Continuing from my last blog post : There are proofs that show that pi is irrational. There are also admissions that it is very difficult to prove that pi is irrational. I have been thinking and watching youtube videos related to pi and infinity and even the number 69 (because my bill this morning came up to 69 dollars). I just want to share my thoughts on where I have reached with my thinking on this topic. Feel free to interject in the comments. My thinking is that just like infinity is a concept and not a number so it is that the irrationality of pi is a concept. And just like we have to use limits to show that any number divided by infinity is zero, I in my current thinking is saying that we have to treat irrational numbers the same way. It is said that "Infinity is a concept, not an actual number, so we can't just divide a number by infinity."
If I take the base case 0.1111 and keep adding 1 to the end we can say that it is repeating. If we take another case 0.0101 and we keep adding 0 then 1 it switches between repeating to non-repeating. If the last number added was 1 then it would be repeating (an even number of 1s and 0s) and if the last number added was 0 then it would be non-repeating (up till that point). But it never stops so that we cannot say if it is repeating or non-repeating. If we look at the starting of pi after the point that is 141 we can say that the next time we see 1 or 14 or 141 it has started to repeat but whenever we see any number besides 4 after 1 then it stops that repeating possibility.
The only conclusion I can draw about pi is that it switches between non-repeating and repeating but since it never ends we cannot say what its final state is. It does not have a final state. My thinking is that the only way we can say that the number is irrational is if it ends in an irrational state which means that it is not infinite length or that infinity has an end. But then the other possibility is that it stops switching state and that can happen if 1 (or another number from 0 to 9 or any groups of numbers) stops appearing or if the last added digit stops changing. If we can show that a number can stop appearing or the last added digit stops changing in pi then we can say it is irrational. If a number stops appearing then it is not infinite so forget about that option. So we are left with the last digit that stops changing and if that happens then we can multiply by 10 until we get repeating digits after the decimal point and so that is now out the door.
That is as much as I want to think about for today. Maybe there will be a part three when I go home and think about it some more. These are just raw thoughts that may or may not add up but are worth sharing and putting to paper.
*My thinking above lends itself to recursion. Is pi recursive? Can we use recursion to prove pi is or not is irrational? How can we relate recursion to pi?
*I named my next blog post in this series "Piping hot topic" and then realised that piping starts with pipi (pi by two). An entry for pipi in wikipedia states that pipi is a transcription (ΠΙΠΙ) into Greek of the Tetragrammaton (the four-letter Hebrew theonym of the name of God in the Hebrew Bible). Also Tetragrammaton reminded me of Strobogrammatic which I learnt of earlier while researching pi and the number 69.
*A cool tidbit is that 69 first occurs at the 42nd digit of pi.
*Pi by two or pi squared is approximately 9.8696 and acceleration due to gravity is close to this. This is a coincidence but there is some gistorical context according to Bard - Early definitions of the meter, the unit of length in the metric system, considered using the length of a pendulum whose half-swing had a period of one second. This would have made the numerical value of gravity equal to π² exactly. However, this definition wasn't adopted.
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