Re: sine and modulus ?
- From: amy666 <tommy1729@xxxxxxxxxxx>
- Date: Mon, 20 Oct 2008 15:23:07 EDT
quasi :
On Sun, 19 Oct 2008 15:54:30 EDT, amy666
<tommy1729@xxxxxxxxxxx>
wrote:
quasi wrote :their
On Sun, 19 Oct 2008 07:17:44 EDT, amy666
<tommy1729@xxxxxxxxxxx>
wrote:
im thinking about a sine defined for modulusalgebra.
Sounds like fun.
sin(a) mod p = b mod p
cos(a) mod p = c mod p
b^2 + c^2 = 1 mod p
where the sine and cosine can be computed by
thetaylor series expanded at 0 and mod p.
But regardless of the modulus, the coefficients of
(division by 0).Taylor series are eventually all undefined
multiplicative inverses ...
huh ?
mod p is chosen such that all elements have
All _nonzero_ elements have multiplicative inverses.
What is the Taylor series for sin(x)? (over the
reals)
x - x^3/3! + x^5/5! - ...
computing divisions and powers is no problem in modulus.
Now try it (mod 2). Next try (mod 3).
Do you see a problem?
i havent tried anything but there are only a few possible problems :
1) convergeance problems
even the lim of the average might not converge
2) the taylor series might be considered not to be a sine at all since :
2a) sin^2 + cos^2 =/= 1
2b) sin is not periodic
2c) sin does not satisfy its addition formula's , multiplication formula's , ...
2d) the related addition formula's cannot be computed if sqrt(1-x^2) cant be computed ...
and the alike ...
i dont think a sine and cosine exists that get satisfies all the conditions ...
we probably will have to choose some ...
most logical choices - seem to me - :
1) sine and cosine = its taylor series
or
2) sine and cosine = satisfies addition formula
( i believe it follows from both that sin^2 + cos^2 = 1 )
but im also intrested in what others ( like quasi ) think of it ...
quasi
regards
tommy1729
.
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