Piping hot topic

Continuing from my last blog post : I have been reading and watching youtubes and chatting with Bard on the topics of pi. The more I pursue this topic the better my understanding gets but at the same time, the more questions I have. I like how I started in one topic of Math and I am having to explore the related topics and disciplines. How could we relate the saying that "information is never lost in the universe" to irrational numbers? Can I create a function that can produce a number from pi at a given position and for a given length? I like this article that shows that the universe is in pi. Comes from the fact that we can find any number within the decimal digits of pi as long as we go out far enough. Also this sum-of-three-cubes puzzle article was a fascinating read. Can every irrational number be expressed as the ratio derived from the measurements of a shape? Maybe even three dimensions and other dimensions.

Is there a negative and positive zero, a sort of -00+? The two zeroes when put together look like the infinity symbol. Pi has me thinking about uncountable and countable infinity and even higher order infinities. Can I represent all numbers as a tree? What if I tried to represent all numbers as a tree starting with 0 at the center. A group of functions are performed on each node to create  other numbers. Does that help me understand countable and uncountable?

If I take any real number the distance from the two closest natural numbers adds to 1. So all real numbers can be paired to form a natural number. Does that help me think about countable and uncountable and infinity? Is 4-pi irrational? yes (irrational + rational = irrational). What if I add two irrationals? Is the result irrational? The answer is yes. If I try to find a rational when added to pi gives me a rational then I can prove that pi is rational or not. Is there any rational when added to pi that gives a rational? The set of rationals and irrationals are closed under addition and subtraction.

Let's say we do not know that pi is irrational and we went looking for a rational number that when added to pi produces a rational number. How can I go about showing that it is impossible to find this number? Bard tells me about proof by contradiction but what if there did not exist a proof that pi was irrational and that was not a known and established fact?

This is how Bard responds:

If there was no established proof that pi is irrational, and it was an open question, you could approach the problem differently. In that case, you might explore various methods to try to find a rational number r such that π+r is rational. This could involve experimentation, mathematical exploration, and potentially discovering new mathematical relationships. However, it's important to note that until proven otherwise, the assumption is that pi is irrational based on historical mathematical knowledge. In the absence of such a proof, the exploration of whether π+r can be rational would be a theoretical pursuit, and any conclusion would be speculative until a rigorous proof is established.

Then I ask, has anyone taken this approach?

The question of whether there exists a rational number r such that π+r is rational has been considered by mathematicians. Historically, attempts to find such a rational number have not been successful. Many mathematical results and proofs are built upon established theories, and the irrationality of pi is a well-established result. While the exploration of mathematical constants and relationships is ongoing, as of my last knowledge update in January 2022, there hasn't been a successful approach to finding a rational r that satisfies π+r being rational. If there have been new developments in this area since then, I would not be aware of them.

*My bill today came up to 97 dollars and I went to see what interesting things I could find related to the topic of pi. Chapter 97 of the book “Life of Pi” is just the title “The story”. In the story the protagonist is challenged to tell a story that will make a person believe in God.

*In the footnotes of my last blog I stated "A cool tidbit is that 69 first occurs at the 42nd digit of pi." Now I come to discover that 97 first occurs just 3 digits before the first occurence of 69 separated only by 1 ... 97169 ... if I minus 1 from 97 I get 96 which is 69 flipped.

*Which leads me to the story of how Archimedes ended up with a 96 sided polygon to estimate the value of pi.

*Also just discovered that P is the 16th letter of the alphabet and I is the 9th letter of the alphabet forming 169. A neat coincidence that ties in nicely with the rest of my footnotes.