Are non-integer moduli useful in modular arithmetic?

[m00npirate]

I've been looking around for this, but I can't find any discussion of non-integer moduli for use in modular arithmetic. Is it not defined simply because it isn't useful? Every source I look at will say "integers a and b are congruent modulo n if blah blah blah." However, it makes just as much sense to say $\pi + \sqrt{2} \equiv \sqrt{2} \hspace{7 mm}(mod \pi)$.

<strike> The reason I'm wondering about this is because every circle would have zero circumference and area $(mod \pi)$ which seemed absurd. Can anyone explain? </strike>Thanks a ton!

EDIT: Just realized I was being silly, as obviously most circles will still be fine .But is there a physical way to interpret this for those circles with 0 circumference but non-zero area etc? Or is modular arithmetic just not useful here?

 

[matt grime]

You say it 'makes as much sense', but does it? What would it mean? Modulo arithmetic is to do with divisibility. You are naturally moving into a field where everything divides everything else (forgetting zero for a second), so it isn't useful: everything would be equivalent to everything else; it isn't telling you anything.

 

[disregardthat]

It would make sense in the meaning of division if we define it like this (which is obviously the definition the original example implies):

$$a=\pi \cdot q+b, 0 \leq b < \pi \Rightarrow a \equiv b (mod \pi)$$ where $$a,b \in \mathbb{R}, q \in \mathbb{Z}$$

 

[matt grime]

Aboslutely right, Jarle: there are ways to make mod work, though there is no reason why b should restricted to be in the range 0 to pi, unless you're making that transitive by fiat.

Normally one sees things "mod 2pi", as that is as natural period occurring when you use trig functions.

Usually, this is formally written in terms of cosets of a subgroup, in this case inside R. The group $\mathbb{R}/2\pi\mathbb{Z}$ is the natural place to do Fourier analysis. Of course it is isomorphic to the groups $\mathbb{R}/\mathbb{Z}$, and $\mathbb{R}/\pi\mathbb{Z}$, which can also be used - then it comes down to personal preference about what constants you want floating around. I remember someone once saying that 2pi is defined to be equal to 1 for the purposes of Fourier series. (Think Planck constant/length/time by analogy I suppose.)

I don't see why just because things differ by a multiple of pi this implies anything about a circle's physical measurements that is 'absurd'.