De Moivre’s Theorem Complex Power (a + bi)^n, Polar Form (r∠θ) & Principal Root Calculator

Compute polar modulus r, argument θ, complex power z^n, and principal root z^(1/n).

Modulus |z^3| = r^3
125 Re=-117, Im=44
Real Part Re(z^n) -117 Re
Imaginary Part Im(z^n) 44 Im
Polar Angle θ of Base z 53.13 degrees
Inputs
⚡ Scenarios:
Re(z)
Real component a of z = a + bi.
Im(z)
Imaginary component b of z = a + bi.
power n
Power / root index n.
Results & Graph
Modulus |z^3| = r^3
125 Re=-117, Im=44
Real Part Re(z^n)
-117 Re
Imaginary Part Im(z^n)
44 Im
Polar Angle θ of Base z
53.13 degrees

Modulus |z^3| = r^3

Graph 1 • De Moivre’s Complex Power & Roots • Domain Architecture & Live Telemetry Maps Real Part A, Imaginary Part B, Exponent Power N through De Moivre’s Complex Power & Roots Formula.
Modulus |z^3| = r^3: 125 Re=-117, Im=44 De Moivre’s Complex Power & Roots • Invariants & Metrics Modulus |z^3| = r^3 125 Re=-117, Im=44 Real Part Re(z^n) -117 Re Imaginary Part Im(z^n) 44 Im Polar Angle θ of Base z 53.13 degrees
Graph 2 • Modulus |z^3| = r^3 Across Real Part A Sweep (40% → 160%) Evaluates r = √(a² + b²); θ = atan2(b, a); z^n = r^n × [ cos(nθ) + i·sin(nθ) ] across 7 operating points (current input highlighted).
0 Re=-11 134.2 Re=-11 268.3 Re=-11 72.83 1.20Re(z) 84.39 1.80Re(z) 101.5 2.40Re(z) 125 3Re(z) 155.8 3.60Re(z) 195.1 4.20Re(z) 243.9 4.80Re(z)
Graph 3 • Joint Sensitivity: Real Part A × Imaginary Part B Compares Modulus |z^3| = r^3 when Imaginary Part B is shifted +25% (solid) vs. −25% (dashed).
0 Re=-11 140.2 Re=-11 280.5 Re=-11 --- Imaginary Part B @ 75% 159.5 -30% 176.8 -15% 198.3 +0% 224.2 +15% 255.0 +30%
Summary: z = 3 + 4i has polar form 5∠53.13°; by De Moivre’s Theorem, z^3 = -117 + 44i.
Summary: z = 3 + 4i has polar form 5∠53.13°; by De Moivre’s Theorem, z^3 = -117 + 44i.
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Formula & Mathematical Methodology: De Moivre’s Complex Power & Roots Formula

r = a² + b²
θ = atan2(b, a)
zn = rn × [ cos(nθ) + i·sin(nθ) ]
r = √(a² + b²); θ = atan2(b, a); z^n = r^n × [ cos(nθ) + i·sin(nθ) ]
  • realPartA: Real Part A (Re(z))
  • imaginaryPartB: Imaginary Part B (Im(z))
  • exponentPowerN: Exponent Power N (power n)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: De Moivre’s Complex Power & Roots

Compute polar modulus r, argument θ, complex power z^n, and principal root z^(1/n).

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: r = √(a² + b²); θ = atan2(b, a); z^n = r^n × [ cos(nθ) + i·sin(nθ) ].
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes De Moivre’s Complex Power & Roots via r = √(a² + b²); θ = atan2(b, a); z^n = r^n × [ cos(nθ) + i·sin(nθ) ].

Frequently Asked Questions

How does the De Moivre’s Theorem Complex Power (a + bi)^n, Polar Form (r∠θ) & Principal Root Calculator compute the solution?

It evaluates r = √(a² + b²); θ = atan2(b, a); z^n = r^n × [ cos(nθ) + i·sin(nθ) ] directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Linear Algebra & Vectors

De Moivre’s Theorem Complex Power (a + bi)^n, Polar Form (r∠θ) & Principal Root Calculator: inputs, method, and how to read the result

Compute polar modulus r, argument θ, complex power z^n, and principal root z^(1/n).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Real Part A: Real component a of z = a + bi.
  • Imaginary Part B: Imaginary component b of z = a + bi.
  • Exponent Power N: Power / root index n.

What the calculation includes

With the default values shown, Modulus |z^3| = r^3 is 125 Re=-117, Im=44. The calculation follows De Moivre’s Complex Power & Roots Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Real Part A from 3 Re(z) to 4 Re(z) changes Modulus |z^3| = r^3 from 125 Re=-117, Im=44 to 181.019 Re=-128, Im=128. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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