Complex Number Calculator

Perform complex number calculations including arithmetic operations, conversions, modulus, and argument calculations.

First complex number z₁
Second complex number z₂
1 + i 3 − 2i i 2 + 2i Conjugate pair (3+2i & 3−2i) cos60° + i sin60°
Privacy first: All calculations are performed locally in your browser. No data is sent to any server.
Results are given in rectangular form (a + bi) and polar form (r ∠ θ) where applicable. Argument θ is displayed in both degrees (°) and radians. Principal argument is in the range (−π, π].

Complex Numbers – Fundamental Concepts

A complex number is expressed as \( z = a + bi \), where \( a, b \in \mathbb{R} \) and \( i = \sqrt{-1} \). The real part is \( a \), the imaginary part is \( b \). Complex numbers extend the one‑dimensional number line to a two‑dimensional complex plane (Argand diagram). They were first introduced by Gerolamo Cardano in the 16th century and later formalized by Rafael Bombelli, becoming essential in algebra, analysis, and applied sciences.

Euler's formula – the most remarkable formula in mathematics:

\( e^{i\theta} = \cos\theta + i\sin\theta \)

When \( \theta = \pi \), we obtain \( e^{i\pi} + 1 = 0 \), linking five fundamental constants (e, i, π, 1, 0). This formula is the foundation of phasor analysis, Fourier transforms, and quantum wavefunctions.

Operations & Formulas (Verified)

Operation Formula (z₁ = a+bi, z₂ = c+di) Example with z₁=3+2i, z₂=1-2i
Addition \( (a+c) + (b+d)i \) 4 + 0i
Subtraction \( (a-c) + (b-d)i \) 2 + 4i
Multiplication \( (ac - bd) + (ad + bc)i \) (3·1 - 2·(-2)) + (3·(-2) + 2·1)i = 7 -4i
Division \( \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2} i \) (3·1+2·(-2))/(1²+(-2)²) + (2·1-3·(-2))/5 i = (-1)/5 + (8)/5 i = -0.2 + 1.6i
Modulus (|z|) \( \sqrt{a^2 + b^2} \) |z₁| = √13 ≈ 3.6055
Argument (arg z) \( \arctan2(b, a) \) (principal value (−π, π]) arg z₁ ≈ 0.5880 rad (33.69°)
Conjugate (z̄) \( a - bi \) z̄₁ = 3 - 2i
Polar form \( z = r(\cos\theta + i\sin\theta) = r e^{i\theta} \) z₁ ≈ 3.6055·e^{0.588i}

Real‑World Applications

Electrical Engineering – AC Circuit Analysis

In alternating current (AC) circuits, voltages and currents are represented as complex phasors. Impedance \( Z = R + iX \) combines resistance R and reactance X. Using complex arithmetic, we compute total impedance, phase angles, and power factors. For example, a series RC circuit with R=4Ω and Xc=3Ω has impedance \( Z = 4 - 3i \) Ω. Its magnitude is 5Ω, and phase angle = -36.87° (current leads voltage). Our calculator can instantly verify these values.

Signal Processing – Fourier Transform

The Discrete Fourier Transform (DFT) converts a time‑domain signal into its frequency components using complex exponentials. Each frequency bin is a complex number representing amplitude and phase. Complex multiplication and addition are the core operations of the FFT algorithm. Engineers rely on complex arithmetic to design filters and analyze spectral content.

Quantum Mechanics – Wavefunctions

The state of a quantum system is described by a complex wavefunction \( \Psi(x,t) \). The Schrödinger equation involves complex coefficients. Observable quantities are obtained via modulus squared (probability). Complex conjugates appear in inner products. Understanding complex numbers is essential for interpreting quantum phenomena.

Common Misconceptions

  • “Complex numbers are just imaginary.” – They combine real and imaginary parts; they are as “real” as real numbers, just extending the number system to solve equations like \( x^2 + 1 = 0 \).
  • “Division by zero is possible.” – No, division by the complex zero (0+0i) is undefined, just as in real numbers. Our calculator warns you.
  • “Argument is unique.” – Argument is multi‑valued (differing by multiples of 2π); we show the principal value (usually in (−π, π]) for consistency.
  • “Multiplication is just like real numbers.” – Multiplication includes cross‑terms because \( i^2 = -1 \); it’s not component‑wise.

Expertise & Authority

This tool is built on formulas verified against authoritative sources: Complex Variables and Applications by Churchill & Brown, Advanced Engineering Mathematics by Kreyszig, and the Wolfram MathWorld entry. The implementation uses double‑precision floating point and the standard Math.atan2 for correct quadrant‑aware argument. Reviewed by GetZenQuery’s Tech team. Last updated March 2026.

Recommended reading:
  • Churchill, R.V. & Brown, J.W. Complex Variables and Applications (9th ed.). McGraw‑Hill, 2013.
  • Kreyszig, E. Advanced Engineering Mathematics (10th ed.). Wiley, 2011.
  • Needham, T. Visual Complex Analysis. Oxford University Press, 1997.

Frequently Asked Questions

The principal argument is the unique value of arg z in the interval (−π, π] (or sometimes [0, 2π)). Our calculator returns the value in radians within (−π, π], which is the most common convention in mathematics and engineering.

Polar form \( r \angle \theta \) means a complex number with modulus r and argument θ. It is equivalent to \( r(\cos\theta + i\sin\theta) \) and to \( r e^{i\theta} \). The angle θ is measured from the positive real axis.

The complex number 0+0i has no direction; its argument is undefined. Our calculator correctly displays “undefined” for the zero complex number.

For powers and roots, use the polar form: \( z^n = r^n e^{in\theta} \). Roots are multi‑valued: \( z^{1/n} = r^{1/n} e^{i(\theta+2k\pi)/n} \). This calculator focuses on basic arithmetic; a dedicated complex power/root calculator is planned.

The conjugate is used to compute the modulus (\( z \bar{z} = |z|^2 \)), to rationalize denominators in division, and to find the real and imaginary parts: \( \Re(z) = (z+\bar{z})/2 \), \( \Im(z) = (z-\bar{z})/(2i) \).
References: MathWorld Complex Numbers; Wikipedia; Kreyszig, E. “Advanced Engineering Mathematics” (10th ed.).