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BodhiAI
Nov. 11, 2019

Cube roots,nth roots of unity & theirs properties:

Cube Roots of Unity

Let z is the cube of unity, such that

either z - 1 = 0

z = 1

and are the cube roots of unity.

Properties of Cube Roots of Unity

P-1 One of the complex cube root of unity is real (i.e. 1) and the other two are conjugate complex of each other.

Sum of all cube roots of unity i ZERO i.e. 1+

Product of cube roots of unity is UNITY i.e.

P-2 Each Complex Cube root of unity is square of other.

P-3 Each complex Cube root of unity is reciprocal of each other

P-4 If a is any +iv number, then has roots

If a is any -iv number, then has roots

Geometrical Interpretation of Cube Roots of Unity

Roots of Unity

1

 

Quadrants

Positive x-axis

II Quadrant

III Quadrant

Modulus

arg(z) =

arg(z) = 0

Distances