The sixth roots of unity are the complex solutions to the equation x6 = 1. To find these roots, we can use Euler’s formula, which states that a complex number can be represented in exponential form as e^(iθ), where θ is the angle in radians.
For sixth roots of unity, we can express them as:
-
1 -
e^{i(2π/6)} = e^{iπ/3} = 0.5 + i(√3/2) -
e^{i(4π/6)} = e^{i2π/3} = -0.5 + i(√3/2) -
e^{i(6π/6)} = e^{iπ} = -1 -
e^{i(8π/6)} = e^{i4π/3} = -0.5 - i(√3/2) -
e^{i(10π/6)} = e^{i5π/3} = 0.5 - i(√3/2)
In summary, the sixth roots of unity in the complex plane are:
-
1 -
0.5 + i(√3/2) -
-0.5 + i(√3/2) -
-1 -
-0.5 - i(√3/2) -
0.5 - i(√3/2)
These six points are evenly spaced on the unit circle in the complex plane, making each root correspond to a 60-degree (or π/3 radians) rotation.