Cube Root of Unity

As mentioned above, the cube root of Unity are 1,𝛚, 𝛚², where 1 is a real root, 𝛚 and 𝛚² are the imaginary roots.
The essential features or properties of the cube root of Unity are:

The imaginary roots 𝛚 and 𝛚², when multiplied together, yields 1

𝛚×𝛚²= 𝛚³=1

The summation of the roots is zero → 1+𝛚+𝛚²=0.

The imaginary root 𝛚, when squared, is expressed as 𝛚², which is equal to another imaginary root.

Fact check: Do you know? The values of Cube root of (-1) are -1, -𝛚, and -𝛚² 

Now, let us find the meaning of 𝛚 here. To find the cube root of Unity, we will make use of some algebraic formulas. We know that, the cube root of unity is represented as ∛1. Let us assume that ∛1= a so,

∛1= a

⇒ 1 = a³

⇒ a³- 1 = 0 

 ⇒ (a - 1)(a²+a+1) = 0              [using a³-b³= (a - b)(a²+a.b+b²)]

⇒a - 1 =0

⇒ a= 1  …………..(1) 

Again, a²+a+1 = 0

    ⇒ a = (-1 ±√(1²–4×1×1)) / 2×1

⇒ a =  (-1 ±√(–3)) / 2

⇒ a =  (-1 ± i√3) / 2

⇒ a =   (-1 + i√3) / 2     …………(2)

Or

a =  (-1 - i√3) / 2        …………(3)

From equation (1), (2), and (3), we get,

The roots are →  1,  (-1 + i√3) / 2 and  (-1 - i√3) / 2

 Hence, 𝛚 = (-1 + i√3) / 2

𝛚²= (-1 - i√3) / 2
 

• Omega - This is a symbol used for depicting the imaginary roots of the cube root of unity. It is represented by 𝛚. 

 

• Complex Number - The numbers which are represented as m+i.n, where m and n are real numbers and “i”, known as iota, is an imaginary number.