Cube Roots of Unity (1 Viewer)

ENFP · Oct 28, 2015

Do the 3 Cube Roots of Unity apply to any cubics or just x³ = 1? Or do they apply to some but not all cubics?

InteGrand · Oct 28, 2015

ENFP said:
Do the 3 Cube Roots of Unity apply to any cubics or just x³ = 1? Or do they apply to some but not all cubics?

$Cube roots of unity means cube roots of the number 1.$

$The following is also true: any non-zero complex number $w$ has exactly three distinct cube roots, where a cube root of $w$ is a complex number $z$ satisfying $z^3 = w$.$

$Moreover, if $w=r\,\mathrm{cis}\,{\theta}$, where $r>0$ and $\theta \in \mathbb{R}$, then the cube roots of $w$ are $z= \sqrt[3]{r}\, \mathrm{cis} \left(\frac{\theta}{3}+\frac{2k\pi}{3} \right)$, $k=0,1,2$.$

ENFP · Oct 28, 2015

InteGrand said:
$Cube roots of unity means cube roots of the number 1.$

$The following is also true: any non-zero complex number $w$ has exactly three distinct cube roots, where a cube root of $w$ is a complex number $z$ satisfying $z^3 = w$.$

$Moreover, if $w=r\,\mathrm{cis}\,{\theta}$, where $r>0$ and $\theta \in \mathbb{R}$, then the cube roots of $w$ are $z= \sqrt[3]{r}\, \mathrm{cis} \left(\frac{\theta}{3}+\frac{2k\pi}{3} \right)$, $k=0,1,2$.$

Thanks for that! I encountered it at the start of my Coroneos book and was curious what it was... The last bit (starting at Moreover) I'll look at later when im done with set 1B

Cube Roots of Unity (1 Viewer)

ENFP

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InteGrand

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