Tom Apostol’s square root of 2 proof is beautiful — This proof was published in 2000 by Tom Apostol. It just might become your new favourite proof that the square root of 2 is irrational. Here’s how it goes. You have a right isosceles triangle, ABC (Fig. 1). Suppose sides AC and BC have integer length. Their ratio will be…

A visual approach — Imagine a rectangle with aspect ratio of √2. Dissect it into a grid of tiny squares: u squares long by v squares wide (Figure 1).

Integral calculus makes it easy — Watch on YouTube! To find the volume of a unit sphere, approximate the sphere with numerous discs, each of thickness dR (Figure 1). We’ll do this with the northern hemisphere only. Then we can double our result.

Why is it the same as the curved strip which surrounds it? — Watch this on YouTube! Grab a perfect sphere. Enclose it in a tube of paper (Figure 1). I claim that the rectangular sheet required to make the tube has the same area as the surface of the sphere. How can this be?

We all have a friend like this. I’m talking about the highly intelligent person who knows a lot about a lot. He speaks with confidence on a variety of topics. He mesmerizes us with his ability to cross reference seemingly unrelated atoms of obscure facts. …

How natural language processing and intelligent tutoring systems are revolutionizing research in the field — As the field of artificial intelligence continues to advance, it is becoming increasingly clear that AI will play a significant role in the future of writing mathematics papers. …

And where’s my £10K? — I have a math channel on YouTube. Videos take effort. And research. People watch my videos. They leave nice comments. Thank you, People. Yet, somehow a 30-second video that took me about a long to create is on its way to being the most-viewed effort on my whole channel. Folks…

Satire. Seriously, this is satire. Do I have to say it again? — (From the Horton Hears a WHO Department) The World Health Organization now recommends cold showers to combat Covid-19. “I read this awesome book by this guy who calls himself the Iceman,” confided a top WHO epidemiologist, who spoke on condition of anonymity. …

How many lights remain on? — You have a numbered string of 1,000 light bulbs. You begin by flicking them each on.