What is a complex number and an imaginary unit?

A complex number is an expression of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by the fundamental property i² = -1.

What is an Argand Diagram?

An Argand diagram is a geometric representation of complex numbers as 2D position vectors on a Cartesian coordinate plane, where the horizontal axis represents the Real part (Re) and the vertical axis represents the Imaginary part (Im).

How do you convert between Rectangular and Polar forms?

From Rectangular (a + bi) to Polar (r ∠ θ): r = √(a² + b²) and θ = atan2(b, a). From Polar (r ∠ θ) to Rectangular: a = r · cos(θ) and b = r · sin(θ).

What is De Moivre's Theorem for powers and roots?

De Moivre's theorem states that for any real number θ and integer n: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). To find the n-th roots of z = r e^(iθ), the roots are given by: wk = r^(1/n) · e^(i(θ + 2kπ)/n) for k = 0, 1, ..., n - 1.

How are complex numbers used in AC electrical circuits?

In alternating current (AC) circuit analysis, sinusoids are represented as complex phasors. Resistance is purely real (R), inductive reactance is positive imaginary (+jωL), and capacitive reactance is negative imaginary (-j/(ωC)), allowing circuit networks to be analyzed using standard Ohm's and Kirchhoff's laws: V = I · Z.

Complex Number Calculator – Arithmetic, Polar Conversion & Argand Diagram

Complex Number Calculator

Free online Complex Number Calculator. Add, multiply, divide, compute powers, find n-th roots of unity, calculate AC RLC impedance, and plot vectors on an interactive Argand plane.

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How to Use Complex Number Calculator

Select your calculation mode: Arithmetic (Z1 op Z2), Analytic Functions (Properties of Z1), De Moivre Roots of Unity, or AC RLC Circuit Impedance.
Enter complex numbers in either Cartesian/Rectangular format (a + bi) or Polar/Phasor format (r ∠ θ°).
Choose the operation: Addition (+), Subtraction (-), Multiplication (×), Division (÷), Complex Exponentiation (^), or Parallel Impedance (∥).
Inspect the interactive Argand Diagram with vector arrows, modulus circles, and root polygon geometry.
Review instant conversions across Rectangular, Polar (degrees/radians), and Euler exponential (r · e^(iθ)) notations alongside step-by-step math derivations.

Fundamental Operations on Complex Numbers

Complex arithmetic follows the standard field axioms of real algebra combined with the identity i² = -1: - **Addition & Subtraction**: Performed component-wise: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. - **Multiplication**: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. - **Division**: Multiplied by the complex conjugate of the denominator: (a + bi)/(c + di) = [(ac + bd) + (bc - ad)i] / (c² + d²).

Euler's Formula and Exponential Representation

Euler's identity e^(iθ) = cos(θ) + i sin(θ) establishes a profound bridge between trigonometry and complex analysis. It allows multiplication and division to be conceptualized as scaling and rotation in the complex plane: multiplying by e^(iθ) rotates any vector counterclockwise by angle θ without altering its magnitude.

Roots of Unity and Regular Polygon Geometry

The n-th roots of unity are the solutions to the equation z^n = 1. In the Argand plane, these n solutions lie on the unit circle (|z| = 1) spaced at equal angular intervals of 2π/n radians, forming the vertices of a regular n-sided polygon centered at the origin.

Frequently Asked Questions

What is a complex number and an imaginary unit?: A complex number is an expression of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by the fundamental property i² = -1.
What is an Argand Diagram?: An Argand diagram is a geometric representation of complex numbers as 2D position vectors on a Cartesian coordinate plane, where the horizontal axis represents the Real part (Re) and the vertical axis represents the Imaginary part (Im).
How do you convert between Rectangular and Polar forms?: From Rectangular (a + bi) to Polar (r ∠ θ): r = √(a² + b²) and θ = atan2(b, a). From Polar (r ∠ θ) to Rectangular: a = r · cos(θ) and b = r · sin(θ).
What is De Moivre's Theorem for powers and roots?: De Moivre's theorem states that for any real number θ and integer n: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). To find the n-th roots of z = r e^(iθ), the roots are given by: wk = r^(1/n) · e^(i(θ + 2kπ)/n) for k = 0, 1, ..., n - 1.
How are complex numbers used in AC electrical circuits?: In alternating current (AC) circuit analysis, sinusoids are represented as complex phasors. Resistance is purely real (R), inductive reactance is positive imaginary (+jωL), and capacitive reactance is negative imaginary (-j/(ωC)), allowing circuit networks to be analyzed using standard Ohm's and Kirchhoff's laws: V = I · Z.