Complex Number Calculator
Solve operations on complex numbers instantly. Enter real and imaginary components in rectangular form to calculate sums, differences, products, and quotients. Determine modulus, argument, conjugates, reciprocals, and use De Moivre’s theorem to evaluate exponents and roots. Includes detailed step-by-step breakdowns and an interactive SVG Argand plane diagram.
Perform arithmetic operations, calculate modulus and argument, and visualize complex numbers on the Argand plane.
Calculation result (z₁ + z₂)
4 + 2i
Why Use Our Complex Number Calculator?
Comprehensive Operations
Add, subtract, multiply, and divide complex numbers, or calculate powers and roots using De Moivre’s theorem.
Interactive Argand Diagram
Visualize input and output complex numbers as vectors on an auto-scaled 2D complex plane coordinate system.
Polar & Exponential Formats
Instantly convert rectangular form coordinates into polar form and Euler’s exponential format with degrees/radians angle scales.
Step-by-Step Explanations
Review detailed mathematical equations showing how components are grouped, conjugated, and solved at every step.
100% Client-Side Privacy
All calculations, plots, and formulas are processed entirely in your web browser. No arithmetic data is ever transmitted online.
Quick Presets Cheatsheet
Load sample calculations with one click to understand multiplication, conjugate division, and multi-root geometry.
Who Uses This Complex Number Calculator?
Students & Homework Checks
Verify solutions to complex algebra, practice conjugates, and study De Moivre’s multi-root angles for class.
Electrical Engineering & AC Circuits
Compute AC voltage, current phases, electrical impedance (R + jX), and convert between rectangular and polar formats.
Quantum Mechanics & Physics
Evaluate state vector operations, probability amplitudes, and wave functions using complex matrices and numbers.
Control Systems & Signal Processing
Determine poles and zeros, map transfer functions, and convert Fourier/Laplace transforms to graphical coordinates.
Tutors & Classroom Prep
Create step-by-step example algebraic solutions and visual Argand vector plots to present in lectures.
Scientific & Academic Research
Double check complex solutions and coordinates before plotting final publication-ready figures.
Understanding Complex Numbers & Argand Diagrams
What is a Complex Number?
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined by the equation i² = -1. The number a is called the real part, and b is called the imaginary part.
Geometric Representation: The Argand Plane
Just as real numbers can be represented as points on a one-dimensional number line, complex numbers can be represented as points or vectors in a two-dimensional coordinate system called the complex plane or Argand diagram. The horizontal axis represents the real component, and the vertical axis represents the imaginary component. For example, the complex number 3 + 4i is plotted as the point (3, 4).
Polar and Exponential Forms
In addition to rectangular coordinates (a + bi), a complex number can also be described using polar coordinates:
- Modulus (r): The distance from the origin to the point, calculated as
r = √(a² + b²). - Argument (θ): The angle formed with the positive real axis, computed as
θ = atan2(b, a).
This gives the polar representation: z = r(cos θ + i sin θ). By applying Euler's formula (e^(iθ) = cos θ + i sin θ), we get the exponential form: z = re^(iθ).
De Moivre's Theorem for Powers & Roots
De Moivre's Theorem states that for any real number x and integer n:
This formula makes it simple to compute high powers and fractional roots of complex numbers. A complex number has exactly n distinct n-th roots, spaced symmetrically at angles of (θ + 2kπ)/n for k = 0, 1, ..., n-1.
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Complex Number Calculator — Frequently Asked Questions
The imaginary unit i is defined as the square root of -1, meaning i² = -1. It is used to solve algebraic equations that have no real solutions, such as x² + 1 = 0. Introducing imaginary numbers allows all polynomial equations to be factored completely.
To convert a complex number a + bi to polar form r(cos θ + i sin θ), first find the modulus r = √(a² + b²). Then find the argument (angle) θ = atan2(b, a). In the polar format, r represents distance and θ represents the direction angle.
To divide two complex numbers z₁ / z₂, you multiply the numerator and denominator by the complex conjugate of the denominator. This process (called rationalizing the denominator) converts the denominator into a real number, allowing you to split the result into real and imaginary parts.
De Moivre’s Theorem states that for any complex number in polar form, zⁿ = rⁿ(cos(nθ) + i sin(nθ)). It allows you to calculate powers and roots of complex numbers without performing long, iterative binomial expansions.
The complex conjugate of z = a + bi is z̅ = a - bi. You find it by reversing the sign of the imaginary component. Geometrically, it represents a reflection of the complex number vector across the horizontal real axis.
No. The complex number calculations and Argand SVG plot run entirely in your web browser using client-side JavaScript. None of your math coordinates or equations are sent to our servers, ensuring your work stays completely private.