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ALG-04

De Moivre's theorem

[r (cos θ + i sin θ)]^n = r^n (cos nθ + i sin nθ).

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ComplexDe Moivre

Governing equation

[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)

θ∈[−180,180]\theta \in \left[-180, 180\right]

where

r
Modulus (—)
\theta
Argument (°)
n
Power (—)
\Re
Real part (—)
\Im
Imag part (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)

θ∈[−180,180]\theta \in \left[-180, 180\right]

θ=−180\theta = -180ℜ=−8\Re = -8

②

Domain of the variable — blue

θmin⁡=−180,θmax⁡=180\theta_{\min} = -180,\quad \theta_{\max} = 180

θ∈[−180,180]\theta \in \left[-180, 180\right]

θ=−180\theta = -180θ=−180\theta = -180

③

Response curve — green

[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)

θ∈[−180,180]\theta \in \left[-180, 180\right]

θ=−180\theta = -180ℜ=−8\Re = -8

④

Rate of change — amber

dℜdθ\frac{\mathrm{d}\Re}{\mathrm{d}\theta}

θ∈[−180,180]\theta \in \left[-180, 180\right]

θ=−177.7\theta = -177.7ℜ′=0.049\Re^{\prime} = 0.049

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-04 — De Moivre's theorem) is the form associated with De Moivre. Working symbols: rr, θ\theta, nn →\rightarrow ℜ\Re, ℑ\Im. Induction on the angle-addition formulae, or exp(iθ)^n = exp(inθ).

Purpose

Purpose: compute ℜ\Re, ℑ\Im from rr, θ\theta, nn in Algebra via [r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta) [r (cos θ + i sin θ)]^n = r^n (cos nθ + i sin nθ). Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given r=2.000 —r = 2.000\,\mathrm{—}, θ=30.000 ∘\theta = 30.000\,\mathrm{^{\circ}}, n=3.000 —n = 3.000\,\mathrm{—}, the governing relation [r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta) yields ℜ=4.899e−16 —\Re = 4.899e-16\,\mathrm{—}, ℑ=8.0000 —\Im = 8.0000\,\mathrm{—}. Polar form, θ in degrees, output the real and imaginary parts. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Real part \Re0.0000 —
  • Imag part \Im8.0000 —
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Narration of this film

Polar form, θ in degrees, output the real and imaginary parts.

Induction on the angle-addition formulae, or exp(iθ)^n = exp(inθ).

Reading speed

Watch on YouTube