Maths > Complex Numbers > 11.0 Cube roots of unity
Complex Numbers
1.0 Definition
2.0 Algebraic operations
3.0 Conjugate of complex number
4.0 Properties of conjugate
5.0 Representation of complex number
5.1 Cartesian form (Geometric Representation)
5.2 Principal value of complex number
5.3 Properties of modulus
5.4 Trigonometric/Polar Representation
6.0 Euler's formulae
7.0 Properties of Argument
8.0 De Moivre's Theorem
9.0 Square root of a complex number
10.0 The ${n^{th}}$ root of unity
11.0 Cube roots of unity
12.0 Rotation
13.0 Geometrical properties
14.0 Locus
15.0 Ptolemy's Theorem
11.1 Properties of cube roots of unity
5.2 Principal value of complex number
5.3 Properties of modulus
5.4 Trigonometric/Polar Representation
1. Sum of cube roots of unity is zero i.e., $1 + \omega + {\omega ^2} = 0$. In general $1 + {\omega ^r} + {\omega ^{2r}} = 0$ where $r \in I$ but is not a multiple of $3$.
2. ${\omega ^{3n}} = 1,{\text{ }}{\omega ^{3n + 1}}{\text{ = }}\omega {\text{, }}{\omega ^{3n + 2}}{\text{ = }}{\omega ^2}{\text{ }}\quad ({\text{where }}n = 0,1,2,3...)$
Question 11. Solve $$4 + 5{\left[ { - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}} \right]^{334}} + 3{\left[ { - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}} \right]^{365}}$$
Solution: $$\begin{equation} \begin{aligned} \Rightarrow 4 + 5{\omega ^{334}} + 3{\omega ^{365}} \\ \Rightarrow 4 + 5{\omega ^{333}}.\omega + 3{\omega ^{363}}.{\omega ^2} \\ \Rightarrow 4 + 5\omega + 3{\omega ^2} \\ \Rightarrow 4 + 5\left[ { - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}} \right] + 3\left[ { - \frac{1}{2} - i\frac{{\sqrt 3 }}{2}} \right] \\ \Rightarrow 4 - \frac{5}{2} + i\frac{{5\sqrt 3 }}{2} - \frac{3}{2} - i\frac{{3\sqrt 3 }}{2} \\ \Rightarrow i\sqrt 3 \\\end{aligned} \end{equation} $$
