The cube roots of unity: ω, its properties and its uses

Unity is another name for the number 1, so the cube roots of unity are the three solutions of Z³ = 1 (in Arabic الجذور التكعيبية للواحد الصحيح). Use last lesson’s method: 1 = cos 0° + i sin 0°, so Z = cos(120°k) + i sin(120°k) for k = 0, 1, 2. The three roots are 1, cos 120° + i sin 120° = −1/2 + (√3/2)i and cos 240° + i sin 240° = −1/2 − (√3/2)i. Algebra gives the same answer: Z³ − 1 = (Z − 1)(Z² + Z + 1) = 0, so Z = 1 or Z is a root of Z² + Z + 1 = 0, namely Z = (−1 ± √3 i)/2. Call the root with amplitude 120° by the Greek letter ω (omega): ω = −1/2 + (√3/2)i. Squaring it, ω² = 1/4 − 3/4 − (√3/2)i = −1/2 − (√3/2)i, which is the third root. So the three cube roots of unity can always be written 1, ω, ω², and ω³ = 1. All three have modulus 1, so they lie on the unit circle, 120° apart, and they are the corners of an equilateral triangle.