What is a complex number?

A number of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = −1. Complex numbers extend the real number line into a two-dimensional plane.

Why do we use the conjugate in division?

Multiplying numerator and denominator by the conjugate (c − di) turns the denominator into c² + d² — a real number. This eliminates i from the denominator so the result can be written in standard a + bi form.

By definition, i² = −1. In multiplication, when you FOIL two complex numbers, the i·i term becomes i² = −1, which flips the sign of that coefficient. This is what makes complex arithmetic different from real arithmetic.

What is the polar form of a complex number?

An alternative representation r∠θ, where r = |z| = √(a² + b²) is the magnitude (modulus) and θ = atan2(b, a) is the angle (argument). Multiplication in polar form is elegant: r₁∠θ₁ × r₂∠θ₂ = r₁r₂∠(θ₁+θ₂).

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IPL 2026

Complex Number Calculator

Perform arithmetic on complex numbers (a + bi) in Cartesian and polar form.

Complete User Guide

Enter the real and imaginary parts for two complex numbers. Select the operation (Add, Subtract, Multiply, Divide) and press Calculate. The result appears in standard form (a + bi), with the magnitude |z|, angle θ in radians, and the equivalent polar notation r∠θ°.

The Mathematical Formula

(a+bi) + (c+di) = (a+c) + (b+d)i

1) Addition: (a+bi)+(c+di) = (a+c)+(b+d)i. 2) Subtraction: (a+bi)−(c+di) = (a−c)+(b−d)i. 3) Multiplication: (a+bi)(c+di) = (ac−bd)+(ad+bc)i (using FOIL and i²=−1). 4) Division: multiply by the conjugate — [(a+bi)(c−di)] / (c²+d²), which eliminates i from the denominator.

Frequently Asked Questions

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Complex Number Calculator

What is a complex number?

Why do we use the conjugate in division?

What happens to i²?

What is the polar form of a complex number?