It looks like you originally posted:
In some high math courses, π is defined as the number such that e^(2π•i) = 1.
But, what is exponential? The application that transforms addition into multiplication (if f : x --> e^x, f(a+b) = f(a)•f(b)).
A classical source (Talmud or Midrash) deduces from a verse of Isaiah that, leatid lavo, tsadiqim will make a great dance in the shape of a circle "around" H-m; while "showing Him" with their hand. And I once heard that, all the tsadiqim, everyone with his own personality and point of view, will understand that everyone was necessary in the Great Plan of the Universe.
So, this can say that, to approach the Unity, we need the whole circumference, in a situation where we can add our particularities to make a product. And all this along the imaginary line, because of the צלם notion.
After this was deleted, it looks like this was your second answer:
2π is generally known as the ratio of circle circumference over its radius.
In some high math courses, π is more rigorously defined as
the (smallest, positive real) number such that exp(2π•i) = 1.
Where exp is the (complex) exponential, and i the basis of the imaginary line. Equation above even get more sense by defining exponential as power series etc.
Okay, but, now, what represents the exponential? An application that transforms additions into multiplications:
exp(a + b) = exp(a) × exp(b)
Now, the Talmud Taanis 31a says
אמר עולא ביראה אמר רבי אלעזר, עתיד הקב''ה לעשות מחול לצדיקים והוא יושב ביניהם בגן עדן וכל אחד ואחד מראה באצבעו שנאמר (ישעיה כה) ואמר כו' זה ה' קיוינו לו Oula Biraa said from Rabi Elazar: Hqb"h will make a circular danse for the righteous, and He sits between them in the Gan Eden, and every one shows ...
Once I heard an explanation as a metaphor. In this world, every one has his own neshama, personality, point of view to Hqb"h. But in the future, tsadiqim will make a circular danse, and every one will range the whole circle, every one will alternatively take all the places, and understand all the points of view. And, as a result, they will 'show' the 'center', they will apprehend and praise Hqb"h.
Okay, so we have an abstract metaphor that, when we could sum up the whole circle, and take a product, we will apprehend the Unity. And this is learned from the verse in Isaiah 25, 9.
Reminds you something? Returning in mathematical notation. What sums up yielding a product? Exponential. Along the imaginary line, because it will transcend limits of this current World. And when you apply this to the Circle , you reach the Unity.
exp(i•x) = 1, so x = 2π
(And because the radius is not mentioned, seems to be taken as the simplest one: 1)
So, circumference of the unit circle is 2π. End of proof.
I wasn’t the one who deleted either one of these answers, so I can’t know for certain what the precise issue was. However, looking at these myself after the fact, I can see some problems with these that may have led to their deletion:
- These answers are very hard to follow. It’s unclear to me in certain parts how you get from one stage to another. Your second, lengthier answer is much better in this regard, but it could be improved further.
- Your bottom line is nice, even if I can’t figure out how you got there. That said, it seems to me that this is your own pshat. Nothing wrong with that, but it could be strengthened a bit. It seems to me that you really wanted this understanding of the Gemara in Taanis to work with the equation e^(iπ)=1. Again, your second answer is better, but it could be better still.
- This one probably didn’t get it deleted, but the actual equation is either e^(iπ)=-1 or e^(2iπ)=1. See further at Wikipedia here.
- Finally, and most importantly, the original question asked for a way that “the irrational and infinite number Pi be derived from the written Torah.” Your answers don’t do that at all. They get it from the oral Torah, based on a verse in the Prophets, but they don’t have a source in the Pentateuch. As such, they doesn’t answer the question, and likely this was the reason they were deleted.
I hope this helps. On behalf of the community, I’m sorry you had this experience, and I hope you decide to stay and share more of your insights here. One of the reasons that I included your posts here, besides to show you some potential problems with them, is so that you don’t have to start entirely from scratch - you can take these and modify them to try to address these concerns, rather than starting from a clean slate, if you’d like.