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The Cantor Problem

Homework on decimal representation and Cantor sets:

1. Rational numbers (numbers that are the ratio of integers) can always be represented by decimals that repeat after some point. For example:

  .3333... = 3/10 + 3/100 + 3/1000 + ...
               = 3( 1/10 + (1/10) + (1/10) + ...  )
               = 3( 1/(1-1/10) - 1)
               = 3(10/9 - 1)
               = 1/3

Use geometric series to express the following rational numbers in the form of p/q for p,q integers.

 a) .313131...
 b) .402402402...
 c) 1.3544444...

2. A cantor set is formed by removing the middle third from the unit interval and repeating this operation for each of the remaining line segments over and over again as shown below for two stages.  The limiting set of points denoted by C  is called the Cantor set.

     Points in a Cantor set are most conveniently represented in the base 3 system.  For example, as shown below, all points in the first interval of C   are the numbers greater that 0 but less than .1 (Base 3), i.e. they take the form .  0xxxx for different values of the x's,  while the points in the second interval are represented by the decimal numbers from .2 (Base 3) to 1 and so take the form, .2xxxx for different values of the x's.

   a) Characterize the decimal representation of all the points in C and C . Can you conjecture what the decimal representation of the points in the actual Cantor set, C  ?

   b) Compute the dimension of C  .


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