Hello everyone, (note there is a summary at the bottom of the post if you're rushed for time)
I'm about to embark on a journey into the study of irrational and transcendental numbers. So I'm basically seeking any guidance for materials, resources, or communities that study this specific topic.
One thing I would like to do is construct a list of known truths about irrational and transcendental numbers that are true in general. So I would specifically be interested in knowing if such a list has already been constructed that I can just obtain a copy of.
For example, I believe the following to be true as almost "self-evident" truths, although it would be nice to obtain formal proofs of these as well.
For example, it seems to me that if is irrational then must necessarily be irrational too since it make no sense that a rational number times itself could produce an irrational number. In fact, isn't the set of rational numbers closed under multiplication? This fact would then be the proof of this?
So I can start my list off with , where is irrational, implies that is irrational.
Also things like where is irrational and is rational, implies that is irrational.
I would like to construct, or find a list, of these types of statements where I can determine from various equations that the number on the left-hand side is necessarily irrational based on what's on the right-hand side.
I would also like to obtain a similar list for transcendental numbers if possible.
For example is transcendental. Is then also transcendental, where n is a natural number? I don't know the answer to this but it seems reasonable, and I would love to use this fact, if indeed it is a fact.
Finally, I would also like to obtain a list of what is known about current popular irrational and transcendental numbers.
For example is irrational but algebraic. But numbers like and , are both irrational and transcendental.
But what about a number like , is that known to be either irrational, or transcendental? Or even ?
Also does anyone know if (the golden ratio of the Fibonacci sequence) is transcendental? I know it's irrational, but has it also been proven to be transcendental, or has it been proven to be algebraic? Or is it yet unknown?
These are the types of things I'd like to have a list of.
Summary in Brief
I'm seeking concise information about relationships between irrational, transcendental, and rational numbers. I don't want to have to re-invent the wheel on this, or have to glean it from reading a ton of math books. Surely someone has already compiled a fairly concise list of this type of information about irrational and transcendental numbers, including both what is known, and what isn't known. And that's what I'm hoping to find. Of course, if there is a book that focuses specifically on these issues, that would be a worthy read to be sure. I have no problem reading books, I just don't want to have to glean information from books that aren't focused on the issues I'm addressing here.
Thank you for your time.