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 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
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’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!
Before these young ladies came up with these proofs, the only way people could come up with to balance the equation was to use things that boiled down to the actual thing they were trying to prove. It’s like saying all things are made of atoms, but then people say well what the heck are atoms made of, smartypants?! So these girls found the Higgs-Boson of trigonometry while in high school as a piece of extra credit on a test. That’s my understanding as a low B, high C math student.
Edit: For any of you that are mathy, here is their actual paper.
From the Conclusion of the paper:
Huh. Reading that paper, I understand. Throw out the nonsensical and focus on the actual and the solutions are right there. Interesting.
You know, I’m still not “mathy”… But it made way more sense just reading the actual paper than it did reading summaries, explanations, or the article.
Thanks, Chief.
It’ll sound like splitting hairs, but I’ll try:
Trigonometry is based on the Pythagorean theory being true. They proved the Pythagorean theorem effectively in reverse without using the theorem itself as a basis. So they used the structure of trigonometry to prove the basic underlying principle of trigonometry. Bad analogy: kind of like if you have an airplane first, and THEN you worked out the physics of lift. You knew it could fly and how to fly it, but never questioned how it worked.