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I have the following decimal number: $0.01001100011100001111...$

Notice how whenever we have one 0, we also have one 1, two 0's, two 1's, etc.

How do you continue it to infinity and prove that this decimal number is not periodic?

From my understanding of periodic numbers, some portion of the decimal must repeat, but in this case it doesn't repeat, but instead number of 0s and 1s grows larger.

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I am assuming the decimal digits are 1 zero then 1 one then 2 zeros then 2 ones then 3 zeros then 3 ones etc..

Suppose it has period $n$ after some point $\omega$, then you can find a string of $n$ 1s in a row, and a string of $n$ 0s in a row: That contradicts periodicity because every string of length n should be the same after $\omega$.

For example if it has period 3 after 20 letters, you can keep going a bit further to find


and I've bracketed the bit that I used to break periodicity.

Conclusion: That decimal is an irrational number.

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