Proprietati

Proprietatea 1: Modulul oricarei radacini de ordinul $n$ a unitatii este egal cu 1.

Exemple:

$\begin{vmatrix}\frac{1}{2}+i\frac{\sqrt{3}}{2} \end{vmatrix}=\sqrt{\begin{pmatrix}\frac{1}{2} \end{pmatrix}^{2}+{\begin{pmatrix}\frac{\sqrt{3}}{4} \end{pmatrix}^{2}}}=\sqrt{\frac{1}{4}+\frac{3}{4}}=\sqrt{\frac{4}{4}}=\sqrt{1}=1$

$\begin{vmatrix}\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2} \end{vmatrix}=\sqrt{\begin{pmatrix}\frac{\sqrt{2}}{2} \end{pmatrix}^{2}+{\begin{pmatrix}\frac{\sqrt{2}}{2} \end{pmatrix}^{2}}}=\sqrt{\frac{2}{4}+\frac{2}{4}}=\sqrt{\frac{4}{4}}=\sqrt{1}=1$

Proprietatea 2: Daca $\varepsilon \in U_{n}$ , atunci $\overline{\varepsilon} \in U_{n}$ .

Proprietatea 3: Orice putere intreaca a unei radacini de ordinul $n$ a unitatii este de asemnea o radacina de ordinul $n$ a unitatii. Altfel spus, daca:

$\left.\begin{matrix} \varepsilon\in U_{n} \\ k\in \mathbb{Z} \end{matrix}\right\}\Rightarrow \varepsilon^{k}\in U_{n}.$

Exemplu:

$\begin{pmatrix}\frac{\sqrt{3}}{2}+i\frac{1}{2} \end{pmatrix}^{5}=\begin{pmatrix}\cos\frac{\pi }{6}+i\sin\frac{\pi}{6} \end{pmatrix}^{5}=\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}=\frac{\sqrt{3}}{2}+i\frac{1}{2}$

Acest exemplu ne-a reamintit formula lui Moivre, si anume: ${\color{Red} (\cos\varphi+i\sin\varphi)^{k}=\cos k\varphi+ i\sin k\varphi, \forall k\in \mathbb{Z}}.$

Folosind aceasta formula am putut ridica mai usor la puterea a cinciea numarul complex scris sub forma algebrica.

In continuare vom demonstra cele trei proprietati enuntate mai sus:

1. Modulul oricarei radacini de ordinul $n$ a unitatii este de forma:

$\begin{vmatrix}\cos \frac{2k\pi}{n}+i\sin \frac{2k\pi}{n} \end{vmatrix} =\sqrt{cos^{2}\frac{2k\pi}{n}+\sin^2\frac{2k\pi}{n}}=1.$

2. $\overline{\varepsilon }^{n}=\overline{\varepsilon ^{n}}=\overline{1}=1\Rightarrow \varepsilon \in U_{n}$

3. $(\varepsilon ^{k})^{n}=(\varepsilon ^{n})^{k}=1\Rightarrow \varepsilon ^{k}\in U_{n}$

Atat din suportul teoretic,cat si din aplicatiile pentru alte cazuri particulare putem observa:

$U_{2}={\begin{Bmatrix}1,\varepsilon \end{Bmatrix}}, \varepsilon=-1;$

$U_{3}={\begin{Bmatrix}1,\varepsilon,\varepsilon^{2} \end{Bmatrix}}, \varepsilon=-\frac{1}{2}+i\frac{\sqrt{3}}{2};$

$U_{4}={\begin{Bmatrix}1,\varepsilon,\varepsilon^{2},\varepsilon^{3} \end{Bmatrix}}, \varepsilon=i;$

$U_{6}={\begin{Bmatrix}1,\varepsilon,\varepsilon^{2},\varepsilon^{3},\varepsilon^{4},\varepsilon^{5} \end{Bmatrix}}, \varepsilon=\cos\frac{2\pi }{6}+i\sin\frac{2\pi }{6};$

$U_{8}={\begin{Bmatrix}1,\varepsilon,\varepsilon^{2},\varepsilon^{3},\varepsilon^{4},\varepsilon^{5},\varepsilon^{6},\varepsilon^{7} \end{Bmatrix}}, \varepsilon=\cos\frac{2\pi }{8}+i\sin\frac{2\pi }{12};$

$U_{12}={\begin{Bmatrix}1,\varepsilon,\varepsilon^{2},\varepsilon^{3},\varepsilon^{4},\varepsilon^{5},\varepsilon^{6},\varepsilon^{7} \end{Bmatrix}}, \varepsilon=\cos\frac{2\pi }{12}+i\sin\frac{2\pi }{12}.$

Proprietatea 4: Orice radacina de ordinul $n$ a unitatii este o putere chiar naturala a uneia dintre ele. Altfel spus: $U_{n}=\begin{Bmatrix}1, \varepsilon ,\varepsilon^{2},\varepsilon^{3},...,\varepsilon^{n-1} \end{Bmatrix}, \varepsilon=\cos\frac{2\pi }{n}+i\sin\frac{2\pi }{n}.$

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