Re: The infintely small number b

On Nov 17, 7:33 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:
On Nov 18, 3:44 am, lwal...@xxxxxxxxx wrote:





On Nov 16, 7:21 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

b is not equal to b in arithmetic sense (addition, subtraction).

I enjoy thinking about these alternate number systems
and how to make them more rigorous, but there can
be no generalization of the real numbers in which we
do not even have b = b.

Otherwise, these numbers remind me a bit more of
Conway's surreal numbers than Robinson's hyperreals --
in particular, the surreals whose birthday is on or
before the first infinite day. For those of you who
are already familiar with the surreals, recall that
these numbers include:

* All the standard real numbers
* an infinite number (Conway's "omega," venkat's "inf"),
* its additive inverse (Conway's "negative
omega," venkat doesn't mention negative numbers)
* an infinitesimal number (Conway's "epsilon," venkat's
"b" of course)
* numbers which differ from a dyadic rational by this
infinitesimal (venkat's "n + b," "n - b" above)

Some of venkat's rules work for these surreals:

b + 0 = b
b - 0 = b
b * 0 = 0
(b / 0) is undefined and not to be taken as inf.
b / b = 1

Many of the venkat rules disagree with the surreals --
and their correct values in the surreals are numbers
whose birthday is beyond the first infinite birthday:

b + b = 2b
b * b = b^2
b - b = 0
n * b = nb
1 / 0 = inf
n / 0 = inf * n

I'm not quite sure why VR defines b / b as 1, but
leaves b - b undefined

Thanks for noticing. Thats an error and I've posted a correction
immediately. b/b is undefined too.

- venkat


Why do you need bee - bee to "bee" (heh) undefined?
Why not bee - bee equal zero?

Also, why don't you like the surreal numbers? They
do what you want, make infinitesimals. And they make
lots more than your bee-number (or whatever that
other silly name was you gave it), and the best thing
is they've got a rigorous definition, which yours
doesn't.
.

Relevant Pages

  • Re: The infintely small number b
    ... the surreals whose birthday is on or ... before the first infinite day. ... Many of the venkat rules disagree with the surreals -- ... whose birthday is beyond the first infinite birthday: ...
    (sci.math)
  • Re: The infintely small number b
    ... be no generalization of the real numbers in which we ... the surreals whose birthday is on or ... before the first infinite day. ... whose birthday is beyond the first infinite birthday: ...
    (sci.math)
  • Re: The infintely small number b
    ... be no generalization of the real numbers in which we ... the surreals whose birthday is on or ... before the first infinite day. ... Why do you need bee - bee to "bee" undefined? ...
    (sci.math)