How to Do Complex Numbers on a TI-84

Press MODE, arrow down to the row that says REAL, scroll right to a+bi, and press ENTER. That is the only setting you need. With that done, your calculator treats every number as a complex number, handles imaginary parts automatically, and will convert between rectangular and polar form on command. This works for the TI-84 Plus, TI-84 Plus Silver Edition, and TI-84 Plus CE. The same steps apply to the TI-83 Plus, though its screen is smaller.

Press MODE. On the screen that appears, you see a line reading REAL, a+bi, re^θi. The default is REAL. Arrow down to that line, scroll right to a+bi, and press ENTER. Then press 2ND + QUIT to return to the home screen.

If you also want polar output, press MODE again, scroll to the same line, and choose re^θi instead. Output shows in polar form as r∠θ. The TI-84 Plus CE guidebook (Texas Instruments, 2023) calls these the two complex modes: a+bi for rectangular and re^θi for polar. You can switch between them at any time without losing your calculations.

On the TI-84, the imaginary unit i is above the decimal point key: press 2ND + . (the key labeled with a small i). To enter 3 + 4i, type 3 + 4 + 2ND + . and press ENTER. The calculator shows 3+4i if a+bi mode is set.

For subtraction, use the minus key, not the negative key: 5 - 2i is 5 - 2 + 2ND + .. The negative key (marked (-)) is for the sign of a number, not subtraction.

Multiplication and division work as you expect: (3+4i)*(1-2i) gives 11 - 2i. Division (3+4i)/(1-2i) gives -1 + 2i. The device uses the division formula from the fact sheet: (a+bi)/(c+di) = (ac+bd)/(c²+d²) + (bc-ad)/(c²+d²)i.

To convert a complex number to polar form on the TI-84, use the MATH menu. Press MATH, then arrow over to the CPX submenu (the second tab). You find three functions that matter:

• abs(, gives the modulus (distance from origin). For 3+4i, abs(3+4i) returns 5.
• angle(, gives the argument (angle from positive real axis). For 3+4i, angle(3+4i) returns about 53.13° (in degree mode) or 0.93 rad (in radian mode). The TI-84 returns the principal argument in the range (-π, π].
• ►Polar, converts rectangular to polar and displays as (r∠θ). Found in the same CPX menu.

To convert from polar to rectangular, use ►Rect in the same menu. If you have a polar number like 5∠53.13°, type 5 + 2ND + . (for the angle symbol) + 53.13 and press ENTER in a+bi mode. The TI-84 shows 3+4i.

The TI-84's angle output depends entirely on whether the device is in degree or radian mode. Press MODE; the top line shows DEGREE or RADIAN. Arrow to it and press ENTER to switch.

If you are working with phasors in electrical engineering, degree mode is standard. For pure mathematics or physics, radian mode is expected. The most common error: entering sin(90) in radian mode gives 0.89, not 1. Check the mode before any angle calculation. The TI-84 does not warn you.

When using angle(, the output unit matches the current mode. A number entered with the degree symbol (found above the APPS key) is treated as degrees regardless of mode, but the device's own output follows the mode setting.

The Casio fx-991EX (and the newer fx-991CW) have a dedicated complex mode. Press MODE, then 2: Complex. The i key is on the bottom row, second from the left. Enter numbers the same way: 3 + 4i shows as 3+4i.

To convert to polar on the Casio, press OPTN, then 2: Angle, then 3: >r∠θ. The result shows as r∠θ. The Casio's default angle range is also (-π, π], matching the TI-84. The Casio fx-991EX user's guide (Casio, 2017) confirms this.

The Casio fx-CG50 (a graphing calculator similar to the TI-84) uses the same complex mode setup: press MENU, select Complex, and use the i key at the bottom. Polar conversion is under OPTN + Complex + ►r∠θ.

Both Casio models and the TI-84 produce the same numerical results, but the button locations differ. If you switch from a TI-84 to a Casio for an exam, practice the key sequence for angle(, on Casio it is OPTN, 2: Angle, 1: arg.

NONREAL ANS Error

NONREAL ANS appears when the TI-84 is in REAL mode but you entered a complex number. The device cannot display the imaginary part. Fix: go to MODE and change to a+bi or re^θi. This is the most frequent error message for students starting complex numbers on a TI-84.

Quadrant and Mode Pitfalls

Quadrant error with angle( happens when you manually compute arctan(b/a) and forget that for a < 0, you must add 180° (or π rad). The TI-84's angle( function does this automatically, but if you write down its output and then adjust it incorrectly, you introduce the error yourself. The device's output is correct within its chosen range.

Wrong mode for output: if you expect a polar answer but the calculator shows a+bi, you are in a+bi mode, not re^θi. Switch modes before converting. The device does not convert automatically; it displays in whatever mode you set.

Radians vs. degrees for angle input: if you type sin(90) expecting 1, and the device is in radian mode, you get 0.89. Always check the mode indicator at the top of the screen. A small RAD or DEG text is visible when the screen is active.

This approach suits precalculus and college algebra students who need to perform arithmetic and convert between rectangular and polar form. It also suits engineering undergraduates who need phasor arithmetic for AC steady-state circuits, and physics undergraduates who need complex exponentials for oscillating functions. Exam revisers for A-level Further Maths or IB will find the key-by-key steps useful for fast, accurate manipulation under time pressure.

Skip this if you are a pure mathematician researching complex analysis, you need topology and contour integrals, not a graphing calculator's numerical output. Also skip it if you are a high school student who has not yet seen the quadratic formula; you lack the prerequisite algebra. Professionals doing high-precision numerical computation should use MATLAB, Python's cmath, or a computer algebra system instead.