Try tapping out this sequence, and see how far you can get.
The 0110 sequence consists of an item followed by its opposite.
Take 0 as the initial item, and 1 as its opposite.
So the sequence starts out
To continue the sequence, take the whole sequence so far and use it as the "item". The "opposite' of an item means switching a 0 to a 1, and a 1 to a 0. So the opposite of 01 is 10.
The continued sequence is:
The opposite of 0110 is 1001, so the sequence continues as:
Taking this sequence as an item, the sequence continues as:
0110 1001 1001 0110Taking this sequence as an item, the sequence continues as:
0110100110010110 1001011001101001Taking this sequence as an item, the sequence continues as:
0110100110010110 1001011001101001 1001011001101001 0110100110010110and so on...
Use your left index finger for 0 and your right index finger for 1.
Or, use an index finger and a middle finger on the same hand.
Or sit at a computer and type two keys in this pattern.
Having done this for a long time, I can tap out the first 256 digits consistently.
There are never more than two 1s or two 0s in a row.
Does this sequence ever repeat? That is, is the number represented by a decimal point followed by the digits in this sequence a rational number?
This sequence is known as the Thue-Morse Sequence, and not only is it irrational, it has been proven to be transcendental.
What fascinates me about this sequence is that it contradicts my simplistic notion that irrational numbers don't have a pattern. This sequence has a very simple pattern -- it is hard to imagine a simpler pattern than an item followed by its opposite -- and yet it is irrational. The pattern is powerful because it is a meta-level pattern.
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