On Wed, 31 Mar 2004 20:37:58 +0700, Susanne Vitoux <[log in to unmask]>
wrote:

>> >In the very
>> >unlikely case you never saw the one on 0.99999(ad infinitum) = 1,
>> >I'll be happy to give it.
>>
>> Please do.  I've never seen it put that way.  But that's a truth
>> in limits or asymptotics.  :-)   Something like the opposite of Z
>> eno's paradox ... 1/2 + 1/4 + 1/8 + 1/16 + ... = 1.


>
>Let's set x = 0.99999()   (since I do not know the proper notation on the
>computer, the () indicates the period 9)

The proper notation would be a dot on top of the last 9, or an umlaut
missing one dot.  :-)

>
>10x = 9.99999()
>10x - x = 9.99999() - 0.99999()
>9x = 9
>x = 9/9 = 1
>
>CQFD :-)
>
>Jean-Marc

That would be a NON-mathematician's proof.  :-)

If 9 is what you like to see, then ONE of the correct way of expressing
your result is 9+ 9/10 + 9/100 + 9/1000 + ...  <the "..." means the
pattern continues ad infinitum>.

The mathematician would immediately recognize that the expression is
an INFINITE GEOMETRIC SERIES, with 1/10 being the ratio and the first
term is 9.

So, by the well-known result that ar + ar**2 + ar**3 + ... = a/(1-r),

we immediately have  9/(1-1/10)) = 9/(9/10)) = 10.  :-)

OR. ..  .9 + .9/10 + .9/100 + ... = .9/(1-1/10)) = 1.   :-))

But the MOST amazing of such results is the infinite series that
sums to "e", the irrational number central to the entire field
of probability and statistics, where ln(e) = 1, where ln is the
natural logarithm.  e = btw = 2.718281828...  (and NO, the next
two digits are NOT "18" but "34".  :-))


Jean-Marc and others, you'll find a FANTASY world of REALITY in
mathematics!   For example, ALL rational numbers (any number that
has a FINITE representation as a fraction, which includes all integers,
and all fractions) have MEASURE ZERO on the real line!

That's if you leave out ALL of those rational numbers, you STILL have
the entire real line that can be measured in terms of distances. :-)

Then there are all kinds of "infinities", countable infinity (the
integers), and uncountable intinity (the irrational numbers), and
other alephs (infinities).  :-)))

A practical application:  The richest man on earth, or if you add up
ALL the wealth of ALL the countries (which is still "countable"),
they add up to precisely NOTHING (ZERO/ZILCH) in the real world of
mathematics.  :-))   That's one reason I don't bother to count how
much dough I have.

-- Bob.