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 constantly amazed by the number of things that we have assumed was true simply because we were taught that in school and never questioned it.
Pythagorean theorem has over 100 proofs, they are just geometric one, this is the first trigonometric one. The reason it is impressive is that the pythagorean theorem is foundational to trigonometry, so any attempt at a trigonometric often calls on the pythagorean theorem implicitly.
One might need to prove that a trigonometric proof isn't equivalent to any geometric proof. Somehow the premise here "a proof based on the sine law" doesn't inspire that confidence for me as sine law has equivalent formulations in geometry.
That said, I'd also say that the boundary between geometry and trigonometry isn't a particularly necessary one, and the work of these young girls exposes this.
This is incredible. High Schoolers solving questions that are thousands of years old.
Actually there was already a known proof of PT that was based on trigonometry, the proof of the students is also unlikely to be accepted as it relies on calculus and not pure trigonometry.
I'm not a mathologist, so this reads to me like "they proved it is what it is because of the way it is. That's pretty neat!"
I can understand it's significant, but that's about it. From my understanding, this doesn't really change anything about math, it's just something we didn't think was possible being proven possible.
Please correct me, mathletes! Hilariously almost all my fields of interest require math... cries in physics
it's just something we didn't think was possible being proven possible.
The theorem was proven thousands of years ago, I think it's the particular method didn't seem possible to be used to prove the theorem. Specifically much of trigonometry uses the pythagorean theorem as a foundation, so the fact the proof was constructed without needing anything that depended on the pythagorean theorem is what was difficult. Definitely a cool start for a math career, it's generally how mathematicians approach math research, i.e. the proof being the the focus even if a theorem is established. I doubt it'll be revolutionary by any means, and it's annoying for media to sell stuff like this, it is extremely impressive to do this, and especially as a high schooler, even if there isn't some quantifiable impact.
I meant more as a "we knew it was possible just not WHY it was possible" kind of way, not "we didn't believe triangles make sense because it's not possible at all" kind of way.
I'm not a wordsmith either so I hope that makes sense...
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?
