Tag Archives: math

My First Peek at Renormalization

I have vaguely known about renormalization since the 1970’s, but had never seriously studied it. Out of curiosity I watched Renormalization and envelopes on YouTube Thursday evening. This was the final lecture of the Asymptotics and perturbation methods course by Prof. Steven Strogatz of Cornell University. I had watched the first two lectures of the course, but none of the others until this one. Fortunately, there were relatively few explicit dependencies on them, so I was able to follow this quite well. Here is the description:

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Significant Figures

I found this post on Facebook: Why is it important to know so many digits of pi?.

My comment:

As someone who started computing with log tables and slide rules, the first question I ask is how many significant digits do the other variables in your calculation have? The smallest such number tells you how many digits of pi you need. With electronic devices there is no harm in using more in your calculation, as many as your device has, but do not let that give you a false idea of the precision of your result.

I learned about significant figures in my high school chemistry in 1967-68. (Thank you, Mr. Wheeler!). Use of appropriate significant figures, also from a chemistry class, clearly explains the concept and its use in practice.

I only first saw Star Trek (TOS) after high school, in reruns. Thanks to that chemistry class I gag every time I hear Mr. Spock reporting some calculation to an absurd number of decimal places. His input data could not possibly be that precise!

Calculating a limit

Today I saw this problem on Medium: Compute the limit

Followed by “Pause the article and attempt a solution now.” (Don’t cheat and look ahead)

So I did. I do not really like factorials, so I immediately thought of Stirling’s approximation:

ln n! ≃ n ln nn as n → ∞

and all of the n‘s cancelled, leaving the result 1/e. I then looked at the author’s solution. My answer was correct, but he used a completely different approach, as you can see. I posted my solution, and got a nice complement from him.

This is Thanksgiving day in the USA. I am thankful that my calculus skills are still pretty good decades after my last formal course in that or any related field .

Columbus and the Flat Earth

Inventing the Flat Earth: Columbus and Modern Historians, by Jeffrey Burton Russell, is the book for the day. Columbus did not show the world that the Earth was round. No educated European in 1492 believed that the Earth was flat. They all knew it was round. As all math geeks know, Eratosthenes of Cyrene had made a good calculation of the circumference of the Earth about 200 BCE.

Catholic church authorities did not say that the plan of Columbus to reach the orient by sailing westward was impossible because the Earth was flat. Their scholastic theology was based on the philosophy of Aristotle, who understood perfectly well that the Earth was round.

There are passages in the Bible that suggest a flat Earth, but almost all theologians of ancient and medieval times knew the evidence for a round Earth was overwhelming, and understood the Bible was not to be taken literally in this and similar cases.

The objection to the plans of Columbus was that, thanks to Eratosthenes, people had a good idea of the distance from the west coast of Europe to the east coast of China, and could easily calculate that no ship of the day could possibly carry enough supplies for the voyage.

Columbus, acting like a 21st century Republican, rejected the best science of the day and chose a smaller alternative value for the circumference that suited his purposes. He was just lucky that the Americas happened to be there. As a result their inhabitants were then horribly unlucky.

The story about Columbus and the flat Earth is a 19th century invention, not history.

Also posted on Facebook.