Re: The infintely small number b

On Nov 18, 9:56 am, mike3 <mike4...@xxxxxxxxx> wrote:
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?

It is not equal to zero because of the first rule: b is not equal to
itself. b is not a single number but the "smallest" number. So it is a
quality of the things (quoting Buddha Thu referring Gauss).


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.

Yes, I'm not formally trained in mathematics, so please pardon the
lack of rigor in my writings. Regarding surreals, I have just now
taken a look at them on wikipedia. Conway's infinitesimal allows b = b
and in general the operations go too far making the infinitesimal look
like a single definite number, but not a symmetric counterpart of
infinity in geometric sense (or multiplicative sense). So I think b
does not adapt to the definition and properties of surreal
infinitesimal. I haven't looked at hyperreals - will take a look and
see if b is superfluous.

- venkat
.

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