Complex Number Calculator
Add, subtract, multiply and divide complex numbers, and find the modulus, argument, conjugate, polar form, powers and roots. Free, runs in your browser.
Complex Number Calculator
How to Use the Complex Number Calculator
Enter two complex numbers
For each of A and B, type the real part and the imaginary part into the two boxes — for example 3 and 4 for the number 3 + 4i. Negatives are fine.
Read the arithmetic
The calculator shows A + B, A − B, A × B and A ÷ B straight away, each as a complex number in the form a + bi.
See A's properties
It also gives the modulus and argument of A, its conjugate, and its polar form — the natural way to describe a complex number by length and angle. The argument is reported as the principal value, in degrees and radians, between −180° and 180°.
Take powers and roots
Enter a power n to compute Aⁿ by De Moivre's theorem, or a whole number n to list all n of the nth roots of A, spaced evenly around a circle. The first root listed is the principal root.
Complex Numbers: Extending the Number Line
The Number That Made the Impossible Possible
Complex numbers begin with a single bold idea: invent a number, called i, whose square is −1. No ordinary number does this, since squaring any real number gives something positive, so for centuries mathematicians dismissed the square roots of negatives as "imaginary". But accepting i turns out to complete the number system in the most beautiful way. A complex number combines a real part and an imaginary part, written a + bi, and with it every polynomial equation finally has a solution — the Fundamental Theorem of Algebra, which guarantees that an equation of degree n has exactly n roots, is true only because complex numbers exist. This calculator does the full arithmetic of complex numbers — addition, subtraction, multiplication and division — each producing another complex number, and it does so instantly as you type. What looks like an abstract curiosity is in fact one of the most practical tools in all of science and engineering.
The deepest insight about complex numbers is geometric. Every complex number can be pictured as a point in a plane, with the real part as the horizontal coordinate and the imaginary part as the vertical one. Seen this way, a complex number has a length — its modulus, the distance from the origin — and a direction — its argument, the angle it makes with the positive real axis. This is the polar form, and it transforms multiplication into something elegant: to multiply two complex numbers you multiply their lengths and add their angles. De Moivre's theorem extends this to powers, so raising a complex number to the nth power means raising its length to that power and multiplying its angle by n — which is exactly how this calculator computes Aⁿ. The same idea, run in reverse, finds the nth roots of a complex number: there are always exactly n of them, equally spaced around a circle, a fact this tool displays directly. The roots of unity that result are the hidden symmetry behind the Fourier transform, the algorithm that powers digital audio, image compression and wireless communication.
"To multiply complex numbers, multiply their lengths and add their angles. This one geometric fact turns rotation into arithmetic — and underlies signal processing, quantum mechanics and electrical engineering."
Why Engineers and Physicists Can't Do Without Them
Complex numbers are indispensable wherever something oscillates or rotates. Electrical engineers represent alternating current as complex numbers, because the modulus captures a signal's amplitude and the argument captures its phase, turning hard differential equations into simple algebra — the impedance of a circuit is a complex number. In physics, quantum mechanics is written in complex numbers at its very foundation; the wavefunction that describes a particle is complex-valued, and it could not be otherwise. Control engineering judges whether a system is stable by where its complex eigenvalues sit in the plane, and signal processing decomposes sounds and images into complex exponentials through the Fourier transform. Even fractals like the Mandelbrot set are defined by iterating a simple rule on complex numbers. Whether you are studying complex numbers for the first time, checking engineering coursework, or exploring polar form and roots, this calculator gives you the complete picture — arithmetic, modulus and argument, conjugate, polar form, powers and roots — using the principal-value conventions of ISO 80000-2 and the NIST Digital Library of Mathematical Functions.
Multiply the lengths, add the angles
The imaginary unit i is defined by i² = −1.
A complex number a + bi is a point in the plane.
To multiply, multiply lengths and add angles.
Every degree-n polynomial has exactly n complex roots, counted with multiplicity.
Every non-zero complex number has exactly n nth roots, evenly spaced on a circle of radius |z|^(1/n).
De Moivre's theorem turns powers into angle-multiplication.
Euler's identity e^(iπ) + 1 = 0 links five fundamental constants.
Electrical engineers write AC signals as complex numbers.
Quantum mechanics is built on complex-valued wavefunctions.
The argument shown is the principal value: −180° < arg z ≤ 180°, so arg(−1) = 180°, never −180°.
Frequently Asked Questions
- A complex number has a real part and an imaginary part, written a + bi, where i is the imaginary unit satisfying i² = −1. It can be pictured as a point in a plane, with the real part horizontal and the imaginary part vertical. Complex numbers complete the number system so that every polynomial equation has a solution.
- Type the real part in the first box and the imaginary part in the second. For 3 + 4i, enter 3 and 4; for 1 − 2i, enter 1 and −2. A purely real number has imaginary part 0, and a purely imaginary number has real part 0.
- The modulus is the number's length — its distance from the origin in the plane — found by the Pythagorean theorem: |a + bi| = √(a² + b²). The argument is its direction — the angle it makes with the positive real axis. Any angle differing by a full turn describes the same direction, so the calculator reports the principal value, −180° < arg z ≤ 180° (−π < arg z ≤ π in radians), the convention of ISO 80000-2. Together they give the polar form r(cos θ + i sin θ), which describes a complex number by length and angle rather than by coordinates.
- Algebraically you expand (a + bi)(c + di) and use i² = −1. Geometrically there is a simpler picture: the lengths multiply and the angles add. That is why multiplying by a complex number of length 1 is a pure rotation, and it is the key idea behind De Moivre's theorem for powers.
- The conjugate of a + bi is a − bi — the same number reflected across the real axis. Multiplying a number by its conjugate gives a real result equal to the square of its modulus, which is exactly how complex division is carried out.
- Because rotating by a full turn returns you to the same point, there are exactly n distinct nth roots of any non-zero complex number, equally spaced around a circle of radius |z|^(1/n) at angles (arg z + 360°k)/n for k = 0, 1, …, n − 1. For example, every non-zero number has three cube roots. The calculator lists all of them, starting with the principal root (k = 0), and you can check that raising each to the power n returns the original number.
- It says that to raise a complex number to a power n, you raise its modulus to that power and multiply its argument by n: (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ). This makes powers of complex numbers easy to compute through the polar form, and it is how this calculator evaluates Aⁿ. For a whole-number n the result is unique; for a fractional or negative n the calculator uses the principal argument of A, so it returns the principal value of Aⁿ — one of the several values a fractional power has.
- No. Every calculation runs in your browser with a small built-in engine — nothing is uploaded to a server or third-party library, and the tool works offline once the page has loaded.
- Everywhere something oscillates or rotates: alternating-current circuits, signal processing and the Fourier transform, control systems, and the very foundations of quantum mechanics. The modulus captures amplitude and the argument captures phase, which turns hard problems into simple algebra.
- Division by zero is undefined for complex numbers just as it is for real ones, so A ÷ B is shown as "undefined (B = 0)" while A + B, A − B and A × B still compute. If A itself is zero, its modulus is 0 and its argument is shown as 0°, although strictly the argument of zero is not defined; results are rounded to six decimal places, so a value such as 1e−7 displays as 0.
Method & sources
How it computes
Complex arithmetic in rectangular form a + bi (quotient via the conjugate; division by zero reported as undefined), modulus |z| = √(a² + b²), argument as the principal value −π < arg z ≤ π, conjugate and polar form. Powers by De Moivre, zⁿ = rⁿ(cos nθ + i sin nθ) with θ the principal argument; the n nth roots as |z|^(1/n) at angles (θ + 2πk)/n, k = 0…n−1, principal root first. Results rounded to six decimal places.
What this tool implements
- Notation and principal value per ISO 80000-2:2019: imaginary unit i with i² = −1, |z| for modulus, arg z for argument taken in (−π, π], z* / z̄ for the conjugate — so arg(−1) = π, never −π.
- Phase and roots per NIST DLMF §1.9: the principal phase −π < ph z ≤ π, De Moivre's theorem for integer powers, and the n distinct nth roots of a non-zero complex number equally spaced on the circle of radius |z|^(1/n).
- For a non-integer exponent n the tool returns the single principal value exp(n·Log z) built from the principal argument; the other values of a multi-valued power are not listed (only the nth-roots function enumerates all n values).
- arg 0 is displayed as 0 although the argument of zero is undefined; A ÷ B with B = 0 is displayed as undefined rather than as Infinity/NaN.
Sources
- ISO 80000-2:2019. Quantities and units — Part 2: Mathematics. Geneva: International Organization for Standardization; 2019. Clause on complex numbers (i, Re z, Im z, |z|, arg z with principal value in (−π, π], z*). https://www.iso.org/standard/64973.html
- NIST Digital Library of Mathematical Functions, §1.9 Calculus of a Complex Variable — 1.9(i) complex numbers, polar representation, principal value of the phase, De Moivre's theorem and nth roots. https://dlmf.nist.gov/1.9
- NIST Digital Library of Mathematical Functions, §4.2(iv) Powers — the general power z^a = exp(a ln z) and its principal value. https://dlmf.nist.gov/4.2#iv
- Brown JW, Churchill RV. Complex Variables and Applications. 8th ed. New York: McGraw-Hill; 2009. Chapter 1: arguments and principal value, products and powers in exponential form, roots of complex numbers.
What can make this go out of date
- None at runtime — the arithmetic and the principal-value conventions are fixed; nothing on this page tracks a published series.
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