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