a+bi Form Calculator — Complex Number Operations
Enter two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i to get their sum, difference, product and quotient in a+bi form, plus the modulus (|z|) and argument (angle) for each.
How does this calculator work?
z₁ = a₁+b₁i and z₂ = a₂+b₂i: addition is component-wise; multiplication uses (a₁a₂−b₁b₂)+(a₁b₂+b₁a₂)i; division multiplies by the conjugate. The modulus |z| = √(a²+b²) is the distance from the origin; the argument arg(z) = atan2(b,a) is the angle. All four operations and both moduli/arguments are shown simultaneously.
Formula
How this is calculated
Complex numbers extend the real line to a two-dimensional plane: the real part a is plotted on the horizontal axis and the imaginary part b on the vertical axis. Addition and subtraction are component-wise: (a₁+b₁i) ± (a₂+b₂i) = (a₁±a₂) + (b₁±b₂)i. Multiplication uses the distributive law together with the definition i² = −1: (a₁+b₁i)(a₂+b₂i) = (a₁a₂ − b₁b₂) + (a₁b₂ + b₁a₂)i.
Division multiplies numerator and denominator by the conjugate of the denominator: (a₁+b₁i)/(a₂+b₂i) = [(a₁a₂+b₁b₂) + (b₁a₂−a₁b₂)i] / (a₂²+b₂²). Division by zero (when both a₂ and b₂ are zero) is undefined. The modulus |z| = √(a²+b²) is the distance of the complex number from the origin in the complex plane, and the argument arg(z) = atan2(b, a) is the angle from the positive real axis measured counterclockwise, returned here in degrees in the range (−180°, 180°].
The vector diagram plots z₁ and z₂ as 2D arrows from the origin; the resultant arrow shows their sum, illustrating why complex addition is geometrically identical to 2D vector addition.
Frequently asked questions
The imaginary part b is the coefficient of i (where i = √−1). By itself b is an ordinary real number; together with the real part a, the pair (a, b) identifies a point — or equivalently an arrow — in the complex plane. Complex numbers arise naturally in AC circuit analysis, quantum mechanics, signal processing and many branches of pure mathematics.
|z| = √(a² + b²) is the distance from the origin to the point (a, b) in the complex plane — the same as the magnitude of the 2D vector (a, b). When you multiply two complex numbers their moduli multiply: |z₁ × z₂| = |z₁| × |z₂|.
The argument arg(z) is the angle the complex number makes with the positive real axis (in degrees or radians). Multiplying two complex numbers adds their arguments: arg(z₁ × z₂) = arg(z₁) + arg(z₂). This is why multiplication in polar form r∠θ is so clean: magnitudes multiply and angles add.
TG we-Calculate Editorial Team. (2026). a+bi Form Calculator — Complex Number Operations [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/a-bi-form-calculator
TG we-Calculate Editorial Team. "a+bi Form Calculator — Complex Number Operations." TG we-Calculate. 2026. https://we-calculate.com/calculator/a-bi-form-calculator.
TG we-Calculate Editorial Team, "a+bi Form Calculator — Complex Number Operations," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/a-bi-form-calculator
@misc{wecalculate_a_bi_form_calculator, title = {a+bi Form Calculator — Complex Number Operations}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/a-bi-form-calculator}}, year = {2026}, note = {TG we-Calculate} }
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