Complex Arithmetic & Phasors

Complex number calculator: addition, subtraction, multiplication, division, form conversion and nth roots (De Moivre)

Input Format

Angle unit:

Complex Number Z₁

Real Part (a)

Imaginary Part (b)

Complex Number Z₂

Real Part (a)

Imaginary Part (b)

Modulus |Z|

4.472

Argument θ

26.57°

Result Representations

Rectangular Form4 + 2i

Polar Form4.472 ∠ 26.57°

Exponential Form4.472·e^(0.4636 rad·i)

Z₁ Representations

Rectangular Form3 + 4i

Polar Form5 ∠ 53.13°

Exponential Form5·e^(0.9273 rad·i)

Z₂ Representations

Rectangular Form1 − 2i

Polar Form2.236 ∠ -63.43°

Exponential Form2.236·e^(-1.107 rad·i)

Argand Plane

Z_1 + Z_2

Fundamentals & Explanation

Complex Arithmetic Fundamentals

A complex number extends the concept of one-dimensional real numbers by introducing the imaginary unit i, where i² = -1. This expansion allows the resolution of algebraic equations that possess no real solutions and provides essential frameworks for disciplines such as electrical engineering (phasors), signal processing, and quantum mechanics.

Euler's Formula stands as the cornerstone of complex analysis, bridging the gap between trigonometry and the complex exponential, enabling fluid transitions between the diverse representations of a complex number.

Euler's Formula

e^{i\theta} = \cos\theta + i\sin\theta

Arithmetic Operations

Addition and subtraction are intuitively performed component-wise utilizing the binomial (rectangular) form. However, for multiplication and division, it is mathematically more elegant and computationally efficient to employ the polar or exponential form:

Multiplication

Z_1 Z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}

Division

\frac{Z_1}{Z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}

De Moivre's Theorem (Nth Roots)

De Moivre's Theorem generalizes powers and roots for complex numbers. It asserts that any non-zero complex number Z yields exactly n distinct nth roots. These roots are uniformly distributed across the Argand plane, forming a regular polygon of n sides inscribed within a circle of radius r^(1/n).

Roots of a Complex Number

z_k = r^{1/n}\left[\cos\!\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\!\left(\frac{\theta + 2\pi k}{n}\right)\right]

Where k ranges from 0 to n-1, r denotes the modulus of Z, and θ its argument.

Tool Usage Guide

Input Mode: Toggle between the Binomial (a + bi) or Polar (r ∠ θ) form. Upon switching, the calculator will automatically synchronize and map the current values.
Angular Units: You can input and observe the angle θ in degrees (deg) or radians (rad). The tool conducts precise conversions without data loss.
Operations Module: Allows adding, subtracting, multiplying, and dividing two complex numbers (Z₁ and Z₂). View the results instantly in multiple formats.
Roots Module: Enter Z₁, pick a degree (n) spanning from 2 up to 12, and the calculator will analytically derive all corresponding nth roots, accompanied by the angle breakdown step by step.

Argand Plane Visualization

The complex plane (or Argand plane) plots the real part (ℜ) on the horizontal axis and the imaginary part (ℑ) on the vertical axis.

Operations Mode

Depicts Z₁, Z₂, and the result as vectors radiating from the origin (phasors). During addition or subtraction, you will witness the parallelogram or resulting triangle law mapped with visual dashed guides.

Roots Mode

Visualizes Z₁ alongside the n roots generated. The chart will dynamically outline a regular polygon traced by these roots, illuminating the symmetry of De Moivre's theorem.

Interactivity

By toggling the chart's legend to hide or show items, the isometric scale (auto-zoom) adjusts automatically to frame the remaining visible components perfectly.

Mathematical Conventions

atan2(b, a) Function: To circumvent quadrant ambiguities intrinsic to the arctan(b/a) function, this tool computes the argument utilizing atan2. This ensures the principal angle of Z resides strictly within the (-π, π] interval.
Origin Singularity: If Z = 0 + 0i, its modulus is zero yet its argument is undefined. Computationally, the calculator resolves this by assigning an argument of 0 and rendering a warning.
Significant Figures: To mitigate floating-point errors when shifting between modes (Binomial ↔ Polar) or units (Degrees ↔ Radians), the system employs roundtrip-safe formatting, preserving up to 10 significant digits under the hood.

Recommended References

[1] Churchill, R. V., & Brown, J. W. (2013). Complex Variables and Applications (9th ed.). McGraw-Hill Education. A foundational, rigorous text on complex variables.
[2] Weisstein, Eric W. "Complex Number." Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/ComplexNumber.html
[3] LibreTexts. (n.d.). De Moivre's Theorem. math.libretexts.org/.../Polar_Form

Complex Arithmetic & Phasors

Input Format

Complex Number Z₁

Complex Number Z₂

Result Representations

Z₁ Representations

Z₂ Representations

Argand Plane

Fundamentals & Explanation

Complex Arithmetic Fundamentals

Arithmetic Operations

De Moivre's Theorem (Nth Roots)

Tool Usage Guide

Argand Plane Visualization

Mathematical Conventions

Recommended References

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Complex Arithmetic Fundamentals

Arithmetic Operations

De Moivre's Theorem (Nth Roots)

Tool Usage Guide

Argand Plane Visualization

Mathematical Conventions

Recommended References