C10 Group, 10th roots unity with complex number multiplication

In summary, the conversation was about 10th root unity with complex number multiplication and working on closure. The speaker had multiplied 2 elements and found that if n+k<=9, there is an element in the set. The next step was to show for n+k>9 and n+k<0. It was suggested to use cosine's periodicity with a period of 360 degrees and to work in radians for this type of problem.
  • #1
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This is for 10th root unity with complex number multiplication. I am working on closure. I have multiplied 2 elements of my set and I have so far that cos[(n+k)360/10] + isin[(n+k)360/10]. Thus I know that if n+k<=9 then there is an element in the set. Now I need to show for if n+k>9 and if n+k<0. If someone could please help me, I would appreciate it. Thanks.
 
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  • #2
If n+ k> 9, then n+ k= 10a+ b where a and b are positive integers and [itex]b\le 9[/itex]. In that case, cos[(n+k)360/10]= cos[(10a+b)360/10]= cos[360a+ b(360/10)]. Now use the fact that cosine is periodic with period 360 (degrees).

If n+ k< 0, then thre exist an integer a such that 10a> -(n+k) so 10a+ n+ k> 0. Again, (10a)(360/10)= 360a, a multiple of 360 degrees.

Is there any specific reason for working in degrees? For problems like this it is almost always standard to work in radians- that is, use [itex]2\pi[/itex] rather than 360.
 

What is C10 Group?

The C10 Group refers to the group of 10th roots of unity, which are complex numbers that, when raised to the 10th power, equal 1. This group is denoted as ℕ10 and can be represented geometrically as a regular decagon on the complex plane.

What are 10th roots of unity?

10th roots of unity are complex numbers that, when raised to the 10th power, equal 1. They can be expressed in the form √1 = e2πik/10, where k = 0, 1, 2, ..., 9. These roots are important in mathematics and physics as they have applications in polynomial equations and signal processing.

How do you multiply complex numbers in C10 Group?

In the C10 Group, complex numbers are multiplied using the property that the product of two 10th roots of unity is equal to the 10th root of the product of the two numbers. This can be represented algebraically as (e2πik/10)(e2πjm/10) = e2πi(k+m)/10. In other words, to multiply two complex numbers in the C10 Group, we simply add the exponents and take the result modulo 10.

Why is C10 Group important?

The C10 Group is important in mathematics as it is a fundamental example of a finite cyclic group. It also has applications in signal processing, cryptography, and other areas of science and engineering. Furthermore, understanding the properties of the C10 Group can help in understanding more complex groups and their applications.

How do we graph C10 Group on the complex plane?

The C10 Group can be graphed on the complex plane by plotting the 10th roots of unity as points on a regular decagon. Each point corresponds to a complex number and the points are evenly spaced around the decagon. This geometric representation helps in visualizing the properties of the group, such as closure and symmetry.

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