RE: [tuning] Strange ears

Sarn wrote,

>There is a theorem related to DeMoivre's theorem, or so I have heard, that
>states that we will not actually get anything more fantastic than imaginary
>numbers from ordinary algebras, in effect, we are "traped" to:

>reals:+/-, imaginarys:i/-i, and complex numbers: R+i, R-i, -R+i, -R-i, and
>hypercomplex numbers cannot arise out of these algebras,

Actually, there is also a 4-component algebra ("quarternions") which does
not obey the commutative property, and an 8-component algebra ("octonions")
which does not obey the commutative or associative properties. But it's been
proved that there are no other possible algebras of n-component continuous
quantities.

Sarn wrote,

>Please, PLEASE, P-L-E-A-S-E, ---if somebody, anybody, could calculate:

>a) (2i^(1/12)^n
>b) (2i^(1/12i)^n
>c) (2^(1/12i)^n
>d) (2i^(1/12)^in
>e) (2i^(1/12i)^in
>f) (2^(1/12i)^in

>for me, I would be endebetted to them!!!!!

I can calculate any expressions like this for you in Matlab -- however some
of your expressions are ambiguous, and none of them have the right number of
parentheses! Please be as clear as possible: for example, you might want

(2*(i^((1/12)*i)))^(i*n)

in which case the answer is

n (2*(i^((1/12)*i)))^(i*n)

0 1.0000
1.0000 0.8461 + 0.5331i
2.0000 0.4316 + 0.9020i
3.0000 -0.1157 + 0.9933i
4.0000 -0.6274 + 0.7787i
5.0000 -0.9459 + 0.3244i
6.0000 -0.9732 - 0.2298i
7.0000 -0.7009 - 0.7133i
8.0000 -0.2128 - 0.9771i
9.0000 0.3409 - 0.9401i
10.0000 0.7896 - 0.6137i
11.0000 0.9952 - 0.0983i
12.0000 0.8944 + 0.4473i
13.0000 0.5182 + 0.8552i
14.0000 -0.0175 + 0.9998i
15.0000 -0.5478 + 0.8366i
16.0000 -0.9095 + 0.4158i
17.0000 -0.9911 - 0.1330i
18.0000 -0.7676 - 0.6409i
19.0000 -0.3078 - 0.9515i
20.0000 0.2468 - 0.9691i
21.0000 0.7254 - 0.6883i
22.0000 0.9807 - 0.1957i
23.0000 0.9340 + 0.3572i
24.0000 0.5998 + 0.8002i
etc.

I wrote,

>Actually, there is also a 4-component algebra ("quarternions") which does
not obey the commutative >property, and an 8-component algebra ("octonions")
which does not obey the commutative or >associative properties.

Whoops, I think I got that wrong. The octonions do obey the associative
property but not the commutative or distributive properties, and then the
16-component algebra ("sedenions") do not obey the commutative,
distributive, or associative properties. And that's all, no other
hypercomplex algebras. Check Tony Smith's web page for the real deal.

I suspect you're thinking of Wedderburn's Theorem, the exact details of
which escape me at the moment. But there is a certain class of algebraic
"extensions" to the real numbers, and the only rings which satisfy the
requirements are the field of complex numbers and the ring of real
quaternions. Octernions don't qualify.

On another note, there is a precedent for use of complex numbers as
*probabilities*. Again, the exact details escape me but the paper is
well-known in queueing theory. I can dig it up if anyone is interested;
queueing theory is one of the things I do for a living.

> -----Original Message-----
> From: Paul H. Erlich [mailto:PERLICH@ACADIAN-ASSET.COM]
> Sent: Tuesday, June 06, 2000 11:28 AM
> To: 'tuning@egroups.com'
> Subject: RE: [tuning] Strange ears
>
>
> Sarn wrote,
>
> >There is a theorem related to DeMoivre's theorem, or so I have
> heard, that
> >states that we will not actually get anything more fantastic
> than imaginary
> >numbers from ordinary algebras, in effect, we are "traped" to:
>
> >reals:+/-, imaginarys:i/-i, and complex numbers: R+i, R-i, -R+i,
> -R-i, and
> >hypercomplex numbers cannot arise out of these algebras,
>
> Actually, there is also a 4-component algebra ("quarternions") which does
> not obey the commutative property, and an 8-component algebra
> ("octonions")
> which does not obey the commutative or associative properties.
> But it's been
> proved that there are no other possible algebras of n-component continuous
> quantities.
>
> ------------------------------------------------------------------------
> Old school buds here:
> http://click.egroups.com/1/4057/1/_/239029/_/960316274/
> ------------------------------------------------------------------------
>
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