About

Complex numbers extend the one-dimensional number line into a two-dimensional plane, enabling solutions for equations involving the square roots of negative numbers. This system is critical in electrical engineering for analyzing AC circuits, where voltage and current phases are represented as complex values.

This tool performs operations on two complex numbers (z₁ and z₂) in rectangular form (a + bi). It calculates the result of addition, subtraction, multiplication, and division instantly. Additionally, it provides the Polar form conversion (r∠θ), which is often required for phasor analysis in physics and engineering tasks.

Formulas

Given two numbers z₁ = a + bi and z₂ = c + di:

Addition: a + c + b + di

Multiplication: ac − bd + ad + bci

Division: ac + bd + bc − adic² + d²

Polar Magnitude (r): √a² + b²

Reference Data

Power of i	Result	Pattern Cycle
i⁰	1	Start
i¹	i	-
i²	-1	Definition
i³	−i	-
i⁴	1	Cycle Repeats
i⁵	i	Cycle Repeats
i^-1	−i	Inverse

Note: In Electrical Engineering, the imaginary unit is often denoted as j to avoid confusion with current i.

Frequently Asked Questions

In electrical engineering, "i" typically represents current. To avoid ambiguity, the imaginary unit is denoted as "j". This calculator supports both notations via the toggle switch.

The complex conjugate of a number z = a + bi is z* = a - bi. It reflects the number across the real axis and is used in division to clear imaginary numbers from the denominator.

Polar form describes the number by its length (magnitude) from the origin and the angle (phase) it makes with the positive real axis. It is written as r∠θ.

Yes, implicitly. The definition of the imaginary unit is that i² = -1, meaning i is the square root of -1. This system allows arithmetic that would be "impossible" in real numbers.