Sum of the Nth Roots of Unity
| |
Let ε0 = 1, ..., εN-1 be the vertices of a regular N-gon inscribed on the unit circle. Show that the sum of all εk, k = 0, ..., N-1, equals zero.
|
After a suitable adjustment (rotation) of the axes, the vertices of a regular N-gon inscribed in a unit circle can be identified with the Nth roots of unity, so that εk = e2πk·i/N. Thus we are looking into the value of the sum
k = 0, ..., N - 1.
In [Trigg, #213] the problem is posed a little differently: Prove that the sum of all vectors from the center of a regular n-gon to its vertices is zero.
Solution
Reference
- C. W. Trigg, Mathematical Quickies, Dover, 1985
Copyright © 1996-2010 Alexander Bogomolny
| |
Let ε0 = 1, ..., εN-1 be the vertices of a regular N-gon inscribed on the unit circle. Show that the sum of all εk, k = 0, ..., N-1, equals zero.
|
Solution
The geometric interpretation is decidedly useful because of its explicit symmetry. A rotation through an angle 2π/N maps the n-gon onto itself. In the realm of complex numbers a rotation through angle α around the origin is enacted by multiplying by eiα cosα + I sinα. The sum ∑ of the vertices of a regular n-gon does not change when multiplied by e2πi/N:
implying that the sum is zero.
The proof is also easy algebraically:
| |
e2πi/N∑k=0N-1e2πi·k/N | = ∑1Ne2πi·k/N |
| | | = ∑1N-1e2πi·k/N + e2πi·N/N |
| | | = ∑1N-1e2πi·k/N + 1 |
| | | = ∑1N-1e2πi·k/N + e2πi·0/N |
| | | = ∑0N-1e2πi·k/N |
Equating to zero separately the real and the imaginary parts, one obtains two well known trigonometric identities:
| |
∑ cos(2πI·k/N) = 0,
∑ sin(2πI·k/N) = 0,
|
where, in both cases the sum is from k = 0 through k = N-1.
Complex Numbers
- Algebraic Structure of Complex Numbers
- Division of Complex Numbers
- Useful Identities Among Complex Numbers
- Useful Inequalities Among Complex Numbers
- Trigonometric Form of Complex Numbers
- Real and Complex Products of Complex Numbers
- Complex Numbers and Geometry
- Remarks on the History of Complex Numbers
- Complex Numbers: an Interactive Gizmo
- Cartesian Coordinate System
- Fundamental Theorem of Algebra
- Complex Number To a Complex Power May Be Real
- One can't compare two complex numbers
- Riemann Sphere and Möbius Transformation
- Problems
Copyright © 1996-2010 Alexander Bogomolny
|
