Does every irrational number contain arbitrarily long sequences of some

Lucille Cummings

Lucille Cummings

Answered question

2022-06-15

Does every irrational number contain arbitrarily long sequences of some digit?

Answer & Explanation

Brendon Fernandez

Brendon Fernandez

Beginner2022-06-16Added 14 answers

Consider the substitution 1  12 ,  2  1 and iterate this. So
1  12  121  12112  12112121  
This has a limit word w this, in ternary, represents an irrational number (the number 0. w)
0.12112121121121211212112112121121121211212112112121121211211212112112121 
The sequence is not eventually periodic, making it irrational. This is shown by demonstrating that the ratio of 1s and 2s in its expansion is the golden ratio.You may also rather simply demonstrate that each 2 is isolated, so it always appears surrounded by two 1s, moreover, there are never more than two consecutive runs 1s.

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