The Maths Thread

[silentstriker]

I learned it in a class a while back (someone else, not me), but it involved showing it with sqrt(2). There's probably a better way though. That way seems ham handed.

[Spark]

What d'you mean? The proof is two lines. It's logically sort of sneaky but it's incredibly clean.

[uvelocity]

[Neil Pickup]

Two line proof. Very, very sneaky proof too. The specific problem, I should say, is proving that there exists two irrationals a, b such that a^b is rational.

Not looking at e^pi*i are you?

[Spark]

i*pi isn't even real let alone rational

The hint is in the wording: Prove there exists. Not find it, just prove it exists. And as SS said, the simplest version of the proof involves sqrt(2).

[silentstriker]

What d'you mean? The proof is two lines. It's logically sort of sneaky but it's incredibly clean.

Meh.

sqrt(2)^sqrt(2) = irrational. sqrt(2)^sqrt(2)^sqrt(2) = rational. So a = sqrt(2)^sqrt(2), b = sqrt(2).

*Saw it in class, just verified it online. I don't like it.

[Spark]

Yeah that's half the proof, because it relies on the knowledge that sqrt(2)^sqrt(2) is irrational which is actually quite tricky to prove. But that's no problem because, well, if you assume sqrt(2)^sqrt(2) irrational, then...

[silentstriker]

Well yea, but it's already been proven so just [cite] that ****. You don't have to do it .

[Neil Pickup]

Fancy explaining in words that would make sense to my Year 8s?

[silentstriker]

Well, the one I mentions relies on the fact that sqrt(2) is irrational. It has also been proven that sqrt(2)^sqrt(2) is also irrational.

However, sqrt(2)^sqrt(2)^sqrt^(2) simplifies to 2. So if a = sqrt(2)^sqrt(2), and b= sqrt(2), you have the scenario where if a and b are irrational, a^b can be rational. But as I said before, it's not a great proof. As spark said, it requires knowing sqrt(2)^sqrt(2) is irrational - which is not obvious and requires its own proof. As a student, you would just [cite] that and be done, but it's not really an elegant method.

I can't think of another way off the top of my head. Spark implied there's a more elegant way, so he might be able to show a better solution.

[silentstriker]

Wait. Duh.

I remembered from class. It doesn't matter.

You can prove it without proving sqrt(2)^sqrt(2).

You know sqrt(2) is irrational.
You know 2 is rational.

Let x be sqrt(2)^sqrt(2).

If x is rational, then the statement is true because a and b can both be sqrt(2).
If x is irrational, then the statement is true because a can be sqrt(2)^sqrt(2) and b can be sqrt(2) - which would equal 2.

Either way, the statement of a^b with a and b being irrational and the result being rational is true.

Non constructive proof FTL.

But it's nice. This is what Spark probably had in mind.

[Neil Pickup]

^ That makes sense.

I can't, however, remember for the life of me how to compute irrational powers or even begin to get my head around why sqrt(2)*sqrt(2)*sqrt(2) = 2... Worth trying?

[Spark]

Wait. Duh.

I remembered from class. It doesn't matter.

You can prove it without proving sqrt(2)^sqrt(2).

You know sqrt(2) is irrational.
You know 2 is rational.

Let x be sqrt(2)^sqrt(2).

If x is rational, then the statement is true because a and b can both be sqrt(2).
If x is irrational, then the statement is true because a can be sqrt(2)^sqrt(2) and b can be sqrt(2) - which would equal 2.

Either way, the statement of a^b with a and b being irrational and the result being rational is true.

Non constructive proof FTL.

But it's nice. This is what Spark probably had in mind.

(y)

^ That makes sense.

I can't, however, remember for the life of me how to compute irrational powers or even begin to get my head around why sqrt(2)*sqrt(2)*sqrt(2) = 2... Worth trying?

Probably best to think of it as (sqrt(2)^sqrt(2))^sqrt(2) = 2.

[silentstriker]

^ That makes sense.

I can't, however, remember for the life of me how to compute irrational powers or even begin to get my head around why sqrt(2)*sqrt(2)*sqrt(2) = 2... Worth trying?

Just rules of exponents. (a^b)^c = a^(b*c)

sqrt(2)*sqrt(2) = 2. So sqrt(2)^2 = 2.

[nightprowler10]

It's 2am so I might be missing something, but I get ln(2/5) evaluating the attached problem but the correct answer apparently is (1/3)ln(5/2). What the hell am I missing?

EDIT: oh god, ignore me.