<p><strong>Take note that</strong></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">\(z_1\) and \(z_2\) are <strong>conjugates</strong>. So, \(z_2\) is a reflection of \(z_1\) in the real axis.</span></div></li><li><div class="list-item-content"><span class="list-item-text">if&nbsp;\(\omega\) is a cube root of unity, then i.e. \(\omega^3 = 1\)</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">\(\omega^m\) is also a cube root of unity, for any positive integral value of \(m\). \((\omega^m)^3 = \omega^{3m} = (\omega^3)^m = 1^m = 1\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(1 + \omega + \omega^2 = 0\).</span></div></li></ol></li><li><div class="list-item-paragraph">the cube roots of unity can be found using GDC Menu:<br>Algebra – Polynomial tools – Complex roots of Polynomial</div></li><li><div class="list-item-content"><span class="list-item-text">the cube roots of unity are represented by the vertices of an equilateral triangle inscribed in a unit circle with centre at the origin, with \(z_0\) on the real axis, and \(z_0\), \(z_1\) and \(z_2\) separated by equal angles of \(\frac{2\pi}{3}\).</span></div></li></ol>

\(z_1\) and \(z_2\) are **conjugates**. So, \(z_2\) is a reflection of \(z_1\) in the real axis.

if \(\omega\) is a cube root of unity, then i.e. \(\omega^3 = 1\)

\(\omega^m\) is also a cube root of unity, for any positive integral value of \(m\). \((\omega^m)^3 = \omega^{3m} = (\omega^3)^m = 1^m = 1\)

the cube roots of unity can be found using GDC Menu:
Algebra – Polynomial tools – Complex roots of Polynomial

the cube roots of unity are represented by the vertices of an equilateral triangle inscribed in a unit circle with centre at the origin, with \(z_0\) on the real axis, and \(z_0\), \(z_1\) and \(z_2\) separated by equal angles of \(\frac{2\pi}{3}\).

Roots of Unity

Complex Numbers

International Baccalaureate Higher Level Mathematics

Complex Numbers - Roots of Unity

Complex Numbers - Roots of Unity