Complex Number Calculator

How to use

1. Enter the real and imaginary parts of two complex numbers and pick an operation (add, subtract, multiply, divide).
2. Read the result: algebraic form, modulus, argument, conjugate and reciprocal.
3. Check the polar form r∠θ, where the argument is the principal value in (−π, π].
4. Copy the result — use the fraction form for exact checking and the decimal form for engineering estimates.

How it works

How the four operations work

Addition and subtraction work component-wise; multiplication and division follow formulas. To add, combine real parts and imaginary parts separately: (3+4i) + (1−2i) = 4 + 2i. To multiply, expand (a+bi)(c+di) = (ac−bd) + (ad+bc)i, where the bd term flips sign because i² = −1: (3+4i)(2−i) = 6 − 3i + 8i − 4i² = 10 + 5i. To divide, multiply top and bottom by the conjugate of the denominator so the denominator becomes the real number c²+d²: (1+i) ÷ (1−i) = (1+i)(1+i) ÷ 2 = i.

Modulus and argument: the two geometric quantities

The modulus |z| = √(a²+b²) is the distance from the point to the origin — |3+4i| = 5 — and the argument arg(z) is the angle measured counter-clockwise from the positive real axis: about 53.13° for 3+4i, 90° for i, 180° for −1, and −45° for 1−i. Written in polar form z = |z| ∠ θ, multiplication becomes multiply the moduli and add the angles, while division becomes divide the moduli and subtract the angles — which is why AC circuits and signal processing almost always use this form.

Why the result is written as fractions

This tool treats real and imaginary parts as fractions and does the arithmetic in BigInt throughout, so (1/2 + 1/3 i) + (1/6 + 1/6 i) comes out exactly 2/3 + 1/2 i rather than an approximation like 0.6666…. Fractional coefficients are written as "1/2 i" with a space so they cannot be misread as 1 ÷ (2i). For decimal values, see the "real part (decimal)" and "imaginary part (decimal)" rows.

Calculation basis: complex arithmetic follows (a+bᵢ)(c+dᵢ) = (ac−bd) + (ad+bc)ᵢ, and division multiplies numerator and denominator by the conjugate. Modulus |z| = √(a²+b²) and the argument uses the principal value of atan2(b, a) in (−180°, 180°]. Real and imaginary parts are computed exactly as BigInt rationals (results are kept as fractions); the decimal form is display only.

Code example

class Cx {
  constructor(re, im) { this.re = re; this.im = im; }
  add(o) { return new Cx(this.re + o.re, this.im + o.im); }
  mul(o) { return new Cx(this.re * o.re - this.im * o.im,
                         this.re * o.im + this.im * o.re); }
  abs() { return Math.hypot(this.re, this.im); }
  arg() { return Math.atan2(this.im, this.re); }   // principal value (-π, π]
}

new Cx(3, 4).abs();   // 5

z = complex(3, 4)

abs(z)            # 5.0 (modulus)
z.real, z.imag    # (3.0, 4.0)
z.conjugate()     # (3-4j)

import cmath
cmath.phase(z)    # 0.9273 (principal argument)
cmath.polar(z)    # (5.0, 0.9273) (polar form)

FAQ