November 18th, 2024

A joke in approximating numbers raised to irrational powers

The blog post explores approximating irrational powers without calculators, emphasizing Padé approximations over Taylor Series. It provides rational functions for key mathematical functions and includes Python code for visualization.

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A joke in approximating numbers raised to irrational powers

The blog post discusses a mathematical exploration initiated by a Facebook group discussion about approximating \( \lfloor 3^{\sqrt{3}} \rfloor \) without calculators. The author, Andrei N. Ciobanu, shares a method to approximate numbers raised to irrational powers using basic arithmetic operations. He presents a complex formula for approximating \( a^{\sqrt{c}} \) and discusses the use of Taylor Series and Padé approximations to achieve better accuracy for small numbers. The author highlights the effectiveness of Padé approximations over Taylor Series in handling exponential behaviors and discontinuities. He provides specific approximations for functions like \( e^x \), \( \ln x \), and \( \sqrt{x} \) using rational functions. The post concludes with practical examples, including calculations for \( 2^{\sqrt{2}} \) and \( 3^{\sqrt{3}} \), demonstrating the accuracy of the approximations. The author also includes Python code for visualizing the approximations against actual functions.

- The post explores approximating numbers raised to irrational powers without calculators.

- It introduces Padé approximations as a more effective method than Taylor Series for small numbers.

- Specific rational functions are provided for approximating \( e^x \), \( \ln x \), and \( \sqrt{x} \).

- The author tests the approximations with examples, showing their practical utility.

- Python code is included for visualizing the accuracy of the approximations.

Link Icon 10 comments
By @sevensor - 2 months

    sin x = x
Half the problems in EE become trivial once you learn this. Sometimes the universe does a bad job of complying with the approximation though.
By @parsimo2010 - 2 months
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'm not even sure that the approximation would compute faster, but I didn't benchmark it).

But even though the approximation has no value in a real world application, the description of getting to the approximation is really good. I've never heard of Pade approximations before, and I liked the lead in from small angle approximations and Taylor series. I'd say this post is accessible to (and can be appreciated by) advanced undergraduates in engineering or math or comp sci.

By @jbmsf - 2 months
Happy to see someone else who watches Michael Penn videos.
By @NameError - 2 months
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:

Take sqrt(2)^sqrt(2), which is either rational or not. If it's rational, we'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!

It feels like a bit of a sleight of hand, since we don't actually have to know whether sqrt(2)^sqrt(2) is rational for the proof to work.

By @enugu - 2 months
One interesting result implies that numbers like 3^(sqrt(3)) will be transcendental (ie no polynomial will evaluate them to 0).

https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theo...

By @nightpool - 2 months
Kinda weird that we don't get a graph for the final "solution"? I was looking forward to seeing how it compared to the other plots!
By @Miniminix - 2 months
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.
By @xdavidliu - 2 months
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.
By @0xml - 2 months
pi^4+pi^5=e^6 lol