Thread: Irrational and Transcendtal Number Newbie Seeks Guidance.

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.

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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.

Have you done a web-search? Look at this page.

Yes, thank you. I've downloaded much of Wiki into pdfs for reference, and I'm currently using those to try to compile my own concise list.

I guess the thing I'm most interested in are the equations I've mentioned in the OP where you can have an expression on the right-hand side of an equation, that contains various irrational, rational, or transcendental numbers, and be able to say with certainty something about the property of the resulting number on the left-hand side.

The reason I need this is because I'm doing work with geometric figures that produce equations like these, and I want to be able to make statements about the results of those equations.

See my thread here for an example of the type of things I'm working on:

Need a very simple proof checked concerning a triangle.

Edited to add:

In my above "proof", I believe I have shown that H must be irrational. Can I also conclude that it must be transcendental too? In other words, if you have a transcendental number and you multiply it by a by a rational number and add a rational number to it will it still be transcendental? Also if you take the square root of a transcendental number can you say that the result must also be transcendental?

In other words, I want to be able to say as much about H (in this particular problem) as I possibly can. And what I can't be sure of I want to know that too.

I'm going to be doing tons of these types of geometric relationships using irrational and transcendental numbers as various lengths, so I want to be able to say as much as I can possibly say about the resulting numbers. And if there are things I can't be sure of, I want to know that too.

So this is why I'm looking to build a nice concise list of these kind of relationships that I can quickly refer to during my calculations. I'll probably end up building my own table so I can refer to them by number as lemmas, propositions, facts, or whatever.

In other words, I'll be able to say "By facts #1, #2, and 3, H must be irrational", etc. Where say, fact #1 is the fact that multiplying an irrational number by a rational number produces an irrational number. And fact #2 is that adding a rational number to an irrational number produces an irrational number. And fact #3 is the fact that taking the square root of an irrational number must produce an irrational number.

I'm going to be doing a LOT of this, so I want to have a nice "fact base" that I can easily refer to in my proofs. And they are going to need to be well-organized so I can easily refer to them by number as I just eluded to.

By the way I just realized that the Golden ratio is indeed an algebraic number because it equals

I should have realized that before. This is great. That's going to be a really interesting number to work with in my geometric studies.