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Posted
on Mon
Oct 25, 2004 5:44 am  Archived at http://www.groupsrv.com/science/about62540.html
slightly edited version
As long as I am surfing the
worldwide web and the newsgroups for mathematical stuff I have been
intrigued by the phenomenon "Is 0.999999.... = 1?". I got the
impression that it is a recurring phenomenon. It seems to coincide more
or less with the start of the education season.
In my opinion 0.999999..... is the most elusive representation of 1,
even more so than 1 = exp(2.pi.i). Let me explain.
0.9999... is a fabricated number. It cannot be obtained in any way as
the outcome of a long division carried out in the usual manner. As far
as I know nobody before noticed this. This may be the reason why it
pops up time and again in newsgroups and similar forums.
But you can perform long division in a different way as follows:
Consider long division A/B
as a process of exhaustion in the style of
Archimedes, i.e. take away from the dividend A as many times of B as
possible. If nothing is left then you are done; otherwise proceed with
taking away from the remainder as many parts B/10 as possible,
etc.
Now look at A/B = 1/1,
but consider A
as 10 * 0.1 instead of 1 * 1, and
take away nine parts B/10
= 0.1 instead of all ten parts. Write down
0.9 in the quotient field to record that you took away nine parts of
size
0.1
Shift down a factor of 10 and treat the remainder like you treated the
original dividend previously. The quotient becomes 0.99 and the new
remainder is 0.01; etc.
Only two things really matter in the process of long division:
(1) maintaining the relation A
= Q_{n} * B + R_{n} where Q_{n} and R_{n} are the
quotient and the remainder at the nth stage of the process;
(2) getting R_{n}
eventually equal to zero, or at least getting R_{n}
arbitrarily close to zero.
I hope that it is clear from the above that 0.9999... has a sensible
meaning and can be equal only to unity.
I have no illusion that this note will spoil the repetitive character
of the phenomenon "0.999999...... = 1?".
Johan E. Mebius
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