From: Jim Choate <ravage@ssz.com>
Date: Sun, 19 Jan 1997 11:12:12 -0800 (PST)
To: cypherpunks@toad.com
Subject: Re: I beg you, PLEASE prove that 0.123456789101112131415 is IRRATIONAL (fwd)
Message-ID: <199701191919.NAA00377@einstein>
MIME-Version: 1.0
Content-Type: text



Hi,

Forwarded message:

> Date: Sun, 19 Jan 1997 12:20:45 -0600
> From: Sir Robin of Locksley <tozser@stolaf.edu>

>  >Is it possible to prove that number 0.1234567891011121314151617181920...
>  >iz irrational? 
>  
> Most definately. All you need to do is prove that the set of this number is
> uncountable, ergo is irrational. If you have friend who have done math in real
> analysis they can explain more.

irrational means 'not expressible as the ratio of two integers', this does
not imply uncountable.

Whether 0.12345678910111213... is irrational or not depends on how we choose
to define a rational number where the denominator and numerator are both
approaching infinity and how quickly those approaches occur.

[ What is your definition of infinity/infinity ?]

As alluded to before,

  0.12345678910111213    =    12345678910111213.... / n 

  (where n = infinity)
 
0.123456789101112... is certainly countable because by the definition of
countable it must be 1-to-1 with the counting numbers (ie non-negative
integers), which this one clearly is since it contains each positive
non-zero integer (ie the set of numbers required to produce the number are
clearly less, 1 less to be exact, than the counting numbers).

> If you show me an algorythm that calculates the real number TT (=3.14.....)
> I'll give you a Nobel Prize personally!

c=2pi*r

so,

pi = c/2r

You have a algorithm (ie recipe) for determining pi to whatever degree of
precision you are willing to go to. If you insist on using a digital
computer (whose domain of operation is limited to rationals by defintion)
then you are doomed to fail. However, move to analog computers (eg a
compass) and it can be quite easy to calculate, though not easy to plug into
an equation.

> Well, that is not entirely true... The length of any arbitrary line can be any
> number, rational, natural, etc. Real numbers are the numbers that defy all
> other categorization (they are not rational, irrational, natural, etc.) They
> are complex and despite any instinctual perception, there are a lot of them!

Complex numbers are not a member of the Reals, rather the Reals are a
sub-set of the Complex.

[ I recognize this is not what you meant but,

"The slovenliness of our language makes it easier to have foolish thoughts."

                                              George Orwell                ]


VNR Concise Encyclopedia of Mathematics, 15th ed.
ISBN 0-442-22646-2
pp. 74

"If every segment is to have a numerical measure as its length, then a new
domain of numbers is needed, an extension to the domain of rational numbers.
This new domain can no longer be constructed, as in the previous case, by
number pairs. But hints for its construction are provided by a theoretical
analysis of the measuring process for segments."

A mathematicaly rigorous defintion of the class of numbers called 'Real'
is that which equates the members of that set to the possible lengths of an
arbitrary line segment.


                                                   Jim Choate
                                                   CyberTects
                                                   ravage@ssz.com