ZERO-DOT-NINE-REPEATED   MAY 3RD 2014
     



THEOREM: 0.999999... = 1

Remark: all numerals in this article are to be read in the common decimal system.

Remark: 0.999999... has the usual meaning of  \(\lim_{N \rightarrow \infty} \sum_{i=1}^{N} 9 \times (0.1)^{i}\).

PROOF

Evaluate 1 / 1 by long division not in the common way: 1 / 1 = 1 remainder 0, but rather as follows:

1.0 / 1 = 0.9 remainder 0.1;
1.00 / 1 = 0.99 remainder 0.01;
1.000 / 1 = 0.999 remainder 0.001;
1.0000 / 1 = 0.9999 remainder 0.0001 etc.

or in the layout as (no longer!?) taught in arithmetic class in elementary school:

1 ) 1.0 0 0 0 . . . ( 0.9999
    0.9 | | |
    --- | | |
      1 0 | |
        9 | |
      --- | |
        1 0 |
          9 |
       ---- |
          1 0
            9
          ---
            1

Meaning: 1.0 / 1 = 0.9999 remainder 0.0001


The evaluation of 1.0 / 1 was halted here after processing four digits beyond the decimal point.
It is clear that the Golden Rule of Division With Remainder: dividend = divisor
quotient + remainder, is satisfied at each stage of the process.

During any process of long division this rule is an invariant relation among the quantities involved.
When our specific long division of 1.0 / 1 is continued indefinitely we observe that the remainder gets arbitrarily small.
In formula:

$$1 = 1 \times \sum_{i=1}^{N} 9 \times (0.1)^{i} + (0.1)^N,$$where N is the number of decimals processed.
When N increases indefinitely we obtain in the limit 1 = 0.999999... .

END OF PROOF


REMARK: Among all purely periodic decimal expansions of positive rationals not greater than 1, the expansion 0.999999... is the only one which cannot be obtained by long division performed in the conventional way, i.e. by subtracting at each stage of the process the greatest possible multiple of the divisor, thus leaving the smallest possible remainder. This explains in my opinion part of the mysteries and controverses associated with "0.999999... = 1".

This proof can easily be generalised to show that terminating decimal fractions with digit d in the end position (for instance: 1.35; d = 5) are equal to their non-terminating counterparts with the digit d − 1 in that position, which is followed by 999... (1.35 = 1.34999...).



 



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