Re: [Phys-L] irrationals

[John Denker <jsd@av8n.com>]

On 09/12/2017 08:00 AM, Chuck Britton wrote:

> If you think that  1.9999... is NOT identical to 2 then it behooves
> you to present a number that falls between these other two numbers.

That's reasonable; see below for additional detail.

> If you think that infinite expansions should not be a part of the
> particular mathematical system that you are currently using, then
> that is fine.

Lost me there.

Here's how my father explained it to me when I was six:
  him: What's 1 and 1/9?
  me:  1.111111....
  him: What's 1 and 2/9?
  me:  1.222222....
  him: What's 1 and 8/9?
  me:  1.888888....
  him: Do you see where this is going?


It turns out that any finite or repeating decimal represents a
rational number, which is trivial to prove.  The converse is
also true:  Any rational number can be represented as a finite
or repeating decimal.  That's about ten times harder to prove,
but still easy.

Therefore IMHO it is not "fine" to leave repeating decimals
out of your number system.  It would leave some rationals
without a representation.

  This would be bad, but perhaps not fatal;  Europeans knew
  about irrational numbers for more than 1500 years before
  they knew about decimal numerals of any kind.

Let's be clear about the contrast:
  1a) For any finite decimal, the representation is non-unique,
   because you can add trailing zeros:  2.0, 2.00, 2.000 etc.
  1b) You can also write 1.99999.... which introduces some
   additional non-uniqueness.  If you think this is a bug,
   I say it is the world's most harmless bug.
  2) Disallowing repeating decimals would introduce vastly
   more serious bugs.  Some rationals would have no
   representation at all.

Last but not least:  For thousands of years, people have
demonstrated an astonishing ability to fool themselves when
arguing about infinity.  Therefore it is useful to observe
that repeating decimals can be defined *without* mentioning
infinity.  If something repeats in groups of three digits,
just /define/ it in terms of the appropriate multiple of
1/999 and be done with it.  This is an extension of the
definition of non-repeating decimals, fully consistent
with the letter and spirit thereof.