha i instead of ⅈ. The result looks almost right but treats i as a variable. Use 2nd CATALOG.
REAL format. If you get Non-real result, change Complex Format in MODE.
Degrees and radians mixed. Polar angles follow the Angle mode. Engineering problems are often in degrees, calculus in radians.
A stored variable. If a letter like r or a holds a value, results change silently. Run NewProb (2nd F1) before a new problem.

In Calc89

Calc89 is intended for use with TI-89 Titanium, and ⅈ, ∠ and ▶ sit on the same keys: 2nd CATALOG, 2nd EE and 2nd MODE. The examples above were computed by the same CAS that runs in the app, and the color display shows vectors in brackets and roots with a bar, the way your textbook prints them.

Circuit problems with long impedance expressions are where the camera scanner saves the most typing: it reads the printed or handwritten problem into the edit line. The AI keystroke tutor shows the key sequence for each step, including where ⅈ and ∠ go.

Frequently asked questions

How do I type i on the TI-89?

Press 2nd CATALOG for the imaginary unit ⅈ. The letter i from the alpha layer is an ordinary variable.

Why does my TI-89 say Non-real result?

Complex Format in MODE is set to REAL. Switch it to RECTANGULAR or POLAR, or use cSolve( and cZeros(.

How do I convert rectangular to polar on the TI-89?

Type the number or vector, then ▶Polar (the ▶ is 2nd MODE). ▶Rect converts back.

Complex Numbers, Polar Form and Rectangular-to-Polar on the TI-89