Complex Number Calculator — Add, Convert & Solve With Steps
Step-by-Step Solution
By Amanda Reeds, Content Researcher ·
A complex number is a number made of two parts — a real part and an imaginary part written as bi, where i is defined as the square root of −1. This complex number calculator takes any value in a+bi form and adds, subtracts, multiplies, or divides it, finds its conjugate and modulus, converts it to polar or exponential form, and applies De Moivre's theorem for powers and roots — with every step shown, not just the final answer. It's built for students, engineers, and anyone who needs to check work involving imaginary numbers without digging through a scientific calculator's menu.
Whether you're checking homework on adding and dividing complex numbers, converting a rectangular value into polar form for an electrical engineering problem, or trying to find the two complex roots a quadratic equation refuses to give you in real numbers, the AceCalculator complex number calculator gives you an answer you can trust — and shows exactly how it got there.
Type the number the way you'd write it by hand in a+bi form, press Solve, and the calculator walks through the same moves you'd use on paper: combining real and imaginary parts, multiplying by a conjugate to clear an imaginary denominator, or applying De Moivre's theorem to spin a number around the complex plane. Five modes live in one tool, so you're not hunting through a scientific calculator's CMPLX menu for a button that isn't there.
What You Can Do With This Complex Number Calculator
Complex Arithmetic Calculator
Arithmetic mode adds, subtracts, multiplies, and divides two complex numbers, showing how the real and imaginary parts combine at each step — including the conjugate trick used to divide.
- Add and subtract complex numbers
- Multiply using the i² = −1 rule
- Divide by multiplying by the conjugate
Complex Conjugate & Modulus Calculator
Conjugate & Modulus mode flips the sign of the imaginary part to find the conjugate, then applies the distance formula to return the modulus (magnitude) and the argument (angle).
- Conjugate of any a+bi value
- Modulus, or absolute value, of the number
- Argument in both degrees and radians
Rectangular To Polar / Exponential Form
Convert Form mode turns a rectangular complex number into polar form r(cos θ + i sin θ) and exponential form r·e^(iθ), plus a small Argand diagram plotting the point on the complex plane.
- Rectangular to polar conversion
- Exponential (Euler) form
- Visual point on the complex plane
De Moivre's Theorem — Powers & Roots
Powers & Roots mode raises a complex number to any integer power, or finds all n complex n-th roots, using De Moivre's theorem so you don't have to expand binomials by hand.
- Any integer power of a complex number
- All n roots of a complex number
- Square, cube, and higher-order roots
Complex Quadratic Equation Solver
Quadratic Roots mode solves ax²+bx+c=0 for real coefficients a, b, and c. When the discriminant is negative, it returns the two complex conjugate roots instead of stopping at "no real solution."
- Handles a negative discriminant automatically
- Returns roots as a complex conjugate pair
- Same formula behind our Quadratic Formula Calculator
Complex Numbers vs. Complex Fractions
Landed here searching for a "complex fraction calculator"? A complex fraction is a nested fraction with no imaginary numbers involved — a different topic from complex (imaginary) numbers, which is what this tool covers.
- Complex numbers combine real + imaginary parts
- Complex fractions are fractions inside fractions
- See the FAQ below for the full distinction
The Rules Behind The Calculator
Every step this calculator produces comes from one of the standard complex number rules below. According to Wikipedia's entry on complex numbers, the field of complex numbers underlies algebra, signal processing, and electrical engineering wherever a real-number system alone can't represent a solution. The power and root rules below follow directly from De Moivre's formula, first published in the early 18th century and still the standard method for raising a complex number to a power.
| Rule | Idea | Used For |
|---|---|---|
| i² = −1 | The imaginary unit squared equals negative one | Multiplying complex numbers |
| Conjugate Multiplication | (a+bi)(a−bi) = a² + b², a real number | Dividing complex numbers |
| Modulus Formula | |z| = √(a² + b²) | Magnitude / absolute value |
| Argument Formula | θ = atan2(b, a) | Angle on the complex plane |
| De Moivre's Theorem | zⁿ = rⁿ(cos nθ + i sin nθ) | Powers and roots |
| Quadratic Formula | x = (−b ± √(b²−4ac)) / 2a | Complex roots when b²−4ac < 0 |
Why AceCalculator Is Different And Better
Most complex number calculators online give you an answer and nothing else. AceCalculator focuses on showing the working, matching classroom notation, and staying fast on any device.
- ✓ Full step-by-step solutions, every mode
- ✓ Five modes in one tool: arithmetic, conjugate/modulus, form conversion, powers/roots, quadratic
- ✓ Built-in Argand diagram for rectangular-to-polar conversions
- ✓ Clean, classroom-style a+bi notation
- ✓ Mobile-friendly and responsive on any device
- ✓ No downloads, no accounts, no signup required
- ✓ Runs entirely in your browser — nothing is stored or sent to a server
More Tools By AceCalculator.com
- Quadratic Formula Calculator — solve quadratics step by step
- Equation Solver — linear, systems, polynomial & more
- Algebra Calculator — full algebra step-by-step solver
- Matrix Calculator — inverse, determinant, eigenvalues
- Graphing Calculator — plot and visualize any function
- Derivative Calculator — basic, implicit & nth-order derivatives
- Limit Calculator — two-sided, one-sided & infinite limits
- Math Calculators Hub — all maths tools in one place
Content Researcher · AceCalculator
Amanda Reeds researches and builds AceCalculator's math tools, with a focus on matching each calculator's step-by-step output to standard classroom notation. She has written and verified the working behind AceCalculator's algebra, quadratic, and equation-solving calculators.
Frequently Asked Questions
Yes. Arithmetic, conjugate and modulus, rectangular-to-polar conversion, De Moivre's theorem powers and roots, and complex quadratic roots are all free, with no account, signup, or download required.
Write it in standard a+bi form, such as 3+4i, −2−5i, or 6i for a purely imaginary number. Use the i key on the keypad to insert the imaginary unit, and leave out any term that equals zero.
Switch to Conjugate & Modulus mode and enter your number in a+bi form. The calculator flips the sign of the imaginary part for the conjugate and applies the square-root-of-sum-of-squares formula for the modulus, showing both results instantly.
Use Convert Form mode. Enter a rectangular complex number like 3+4i and the calculator returns the polar form r(cos θ + i sin θ), the exponential form r·e^(iθ), and a small Argand diagram showing where the point sits on the complex plane.
De Moivre's theorem states that raising a complex number in polar form to a power n multiplies its angle by n and raises its modulus to the nth power. Powers & Roots mode applies this rule directly to compute any integer power or all n roots of a complex number.
Yes. Enter the coefficients a, b, and c from ax²+bx+c=0 in Quadratic Roots mode. When the discriminant b²−4ac is negative, the calculator returns the two complex conjugate roots instead of stopping at "no real solution." For real-root quadratics, our dedicated Quadratic Formula Calculator also shows the vertex and axis of symmetry.
Most scientific calculators, including common Casio and Texas Instruments models, have a CMPLX or complex mode buried in the settings menu that handles basic arithmetic. This tool skips that menu digging and adds full step-by-step working, polar conversion, and De Moivre's theorem in one place.
No. A complex number combines a real and an imaginary part, like 3+4i. A complex fraction (sometimes called a compound fraction) is a fraction that contains another fraction in its numerator or denominator, with no imaginary numbers involved at all.
Important Disclaimer
AceCalculator's complex number calculator is provided for informational and educational purposes only. While every step follows standard complex number rules, results are not a substitute for verification by a qualified instructor or professional where accuracy is critical. See our full Disclaimer and Terms and Conditions.