A joke in approximating numbers raised to irrational powers

<p><pre><code> sin x = x </code></pre> Half the problems in EE become trivial once you learn this. Sometimes the universe does a bad job of complying with the approximation though.

After following the correct link from @nomemory in the comments, this is good for a bit of a chuckle once you see the formula. If you can evaluate the formula you probably have a calculator or computer on hand and could compute the original value to double precision (I&#x27;m not even sure that the approximation would compute faster, but I didn&#x27;t benchmark it).<p>But even though the approximation has no value in a real world application, the description of getting to the approximation is really good. I&#x27;ve never heard of Pade approximations before, and I liked the lead in from small angle approximations and Taylor series. I&#x27;d say this post is accessible to (and can be appreciated by) advanced undergraduates in engineering or math or comp sci.

Happy to see someone else who watches Michael Penn videos.

Reminds me of a cool proof I saw recently that there are two numbers a and b such that a and b are both irrational, but a^b is rational:<p>Take sqrt(2)^sqrt(2), which is either rational or not. If it&#x27;s rational, we&#x27;re done. If not, consider sqrt(2) ^ (sqrt(2) ^ sqrt(2)). Since (a^b)^c = a^bc, we get sqrt(2) ^ (sqrt(2))^2 = sqrt(2)^2 = 2, which is rational!<p>It feels like a bit of a sleight of hand, since we don&#x27;t actually have to know whether sqrt(2)^sqrt(2) is rational for the proof to work.

One interesting result implies that numbers like 3^(sqrt(3)) will be transcendental (ie no polynomial will evaluate them to 0).<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Gelfond%E2%80%93Schneider_theorem" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Gelfond%E2%80%93Schneider_theo...</a>

Kinda weird that we don&#x27;t get a graph for the final &quot;solution&quot;? I was looking forward to seeing how it compared to the other plots!

Secondly, I remember watching a few months ago a video from Michael Penn, about something called Padé Approximations: Pade Approximation – unfortunately missed in most Caclulus courses. It was a subject worth exploring.

for a second I thought 404 was the joke. Tried thinking hard for maybe 10 seconds to figure out why it was the joke, but then realized it was not.

pi^4+pi^5=e^6 lol