Here is a lovely geometric proof that sqrt(2) is irrational.

The proof begins by looking at an isosceles right triangle whose legs have length 1. By the Pythagorean theorem, this triangle's hypotenuse will have length sqrt(2).

https://twitter.com/LongFormMath/status/1795862192681521326

Now let's assume for a contradiction that sqrt(2) is rational, and that it equals a/b for some integers a and b.

(1/8) One of the first things Ramanujan ever published was a challenge problem to readers of the Journal of the Indian Mathematical Society. He asked for the value of the following. No reader found the answer. So what does this equal? A short 🧵👇 (2/8) The answer that Ramanujan himself offered was 3. His reasoning was this:

(1/5) There are hundreds of proofs of the Pythagorean theorem, some remarkably simple. Here’s on I just typed up.

Start with any right triangle: (2/5) Now scale this triangle three times: once by a factor of b, once by a factor of c, and once by a factor of a: