We are never finished learning and relearning math

“Astronomer Copernicus, conversation with God” by Jan Matejko (1872). Image from Wikimedia Commons

In school, you learned that the Earth was the center of the universe. At least, you learned that people used to think that. Your science teacher then explained to you that actually, the Earth revolves around the sun instead. A man named Nicolaus Copernicus proved it during the Renaissance, and set off a scientific revolution.

If a teacher just wanted to teach you about orbits and the solar system, there would be no point in telling you that people used to be wrong about the sun and the Earth. There would be no point mentioning Copernicus, or Kepler, or Galileo…


The difference between “knowing something and knowing the name of something.”

Photo of Richard Feynman in 1959. Image Source (public domain)

Sunny Balwani did not know much about engineering. This normally would not be a problem — most of us don’t know much about engineering. But Sunny was the president of Theranos, a medical startup in the early 2010s promising to make cutting-edge blood tests, devices which would need engineering breakthroughs just to work properly.

Sunny had a big ego. He had too much pride to admit he didn’t understand what his engineers talked about. So instead, he hid behind jargon, repeating technical terms he overheard from other people, regardless of whether the word made sense or not in context. Sunny…


How we know there are infinitely many prime numbers, explained briefly

Photo by Graham Holtshausen on Unsplash

Last week I wrote about sexy primes, a type of prime number. I mentioned that we’ve known that there are infinitely many prime numbers since about 300 B.C., thanks to a proof documented by Euclid. It’s a nice proof! It’s also a great example of a proof by contradiction. So I’m going to go over Euclid’s proof of infinitely many primes.

What’s a Proof by Contradiction?

Euclid’s proof is a type of proof called “proof by contradiction.” A proof by contradiction works in 3 steps:

  1. Assume the opposite of whatever you’re trying to prove.
  2. Show that assuming the opposite leads to some paradox, absurdity, or…


I Promise This is About Math

Photo by Marionel Luciano on Unsplash

Personally, I like to think that all math is sexy (except real analysis¹). However, I’m well aware that to most people, math is quite unsexy. Have you tried putting the word “math” anywhere near your Tinder bio? It’s not a good idea, I don’t recommend it.

But if there’s one thing in math that’s undoubtedly sexy, it’s sexy primes, which is a real term I did not make up! So let’s spice up this romantic evening (or morning, or whenever you’re reading this) and go over a bunch of stuff we know about sexy primes.

What are sexy primes?

You might remember that a…


What Range by David Epstein teaches us about AI

“Theseus Fighting the Centaur Bianor,” by Antoine-Louis Barye. Image via Wikimedia Commons (public domain)

Range is a good book! It’s about learning skills like a generalist rather than a specialist.

We were taught to specialize. If we truly learned anything from Outliers and all those childhood piano lessons, it was that you should start mastering 1 skill as early as humanly possible, grinding your way to 10,000 hours like Tiger Woods, right? Range says that while that’s not necessarily wrong, the idea has caveats. Many experts started late, and many experts are generalists.

I’m going to talk about one of my favorite parts of the book, which has to do with “centaurs,” AI, and…


Euler was wrong!

Photo by Gabriella Clare Marino on Unsplash

Leonhard Euler was the Mozart of mathematics. If you’ve ever seen the number e, you know what the “e” stands for? Euler. You know the “e” in e-mail? That doesn’t stand for Euler, but it should. Look at this guy:


Explaining and Simulating Binomial Distributions in R

Photo by Nguyen Dang Hoang Nhu on Unsplash

Cheating is as old as humanity itself. It manages to creep into every corner of life, even into something as simple as a coin flip.

Suppose your friend Dave has a coin, and he makes you a bet: if he flips the coin and it lands on tails, you win $5, but if it lands on heads, you lose $5. You take the bet. The coin lands on heads. Dave wins. You take the bet again. It lands on heads again. You keep taking the bet a dozen times, and you get really suspicious of the coin. The coin doesn’t…


“This pill will make you 20% happier”

Photo by Bofu Shaw on Unsplash

You know when you go to a doctor and they try to push specific medications onto you?

I remember going to a specialist doctor a few years ago (I’ll call him Dr. P), and he really, really wanted me to buy this medication, which I’ll just call Dolusol. He told me that Dolusol would not only help me with my problem, it would also increase my quality of life and decrease anxiety. He proclaimed that people on Dolusol “reported 20% more happiness! They reported 35% more calmness!” Then he gave me a snazzy bar graph and like 30 pages of…


A little moment that changed my life

Photo by Inesa Cebanu on Unsplash

I usually didn’t study for math tests in college. It wasn’t because I was a savant who could get every question right without studying. It was because the night before a test, I would usually tell myself “I’ll start studying at 9pm” and then I would “start” around 1am and then sleep at 2am. I slacked off. Also, I majored in math in college, so I pretty much only took math tests.

In my second year, I had yet another midterm where I didn’t study much and got most of the questions wrong. We had to write proofs of geometry…


Too much gamified learning will hold you back

Photo by Paul Morley on Unsplash

Deep down, we all know that you don’t get fluent in a language from only Duolingo. Same goes if you’re only using Rosetta Stone, or any other gamified language-learning app. They supplement your learning, for sure, and they can be good for certain things like vocabulary. But we all understand that clicking buttons and regurgitating the word for “apple” thousands of times will not, on its own, teach you Vietnamese.

The barrier to entry for a foreign language is fear. That’s really it. Everyone is capable enough to memorize words, to understand grammar, to speak in full sentences. We already…

Mike Beneschan

A human, writing (mostly) about math | California | If you want to reach out beneschan@gmail.com

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