A "The 7 Strangest Coincidences in the Laws of Nature" (S. Hossenfelder)

[Bosko]

Everybody knows that's because there are π×10⁷ seconds in a year.

Once a friend of mine, who considered math to be some kind of "intellectual martial art",
began an attack with :
"What is bigger, $\pi^2$ or $2^{\pi}$ ?"
$2^{\pi} \lt g \approx \pi^2$
"Prove it ! - no calculator "
Hahaha ... It's useful though

 

[Herman Trivilino]

$\pi \approx \sqrt{3}+\sqrt{2}$

The true value of $\pi$ equals the diagonal of a unit cube plus the diagonal of a unit square. The rest of the world has it wrong. The value of $\pi$ in common usage is wrong because it's based on unit circles instead of unit cubes. There's a lot more to it. It was all explained to me but I didn't understand. I just stated that it was the diagonal of a unit cube plus the diagonal of a unit square.

The response was "Well, yeah". As if it were an obvious conclusion to the gibberish I'd just heard.

 

[haushofer]

And I’m sure I’m supposed to believe that you picked the number 42 out of thin air, am I right? :)

Yes. Pure coincidence :P

 

[billtodd]

$\pi \approx \sqrt{3}+\sqrt{2}$

If that was exact, you would bound to say:"WTF?!".

 

[martinbn]

If that was exact, you would bound to say:"WTF?!".

How could it be! One is trancendental and the other is not.

 

[Herman Trivilino]

Maybe it is exact and everybody else is using the wrong value!

 

[ohwilleke]

How could it be! One is trancendental and the other is not.

Actually both sides are transcendental numbers.

 

[martinbn]

Actually both sides are transcendental numbers.

$\sqrt3+\sqrt2$ is not.

 

[billtodd]

$\sqrt3+\sqrt2$ is not.

Yes, it's algebraic. $x=\sqrt{2}+\sqrt{3}$ $x^2=5+2\sqrt{6}$, and then again: $(x^2-5)^2-24=p(x)$, this is a polynomial that has this number its one of its roots, thus it's algebraic. $\pi$ can't be a root to a polynomial with rational coefficients.

 

[billtodd]

Actually both sides are transcendental numbers.

No, both are irrational numbers.

 

[billtodd]

How could it be! One is trancendental and the other is not.

I have a question to you.
Can a Transcendental number to the power of algebraic number be a Trascendental number?

For example take $e^{\varphi}$, where $\varphi$ is the golden number solution to $x^2+x+1=0$, how would one prove that it's transcendental?

Can $\pi$ be a Transcendental number raise to the power of an algebraic number?

That would be interesting if it's possible, and if it's not then I would welcome a proof/argument why it's not.

 

[Vanadium 50]

Can π be a Transcendental number raise to the power of an algebraic number?

π = π¹.

 

[jedishrfu]

Looks like its that time again. This thread has run its transcendental course back to itself and so its a good time to close it.

Thank you all for contributing here.

Jedi