Complex numbers — those involving the imaginary unit i, defined as the square root of negative one — appear throughout advanced mathematics and have practical applications in electrical engineering, signal processing, quantum mechanics, and control theory. Students encounter them first in algebra when solving quadratics with negative discriminants, then again in trigonometry through De Moivre's theorem, and later in calculus and differential equations.
Introduction
The TI-84 handles complex arithmetic natively in a+bi mode, making operations that seem intimidating in textbook notation completely straightforward on the calculator. Addition, subtraction, multiplication, division, powers, roots, and the extraction of modulus, argument, and conjugate are all directly supported. The calculator also automatically returns complex roots when solving polynomial equations with the PlySmlt2 app in complex mode.
This guide covers the essential complex number capabilities of the TI-84: setting up complex number mode, entering complex numbers, performing arithmetic, computing modulus and argument, working with polar form, and solving polynomial equations with complex roots.
Setting Up: Complex Number Mode
Press MODE and look for the row containing REAL, a+bi, and re^thetai. Select a+bi for standard rectangular complex number mode. In this mode, the calculator displays complex results in the form a + bi, where a is the real part and b is the imaginary part. For example, the square root of negative 9 displays as 3i in a+bi mode, rather than producing an error as it would in REAL mode.
The re^thetai option selects polar form, where complex numbers are displayed as r times e to the power of i theta. This form is natural for applications in electrical engineering and physics where phasors and AC circuit analysis use polar representation. In polar mode, the modulus r and argument theta (in the current angle units) are the primary descriptors of a complex number.
For most algebra and pre-calculus coursework, a+bi mode is the appropriate choice. Switch to re^thetai for AC circuit analysis, De Moivre's theorem applications, or any problem where multiplication and division of complex numbers is simplified by working with magnitudes and angles. The two modes represent exactly the same mathematical objects — they differ only in how those objects are displayed.
Entering Complex Numbers
The imaginary unit i is accessed by pressing 2ND + the decimal point key. The letter i appears above the decimal key and is the standard notation for the imaginary unit. Confirm you are in a+bi mode before entering complex number calculations.
Enter complex numbers in rectangular form a + bi. For the complex number 3 + 4i, type 3 + 4i using the 2ND + decimal key for i. For negative imaginary parts like 2 - 5i, type 2 - 5i. For pure imaginary numbers like 7i, type 0 + 7i or simply 7i. For real numbers used in complex calculations, type them normally — the calculator treats them as complex numbers with zero imaginary part.
You can also enter complex numbers in polar form using the re^thetai mode or by using Euler's formula: r times e to the power of i*theta. For example, 5 times e^(i*pi/4) represents the complex number with modulus 5 and argument pi/4 radians, which equals 5 times (sqrt(2)/2 + i*sqrt(2)/2) in rectangular form. The calculator converts between forms automatically depending on your current MODE setting.
Complex Arithmetic
Standard arithmetic operators work directly on complex numbers in a+bi mode. Addition: (3+4i)+(1-2i) returns 4+2i — add real parts together and imaginary parts together. Subtraction: (3+4i)-(1-2i) returns 2+6i. Multiplication: (3+4i) times (1-2i) uses the FOIL method automatically, returning 3-6i+4i-8i^2 = 3-2i+8 = 11-2i. Division: (3+4i)/(1-2i) automatically multiplies numerator and denominator by the conjugate, returning -1+2i.
Powers work correctly for complex numbers: (1+i)^4 returns -4, consistent with the result of applying De Moivre's theorem. Square roots of negative numbers work correctly in a+bi mode: sqrt(-9) returns 3i. More complex expressions like (2+3i)^(1/2) return the principal square root of the complex number.
The TI-84 respects the correct algebraic rules for complex arithmetic automatically. You do not need to manually apply the rule i^2 = -1 or work through binomial expansion — the calculator handles all of this internally. This makes it an excellent tool for verifying complex number calculations worked by hand, and for handling complex expressions in applications where hand computation would be prohibitively tedious.
Modulus, Argument, and Conjugate
Complex number utility functions are in the MATH → CPX menu. The abs( function returns the modulus (magnitude) of a complex number: abs(3+4i) returns 5, matching the distance formula sqrt(3^2 + 4^2) = sqrt(25) = 5. The modulus represents the distance of the complex number from the origin in the complex plane.
The angle( function returns the argument of a complex number: angle(1+i) returns 45 in Degree mode, matching the expected argument of pi/4 radians. The conj( function returns the complex conjugate: conj(3+4i) returns 3-4i. The real( and imag( functions extract the real and imaginary parts respectively.
The polar conversion functions are available through the ANGLE menu (2ND+APPS): R to Pr and R to Ptheta convert rectangular coordinates to polar, while P to Rx and P to Ry convert polar to rectangular. These functions work on real number pairs representing complex numbers, which is useful when working in an application that uses the (r, theta) representation of complex numbers rather than the a+bi notation.
Solving Equations with Complex Roots
When using the TI-84's PlySmlt2 polynomial root finder in a+bi mode, it automatically returns complex roots for equations with negative discriminants. For x squared + x + 1 = 0, where the discriminant equals 1 - 4 = -3, set MODE to a+bi and press APPS → PlySmlt2 → Poly Root Finder. Set degree to 2 and enter coefficients 1, 1, 1. Press SOLVE to get the roots: -0.5 + 0.866i and -0.5 - 0.866i.
These values are exactly negative one-half plus or minus the square root of 3 divided by 2 times i — the two primitive cube roots of unity. The decimal values 0.866 and -0.866 are immediate recognizable as ± sqrt(3)/2. This combination of numerical calculator output and mathematical pattern recognition is the optimal way to use the TI-84 for complex equation solving.
For higher-degree polynomials, the complex mode ensures all roots are found, including those that do not appear on a real graph. A degree-4 polynomial may have zero, two, or four real roots, with the remaining roots appearing in complex conjugate pairs. PlySmlt2 finds all roots regardless of type, providing a complete factorization picture.