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Complex Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
An interesting way of finding all nth (for n > 2) complex roots of unity (real number 1) is by inscribing a regular polynomial with n vertices (each vertex on the circumference of the unit circle), vertices in unit circle with one vertex (1,0) as (1,0) is one of the n roots of 1. Each vertex of this polynomial is nth root of unity.
Complex Representations of Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
We can locate these cube roots of unity in the complex plane. In Figure 7.3, we see that these points lie on the unit circle and are at the vertices of an equilateral triangle. In fact, all nth roots of unity lie on the unit circle and are the vertices of a regular n-gon with one vertex at z = 1.
Complex Analysis
Published in Russell L. Herman, An Introduction to Fourier Analysis, 2016
We can locate these cube roots of unity in the complex plane. In Figure 4.3, we see that these points lie on the unit circle and are at the vertices of an equilateral triangle. In fact, all nth roots of unity lie on the unit circle and are the vertices of a regular n-gon with one vertex at z = 1.
Geometry of symplectic partially hyperbolic automorphisms on 4-torus
Published in Dynamical Systems, 2020
The case of two rationally independent relations for partially hyperbolic matrix is possible. It is sufficient to choose, for example, a block-diagonal integer matrix A composed of two integer -blocks, one of which has eigenvalues , and the second block has two complex conjugate eigenvalues on the unit circle. Note in this case, that the characteristic polynomial of such a matrix is the product of two monic polynomials of second degree with integer coefficients, i.e. it is reduced over the field of rational numbers and the numbers are roots of unity (of degree 3,4,6).