Imaginary Number Calculator — Complex Number Arithmetic
Perform arithmetic on two complex numbers (a + bi and c + di). Choose an operation — addition, subtraction, multiplication or division — and instantly get the result in rectangular form, plus the modulus, argument in degrees, and complex conjugate.
Operation
How does this calculator work?
Complex number arithmetic: add/subtract component-wise; multiply as (ac−bd)+(ad+bc)i; divide by multiplying by the conjugate of the denominator. The modulus |z| = √(a²+b²) is the vector length; the argument is the angle from the real axis. The complex conjugate of a+bi is a−bi.
Formula
How this is calculated
A complex number z = a + bi has two parts: the real part (a) and the imaginary part (b), where i is the imaginary unit satisfying i² = −1. These two numbers together represent a point — or an arrow (vector) — in the complex plane, where the horizontal axis is the real axis and the vertical axis is the imaginary axis.
Arithmetic on complex numbers follows natural rules. Addition and subtraction are component-wise: real parts combine with real parts, imaginary with imaginary. Multiplication uses the FOIL rule and the identity i² = −1 to simplify: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac−bd) + (ad+bc)i. Division is done by multiplying both numerator and denominator by the conjugate of the denominator (c−di) to eliminate the imaginary part in the denominator.
The modulus |z| = √(a² + b²) is the distance of the result from the origin in the complex plane — equivalent to the vector magnitude. The argument arg(z) = atan2(b, a) is the angle from the positive real axis, measured in degrees here. Together they form the polar form |z|∠θ. The conjugate z̄ = a − bi reflects the result across the real axis.
Frequently asked questions
An imaginary number is a multiple of i, where i is defined as √(−1). Since no real number squared gives a negative result, i is "imaginary." A complex number a + bi combines a real part (a) and an imaginary part (b·i). Together they form a two-dimensional number system used extensively in engineering, physics, and signal processing.
Multiply (a + bi)(c + di) using FOIL: ac + adi + bci + bdi². Since i² = −1, the bdi² term becomes −bd. Collect real and imaginary parts: real = ac − bd, imaginary = ad + bc. So the result is (ac − bd) + (ad + bc)i.
The modulus (or absolute value) |z| of a complex number a + bi is its distance from the origin in the complex plane: |z| = √(a² + b²). This is the same as the magnitude of the vector (a, b). For multiplication, |z₁ × z₂| = |z₁| × |z₂|; for division, |z₁ / z₂| = |z₁| / |z₂|.
Also known as
TG we-Calculate Editorial Team. (2026). Imaginary Number Calculator — Complex Number Arithmetic [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/imaginary-number-calculator
TG we-Calculate Editorial Team. "Imaginary Number Calculator — Complex Number Arithmetic." TG we-Calculate. 2026. https://we-calculate.com/calculator/imaginary-number-calculator.
TG we-Calculate Editorial Team, "Imaginary Number Calculator — Complex Number Arithmetic," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/imaginary-number-calculator
@misc{wecalculate_imaginary_number_calculator, title = {Imaginary Number Calculator — Complex Number Arithmetic}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/imaginary-number-calculator}}, year = {2026}, note = {TG we-Calculate} }
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