Name and literature on a mathematical object

[Linguofreak]

I've been musing on a certain mathematical object recently, and have not been able to find any information on it via Google. I'm trying to figure out if it has a name that might be easily searchable ("integers mod 2pi" and similar searches don't return anything relevant), and any mathematical literature that might exist on it.

The object is that which can be represented by:

The integers, modulo 2*pi.

The set of complex numbers satisfying e^(i*pi*n), with n being any integer. EDIT: D'oh. This should be "The set of complex numbers satisfying e^(i*n)" (without the pi).

Or, equivalently, the set of all angles consisting of an integer number of radians.

Has anybody ever run across this before?

 

[RGClark]

...
The object is that which can be represented by:
The integers, modulo 2*pi.
The set of complex numbers satisfying e^(i*pi*n), with n being any integer.
Or, equivalently, the set of all angles consisting of an integer number of radians.
Has anybody ever run across this before?

What you described are two different things. For the first, I don't know if there is a name for them, but they are most commonly described as multiples of 2*pi.

For the second it turns out that depending on whether n is even or odd, the result is +1 or -1. This comes from Euler's formula:

For x any multiple of pi, sinx is 0. And cosx is +1 for even multiples of pi, and -1 for odd multiples of pi.

Bob Clark

---------- Post added at 02:00 PM ---------- Previous post was at 01:51 PM ----------

Perhaps by "the integers, modulo 2*pi" you meant the real numbers modulo 2*pi. In that case since two numbers that differ by a multiple of 2*pi are considered equivalent, this like looking at the interval [0, 2*pi).


Bob Clark

 

[Linguofreak]

What you described are two different things. For the first, I don't know if there is a name for them, but they are most commonly described as multiples of 2*pi.

Nope, not multiples of two pi. I'm talking the numbers ..., -1, 0, 1, 2, 3, 4, 5, 6, .7168... (= 7-2pi), 1.7168..., 2.7168..., ...

For the second it turns out that depending on whether n is even or odd, the result is +1 or -1. This comes from Euler's formula:

For x any multiple of pi, sinx is 0. And cosx is +1 for even multiples of pi, and -1 for odd multiples of pi.

See my edit above, the factor of pi should not be in there. I meant the complex numbers satisfying e^(i*n), for integer n.

Bob Clark

---------- Post added at 02:00 PM ---------- Previous post was at 01:51 PM ----------

Perhaps by "the integers, modulo 2*pi" you meant the real numbers modulo 2*pi. In that case since two numbers that differ by a multiple of 2*pi are considered equivalent, this like looking at the interval [0, 2*pi).


Bob Clark

Nope, I meant the integers modulo 2*pi. Or, perhaps, to try to be more clear, consider the integers within the real line. Then consider what happens to the integers when you apply the modulus of 2*pi to the reals.

Alternatively, given a circle of circumference 2*pi, the object I am talking about is the set of points on the circle that can be reached by moving an integer distance around the circle.

 

[RGClark]

...
Alternatively, given a circle of circumference 2*pi, the object I am talking about is the set of points on the circle that can be reached by moving an integer distance around the circle.

OK. I haven't heard of that group of numbers being given a name. You might want to ask on sci.math or the Reddit math subreddit.

Was there some research question on these numbers you wanted to address?

Bob Clark

 

[Linguofreak]

Nothing in particular, it just strikes me as interesting. There are a few things that seem intuitively likely that I don't really know how to confirm. For example, I would assume that for any rational fraction of the circle, there would be an integer that would map arbitrarily close to it (but never exactly, except at 0).

 

[martins]

For example, I would assume that for any rational fraction of the circle, there would be an integer that would map arbitrarily close to it (but never exactly, except at 0).

That seems fairly straightforward if you accept that pi is not a rational number (for which thera are proofs):

If [math]\frac{m}{n} \pi = k[/math] with m, n, k integer > 0, then [math]\pi = \frac{nk}{m}[/math] but since the rhs is rational this equation doesn't hold. qed.

As for getting arbitrarily close, simply use the usual iterative bisection method. Find an initial bracket

[math]\frac{m_1}{n} < \pi < \frac{m_2}{n}[/math]
e.g. [math]m_1=3,\; m_2=4,\; n=1.[/math] Then set [math]\bar{m} = m_1 + m_2, \; \bar{n} = 2n[/math]
[math] \begin{array}{ll} \mathrm{If} \frac{\bar{m}}{\bar{n}} < \pi & m_1 \leftarrow \bar{m},\; m_2 \leftarrow 2m_2,\; n \leftarrow \bar{n} \\ \mathrm{else} & m_1 \leftarrow 2m_1,\; m_2 \leftarrow \bar{m},\; n \leftarrow \bar{n} \end{array} [/math]
and iterate to infinity, halving the interval each time.

 

[Linguofreak]

That seems fairly straightforward if you accept that pi is not a rational number (for which thera are proofs):

If [math]\frac{m}{n} \pi = k[/math] with m, n, k integer > 0, then [math]\pi = \frac{nk}{m}[/math] but since the rhs is rational this equation doesn't hold. qed.

The "not exact except at zero" part is not just straightforward but blindingly obvious from the fact that pi is irrational, which is why stated it parenthetically. The "arbitrarily close" part was what I wasn't quite sure of. If the distribution of the integers here were uneven, so that there were certain intensely they didn't appear in, or something like that, it might not hold.

One interesting property I see in the integers mod 2*pi is that it gives us other ways of ordering the integers (and, I believe, other ways of totally ordering the integers).

But part of the reason I have for asking if there's any existing literature is that I'm wondering about what non-obvious or counterintuitive properties the integers mod 2*pi might have.