Are There Simplification Rules for Modulo 1 Arithmetic?

[svensl]

I was wondering whether there are any simplification/rules when doing modulo 1 arithmetic.

For example: <a+b>*<c+d> = ? or <a+b>^2=?
Here, <> is the fractional part, i.e <4.2> = 0.2

There is plenty I could find on mod N arithmetic with n>1 but nothing on mod 1.

thanks for any tips.

Svensl

 

[AKG]

mattmns, read his post, he's obviously not just talking about integers. He's talking about the reals modulo 1, i.e. arithmetic of the fractional parts of reals.

svensl, what do you mean by "simplification rules"? For example, what could you find on mod N arithmetic?

 

[mattmns]

mattmns, read his post, he's obviously not just talking about integers. He's talking about the reals modulo 1, i.e. arithmetic of the fractional parts of reals.

Woops! I knew I had to be missing something, sorry.

 

[svensl]

Thanks for the reply.

With modulo N arithmetic I was referring to:
http://en.wikipedia.org/wiki/Modular_arithmetic for example.

If N=12 as in hour clock system, I could say that (13 mod 12)*(14 mod 12) = 2 = ((13*14) mod 12) = 2. This I meant by simplification. However, these rules do not work for cases for mod 1. So, (1.222 mod 1)*(5.111 mod 1) is not equal to ((0.222*0.111) mod 1). I was wondering whether there are rules for this sort of thing.

For example, I can write <a + b > = <<a>+<b>>. Again <> stands for mod 1.
Or, <-a> = 1 - <a>.

Are there any for multiplication?

thanks,
svensl

 

[uart]

If N=12 as in hour clock system, I could say that (13 mod 12)*(14 mod 12) = 2 = ((13*14) mod 12) = 2. This I meant by simplification. However, these rules do not work for cases for mod 1. So, (1.222 mod 1)*(5.111 mod 1) is not equal to ((0.222*0.111) mod 1).

Yes but it's not the difference between "1" and "12" that's operating here, it's the far more fundamental difference that in one case you're doing modulo arithmetic over the integers and in the other case you're doing it over the reals.

Try doing modulo 12 arithmetic over the reals and see how many of those results still hold.

 

[svensl]

Thanks for pointing this out uart.

Do you know of any literature which talks about mod 1? There is a wealth of literature on equidistributed mod 1 sequences, number theory, ergodic theory...but I have not found helping me whith my problem.

Cheers,
svensl