Applications
One application of 0.999... as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include:
- 1/7 = 0.142857142857... and 142 + 857 = 999.
- 1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called Midy's theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If it can be proved that a decimal of the form 0.b1b2b3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem. Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest fractals, the middle-thirds Cantor set:
- A point in the unit interval lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.
The nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.
Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying his 1891 diagonal argument to decimal expansions, of the uncountability of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines. A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.
Read more about this topic: 0.999...