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Posted on Mon Oct 25, 2004 5:44 am - Archived at  
slightly edited version

As long as I am surfing the world-wide 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 = Qn * B + Rn where Qn and Rn are the quotient and the remainder at the n-th stage of the process;

(2) getting Rn eventually equal to zero, or at least getting Rn 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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