Complex Number Calculator
Calculate with complex numbers: addition, multiplication, division, polar form, conjugate. Visualize on Argand diagram. Free complex number calculator
Complex numbers — a + bi where i² = -1 — are taught in high school as algebra and immediately abandoned. They are then used heavily in signal processing, AC circuit analysis, quantum mechanics, control theory, and computer graphics (quaternions are extended complex numbers). This calculator handles arithmetic (addition, multiplication, division), conversion between rectangular and polar forms, calculations of magnitude / phase / conjugate, and visualization on the Argand diagram.
Two representations of the same number
- Rectangular (Cartesian) form — a + bi. a is real part, b is imaginary part.
- Polar form — r·(cos θ + i·sin θ) = r·e^(iθ). r is magnitude, θ is phase angle.
- Conversion: r = √(a² + b²), θ = atan2(b, a). And: a = r·cos(θ), b = r·sin(θ).
- Euler's formula: e^(iπ) + 1 = 0. The famous identity connecting e, i, π, 1, and 0 in one equation. Cited as the most beautiful in mathematics.
Working example
Input
z₁ = 3 + 4i, z₂ = 1 - 2i
Output
Sum: z₁ + z₂ = (3+1) + (4-2)i = 4 + 2i Difference: z₁ - z₂ = (3-1) + (4+2)i = 2 + 6i Product: z₁ · z₂ = (3·1 - 4·-2) + (3·-2 + 4·1)i = 11 - 2i (computed as (ac-bd) + (ad+bc)i) Quotient: z₁ / z₂ = (3·1 + 4·-2)/5 + (4·1 - 3·-2)/5 · i = -1 + 2i (multiply numerator and denominator by conjugate of denominator) Magnitudes: |z₁| = √(9 + 16) = √25 = 5 |z₂| = √(1 + 4) = √5 ≈ 2.236 Polar form of z₁: r = 5 θ = atan2(4, 3) = 0.927 rad ≈ 53.13° z₁ = 5·e^(0.927i) = 5·(cos(53.13°) + i·sin(53.13°)) Conjugate of z₁: 3 - 4i Conjugate of z₂: 1 + 2i
The 3-4-5 triangle shows up here too — z = 3 + 4i has magnitude 5. Complex-number magnitudes are just Pythagorean distances in the 2D plane.
Why complex numbers matter in practice
- AC electrical circuits — voltage, current, impedance are complex-valued. Ohm's law V = IZ works with complex Z. Resistors are real; capacitors and inductors are imaginary.
- Signal processing — Fourier transform converts time-domain signals to frequency-domain (complex). DSP, audio, image processing all use complex math heavily.
- Control theory — transfer functions, frequency response, stability analysis (poles and zeros in complex plane). The right-half plane is unstable; left-half plane is stable.
- Quantum mechanics — wave function ψ is complex-valued. Probabilities are |ψ|².
- Computer graphics — 2D rotation is multiplication by e^(iθ). Quaternions (4D extension of complex) used for 3D rotations.
- Fractals — Mandelbrot and Julia sets are iterations of complex functions. Beautiful and computationally interesting.
When to reach for this tool
- You are solving electrical engineering problems involving impedance, phasors, or AC analysis.
- You are taking a control systems course and need to compute pole/zero locations.
- You are studying quantum mechanics and want to verify wave function calculations.
- You are implementing FFT or DSP code and need to verify complex arithmetic against expected output.
- You are debugging fractal-rendering code and need to check whether specific complex values diverge or stay bounded under iteration.
What this tool will not do
- It will not do symbolic algebra. For symbolic manipulation of complex expressions, use SymPy / Mathematica.
- It will not handle quaternions or higher-order extensions. Quaternion calculators exist separately.
- It will not solve complex equations symbolically. Roots of polynomials with complex coefficients require numerical methods (separate equation solver tool).
- It will not visualize complex functions in their full glory (domain coloring, Riemann surfaces). Specialized math visualizers handle this.
Frequently asked questions
Why is i defined as √(-1)?
Historical convenience. Solving quadratic equations like x² + 1 = 0 requires "the number whose square is -1". Mathematicians initially called these numbers "imaginary" (Descartes was dismissive) but found they obeyed consistent algebra rules. Today, complex numbers are as "real" mathematically as the reals; the names are just historical.
What does multiplication by i do geometrically?
Rotation by 90° counterclockwise in the complex plane. (a + bi) · i = -b + ai. The real part becomes the new imaginary part; the old imaginary part (with sign flip) becomes the new real part. This is why complex multiplication is so useful for 2D rotation.
What is the conjugate?
Flip the sign of the imaginary part. Conjugate of (a + bi) is (a - bi). Useful for: division (multiply numerator and denominator by denominator's conjugate to rationalize); computing magnitudes (z · z̄ = |z|²); finding real solutions to equations with complex roots (they always come in conjugate pairs for polynomials with real coefficients).
Are complex numbers used in everyday programming?
Not directly in most applications. Many fields (signal processing, control systems, computer graphics, scientific computing, electrical engineering software) use them heavily. Python's complex type and NumPy support are first-class; many JavaScript / C# / Java apps avoid them.
What is e^(iπ) = -1?
Euler's identity, often considered the most beautiful equation in mathematics. From e^(iθ) = cos(θ) + i·sin(θ), at θ = π: cos(π) + i·sin(π) = -1 + i·0 = -1. Rearranged: e^(iπ) + 1 = 0, connecting five fundamental constants in one equation.
Why does AC circuit analysis use complex numbers?
Sinusoidal voltages and currents have amplitude AND phase. Real numbers represent amplitude only. Complex numbers (phasors) represent both: magnitude = amplitude, phase = the timing offset. Linear circuit elements (resistor, capacitor, inductor) have well-defined complex impedances; Ohm's law V = IZ then works with complex algebra to handle phase relationships automatically.
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Last updated · E-Utils editorial team