Re: rational or irrational?

In <1119036683.821783.266310@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
06/17/2005
at 12:31 PM, stush@xxxxxxxxxxxxxx said:

>Is it that simple?

Yes. See, e.g., the posts by Robert Maas and Arturo Magidin.

>The square root of 2 is a real number, but its decimal expansion (as
>in a sequence of base 10 digits) does not exist in our universe.

You don't know that.

>Yet the sequence of digits 0.000000...000001 does exist--you are looking at
>in on your screen right now!

Wrong: "0.000000...000001" is not a sequence of digits. What I saw on
my screen was a finite sequence of characters chosen from "0", "1" and
".". You may have intended that sequence to represent an infinite
sequence of digits, but there is no plausible way of reading it as a
reference to such a sequence. Specifically, it is meaningless to talk
of an infinite sequence of digits terminating in "1" or anything else.

>Consider 0.33333...33333 where ... replaces an infinite amount of
>three's.

ITYM 0.33333...; those trailing 3s are meaningless.

>Is this 1/3?

The string "0.33333..." is an abbreviation for limit n->oo Sigma i=1
to n 3*10^{-i}, which equals 1/3.

>I do not think this is a trivial question.

But only because you do not understand the symbols that you are
throwing around.

>One would typically script the decimal expansion of 1/3

Script? I don't see no stinking script.

>0.33333..., i.e. a zero, a point, then an infinite amount of three's.

No. I see a zero, a point, 5 3s and three more points.

>Yet 0.33333...3333

Meaningless.

>has the same description.

No.

>Is 0.333...333 the same as 0.333333...?

First define "0.333...333" and then ask the question.

>If 0.9999... + 1.0 = 2.0, could 0.000...0001 + 1.0 = 1.0?

Not unless you defined "0.000...0001" as meaning 1.0; it certainly
doesn't have any conventional meaning.

>Is 0.000...0001 infinitesimal?

It isn't anything until you define it.

>It should not be immediantely obvious, for example what is
>0.000..0001 < e for some real e really saying?

Meaningless.

>Consider the complex number i, is i < 2?

No. C is not an oredered field, so "i < 2" is meaningless unless you
define it.

>My point here is infinite digits is not void in subjects of reals,

You don't understand what a decimal expansion is and you don't
understand what an infinite sequence is.

>but the answer is not so simple as yes or no IMHO,

Your opinion is neither informed nor relevant, and is certainly not H.
The answer is quit simple.

>I hope he/she attempts to *understand* the
>answer and not just *know* the answer.

The first step is to understand the question ;-)


In <1119099976.823470.310390@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
06/18/2005
at 06:06 AM, stush@xxxxxxxxxxxxxx said:

>But what is it?

A string of characters with no referent.

>For example consider discarding the n-adics just becaise they are
>not intergers.

The p-adics are well defined; "0.000...0001" is not.

>It's easy to form infinite strings with fixed prefixes and suffixes
>using say grammers.

No. The best that you could do would be to define a set of mappings
from ordinals into symbols. They wouldn't be strings for large
ordinals.

Now, there might be some Mathematical interest in functions from
arbitrary ordinals, ordered sets or even partially ordered sets into
symbols, but it's incumbent on you to both define what you are talking
about in an unambiguous fashion and to show why it is of interest. So
far you have done neither.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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