Complex Number Polar Phasor & De Moivre Power Calculator

Compute |z| = √(a²+b²), phase angle arg(z), and polar power r^n·e^(inθ).

Complex Modulus |z|
5
Phase Argument θ 53.13 deg
De Moivre Real Part Re(z^2) -7
De Moivre Imaginary Part Im(z^2) 24
Inputs
Real part Re(z) = a.
Imaginary part Im(z) = b.
De Moivre exponent n.
Results & Graph
Complex Modulus |z|
5
Phase Argument θ
53.13 deg
De Moivre Real Part Re(z^2)
-7
De Moivre Imaginary Part Im(z^2)
24

Complex Modulus |z|

Graph 1 • Real A Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 3.3 6.5 50%: 4.272 4.3 50% 75%: 4.589 4.6 75% 100%: 5 5 100% 125%: 5.483 5.5 125% 150%: 6.021 6.0 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Complex Modulus |z| 5 Phase Argument θ 53.13 deg De Moivre Real Part Re(z^2) -7 De Moivre Imaginary Part Im(z^2) 24
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Complex Modulus |z|5Phase Argument θ53.13 degDe Moivre Real Part Re(z^2)-7De Moivre Imaginary Part Im(z^2)24
Summary: z = 3 + 4i has polar modulus r = 5, phase angle θ = 53.13°, and z^2 = -7 + 24i.
Summary: z = 3 + 4i has polar modulus r = 5, phase angle θ = 53.13°, and z^2 = -7 + 24i.

Formula & Mathematical Methodology: Complex Numbers Mathematical Identity

|z| = a² + b²
θ = atan2(b, a)
zn = rn(cos(nθ) + i·sin(nθ))
|z| = √(a² + b²); θ = atan2(b, a); z^n = r^n(cos(nθ) + i·sin(nθ))
  • realA: Real A (numeric parameter)
  • imagB: Imag B (numeric parameter)
  • powerN: Power N (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Complex Numbers

Compute |z| = √(a²+b²), phase angle arg(z), and polar power r^n·e^(inθ).

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: |z| = √(a² + b²); θ = atan2(b, a); z^n = r^n(cos(nθ) + i·sin(nθ)).
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Complex Numbers via |z| = √(a² + b²); θ = atan2(b, a); z^n = r^n(cos(nθ) + i·sin(nθ)).

Frequently Asked Questions

What mathematical formula does the Complex Number Polar Phasor & De Moivre Power Calculator use?

It directly evaluates |z| = √(a² + b²); θ = atan2(b, a); z^n = r^n(cos(nθ) + i·sin(nθ)) in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Algebra & Equations Guide

How to Use the Complex Number Polar Phasor & De Moivre Power Calculator: Practical Guide & Visual Analysis

Convert a complex number z = a + bi into polar modulus r and argument θ, complex conjugate, reciprocal 1/z, and De Moivre power z^n. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching complex number polar phasor & de moivre power calculator, complex numbers, complex number polar phasor de moivre, algebra & equations calculator, this interactive tool helps you convert a complex number z = a + bi into polar modulus r and argument θ, complex conjugate, reciprocal 1/z, and De Moivre power z^n.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Real A, Imag B, Power N, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Complex Numbers), Evaluates Complex Numbers via |z| = √(a² + b²); θ = atan2(b, a); z^n = r^n(cos(nθ) + i·sin(nθ)). Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).