# Cantor set fun

1. Nov 29, 2011

### spitz

1. The problem statement, all variables and given/known data

I am trying to find if 5/27 and 8/9 are in the Cantor set.

2. Relevant equations

$$C_2=[0,1/9]\cup[2/9,3/9]\cup...\cup[8/9,1]$$

$$C_3=[0,1/27]\cup[2/27,3/27]\cup[4/27,5/27]\cup...\cup[26/27,1]$$

3. The attempt at a solution

I have: $$8/9=(0.22)_3$$
and it is an endpoint in one of the closed sets of C2, so it is in the Cantor set.

I also have: $$5/27=(0.012)_3$$
It is an endpoint in C3, but doesn't the 1 in the base 3 expansion mean that it isn't part of the Cantor set?

Last edited: Nov 29, 2011
2. Nov 29, 2011

### micromass

Staff Emeritus
Your $C_3$ is wrong. The interval $\left[\frac{4}{27},\frac{5}{27}\right]$ is not part of the third stage of the Cantor set. Can you think about what would be the correct interval?

3. Nov 29, 2011

### spitz

Oh yes, how stupid of me.

It's:

$$[0,1/27]\cup[2/27,3/27]\cup[6/27,7/27]\cup[8/27,9/27]\cup...$$

So 5/27 has been removed. Thanks.