Two students who discovered a seemingly impossible proof to the Pythagorean theorem in 2022 have wowed the math community again with nine completely new solutions to the problem.
While still in high school, Ne'Kiya Jackson and Calcea Johnson from Louisiana used trigonometry to prove the 2,000-year-old Pythagorean theorem, which states that the sum of the squares of a right triangle's two shorter sides are equal to the square of the triangle's longest side (the hypotenuse). Mathematicians had long thought that using trigonometry to prove the theorem was unworkable, given that the fundamental formulas for trigonometry are based on the assumption that the theorem is true.
Jackson and Johnson came up with their "impossible" proof in answer to a bonus question in a school math contest. They presented their work at an American Mathematical Society meeting in 2023, but the proof hadn't been thoroughly scrutinized at that point. Now, a new paper published Monday (Oct. 28) in the journal American Mathematical Monthlyshows their solution held up to peer review. Not only that, but the two students also outlined nine more proofs to the Pythagorean theorem using trigonometry.
I'm not that great either but to my understanding you are right. The thing is by giving a solid proof foundation to what was mostly glued together by basic understanding we can now build over it and arrive to new things.
Neat!
So super simplistic paraphrasing, once you know the shape of the box, you can start mapping around it? Maybe?
I don't know the specifics, but there are a few reasons why new proof methods for known results are interesting.
First and foremost, every new proof is, in and of itself, a new mathematical discovery. This is how the field expands.
More specifically, proofs that require fewer other results can often be generalized to other systems/branches of math where other proofs don't work for some reason. Like, lots of math is based on the Riemann hypothesis, but it's yet to be proven, so everything built on top of it is, essentially, a house of cards that could come tumbling down if it's ever disproven. And, even if it's not untrue, we can't fully accept the results since they aren't fully proven yet.
I wonder about this one, though; someone else mentioned they used calculus, but many parts of trigonometric calculus use the Pythagorean Theorem somewhere in the proof chain. Which would then mean this proof is already using the existence of itself to prove itself. It passed peer review, though, so either my doubt is unfounded or someone else has previously proven the relevant results in calculus without using the Pythagorean Theorem... which is a great example of why proof using fewer assumptions being useful!
Best comment explanation I've read yet as to why it's important!
Thanks!