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:


No, 2+2 does not equal 5, but that was also never the point

Photo by Brett Jordan on Unsplash

Back in August 2020, an article showed up in Popular Mechanics called “Why Some People Think 2+2=5,” with the tantalizing subtitle, “…And why they’re right.” Spicy!

It was essentially an article about one Twitter thread from Harvard PhD candidate Kareem Carr, where he explains why he thinks people who say 2+2=5 are making “a tremendously deep point.” Some people got very, very pissed off from this thread.

2+2 is not equal to 5. But I believe the most important part of Carr’s thread is actually not the whole “2+2=5” thing. A lot of people (critics and supporters) misunderstand his points…


Don’t finish awful books

Modified version of “Brillenapostel” (1403) by Conrad von Soest. Original image source: Wikimedia Commons. Modified version by author.

Books are an old medium — older than TV, podcasts, and virtual reality rhythm games. I started a reading habit when I was an insecure college freshman. I thought it would make my life better. And truthfully, it did. Books are vessels for life-changing ideas. Books, like sit-ups or salads, are something that deep down many of us want more of in their lives. But we hesitate. We can’t help but recoil at the thought of reading, because of all the dreadful memories of boring high school assignments.

I often get asked by friends how to read more. So these…


The BTS Chicken McNugget Meal, Why you can’t buy 43 McNuggets at McDonald’s, and a math problem from the 1800s.

Photo by Food Photographer David Fedulov on Unsplash

Throughout the past month, the BTS Chicken McNugget meal has been everywhere.

The collab between McDonald’s and K-Pop group BTS launched in the U.S. on May 26, 2021, and for a brief moment it felt like BTS and McNuggets were inseparable from each other. The BTS meal was sold out at nearly every McDonald’s. Artist Josiah Chua turned the packaging into these fire sneakers. And of course, there was that McNugget from the BTS meal that sold for nearly $100,000 on eBay because it looked like an Among Us character, which really makes me think McDonald’s should’ve shaped the BTS…


Why an old email I got teaches us how to think more clearly (even though it was wrong)

Photo by Martin Sanchez on Unsplash

Maybe a year ago, I got an email from an older guy I know, who complained about his daughter’s online math homework. He was convinced that the learning software was wrong, and he was right. I’ve known a few parents over the years who complained about their children’s online software being wrong, and I can’t remember a single time where the parent was right. But still, I wanted to listen.

His daughter was in middle school, and the question was straightforward:

Can you create a triangle with sides of lengths 6, 7, and 13?

His daughter’s answer was yes, and…


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…


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…

Mike Beneschan

A human, writing (mostly) about math | California | If you want to reach out mikebeneschan@gmail.com | Get the newsletter here: https://bit.ly/3Ahfu98

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