ALG-04

De Moivre's theorem

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

Reading speed

ComplexDe Moivre

Governing equation

[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)

where

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

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: $r$ , $\theta$ , $n$ $\rightarrow$ $\Re$ , $\Im$ . Induction on the angle-addition formulae, or exp(iθ)^n = exp(inθ).

Purpose

Purpose: compute

\Re

\Im

from

r

\theta

n

in Algebra via

[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\,\mathrm{—}

\theta = 30.000\,\mathrm{^{\circ}}

n = 3.000\,\mathrm{—}

, the governing relation

[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)

yields

\Re = 4.899e-16\,\mathrm{—}

\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

Modulus r2 —Argument \theta30 °Power n3 —

Outputs

Real part \Re0.0000 —
Imag part \Im8.0000 —

Reading speed

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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