Re: Treating Magnitude as Fundamental

lwal...wrote:



What about P4? When Tim first posted
about the polysigned numbers here
at this newsgroup years ago, it was
Robin Chapman who pointed out that
P4 is isomorphic to R x C. Mr.
Chapman proved this by using the
following isomophism:

-1 becomes (-1,i)
+1 becomes (1,-1)
*1 becomes (-1,-i)
#1 becomes (1,1)

and then use componentwise multiplication
in the ring R x C.

I did know this one, it's nice to know.
The four points ( -1, +1, *1 and #1) form a tetraeder in 3D.
So You have 1D redundancy. This makes it interesting for calculations,
it's a kind of error-correction ( Think of gyro-platforms). And one
has only positive numbers (and zero) - another safety. (when one
treats them as ordered four-tuples).

Thus unlike P2 and P3,
P4 is not a field since (-1+1)(-1*1)
equals -1+1*1#1 which is zero. Indeed,
it's easy to show that Pn is never a
field if n is composite.

There are not so many fields possible.
(NB:Tim never claimed this to be a field)

Still, it is a nice multiplication to investigate.
And he gave some beautiful pictures from iterations too.

So, i like Your encouragement of Tim.

With friendly greetings
Hero

.

Relevant Pages

  • divisors of zero
    ... If you can get the URL above to work you can see that Robin Chapman and ... John Rickard found divisors of zero for B_5 numbers in about a day. ...
    (sci.math.symbolic)
  • Re: Integral of a factorial
    ... Please see a remark by Robin Chapman (who is much missed in this ... Huh? ... to this newsgroup for quite some time, ...
    (sci.math)
  • Re: .999... ?= 1
    ... Robin Chapman wrote: ... I know that there is seemingly no need for treating it like a number ... Do not object against zero as the reciprocal. ... Is it of any importance? ...
    (sci.math)
  • Re: Treating Magnitude as Fundamental
    ... Hero wrote: ... at this newsgroup years ago, ... Robin Chapman who pointed out that ... it's not an equal edged tetra. ...
    (sci.math)
  • Re: Integral of a factorial
    ... Please see a remark by Robin Chapman (who is much missed in this newsgroup!) ... Huh? ...
    (sci.math)