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Need a very simple proof checked concerning a triangle.

I wrote this up in the attached PDF. It's a very short "proof".

I would just like to know if it qualifies as a proof.

All I'm doing is using Archimedes equivalence between a circle and a triangle. And then looking at the hypotenuse of that triangle to see if I can draw any conclusions about the nature of that length. It appears to me that I can safely conclude that the hypotenuse must be an irrational length.

I would just like to know if this is a valid proof and conclusion.

Thank you for your time.

Edited to add: The French mathematician Adrien-Marie Legendre proved in 1794 that pi squared is also irrational. I guess I should cite that in the proof as well huh? Since I have no way of isolating pi in my formula. But since pi squared is also irrational this should still work, right?

Re: Need a very simple proof checked concerning a triangle.

Just to make it clear, are you trying to prove that H is irrational in the specific case you described ?

In general I would suggest you to rely on the proof by contradiction approach for this type of problem.

So the first step for you should be an assumption that H is rational. And then work your way into contradiction using similar set of equations that you used.

As far as your solution goes it appears to be OK especially for the pre-undergrad level.

Re: Need a very simple proof checked concerning a triangle.

Hi, thank you for the reply. Here is more information:

Quote:

Originally Posted by

**kmfdm92** Just to make it clear, are you trying to prove that H is irrational in the specific case you described ?

I'm not really trying to prove anything specific about H. I'm just trying to determine what I can say about it with confidence. In other words, I don't care whether H is irrational or rational. I just want to be able to say which it is with confidence and be able to show that this is true (i.e. have a sufficient "proof" that this is indeed the correct result).

Quote:

Originally Posted by

**kmfdm92** In general I would suggest you to rely on the proof by contradiction approach for this type of problem.

A proof by contradiction requires that I guess the nature of H first.

However, is there any advantage in this particular situation to go with a proof by contradiction versus a direct proof showing that it simply must be irrational?

Quote:

Originally Posted by

**kmfdm92** So the first step for you should be an assumption that H is rational. And then work your way into contradiction using similar set of equations that you used.

How would that differ from what I've already done? I would be interested in seeing a specific example if possible.

Quote:

Originally Posted by

**kmfdm92** As far as your solution goes it appears to be OK especially for the pre-undergrad level.

What could be added to this to make it more professional? What's missing? I would really like to know because I'm writing this up in a paper about irrational and transcendental numbers in general. So I would indeed like to have these "proofs" up to par on a professional level.

So anything you can suggest that would improve the presentation of this result would be greatly appreciated.

I confess that I don't understand how a proof by contradiction would be much different in this case.

Currently my proof goes as follows:

1. Can I say anything about the nature of the number H?

2. I show directly by Pythagorean theorem that H must be irrational.

3. I conclude that H must be irrational.

What would change if I did this by contradiction?

1. Let's say that H is rational.

2. I show directly by Pythagorean theorem that H must be irrational

3. Oops! I was wrong, H must be irrational.

How is this second proof any better from the first proof?

It's a serious question. I don't understand the difference in this particular proof.

I don't see where writing H as p/q where p and q are natural numbers is going to help in this situation.

I don't mean to be difficult. If the proof can be improved upon and appears to be "pre-undergrad level", or lacking in any way, I would love nothing more than to learn how to improve the proof. It's just that, in this case, it's not obvious to me how the proof can be improved upon. So I would sincerely appreciate an example of that if possible.

Thank you.