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.
Do check out the quote from the girls’ research paper I put in the other reply thread. Even they couldn’t believe they actually did it. From my very basic understanding of what they achieved, it is something pretty monumental.
This is incredible. High Schoolers solving questions that are thousands of years old.
I wish I was high on potenuse
I WISH I WAS HIGH ON POTENUSE
IF I SAY THE JOKE AGAIN, BUT I SAY IT, IT’S MORE FUNNIER
Neeeeeerrrds!!
Jk jk that’s super impressive
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.
Can’t fool me. They’re actually searching for the holy ratio that Pythagoras discovered and kept from his cult members.
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
By proving Pythagoras’ theorem using trigonometry, but without using the theorem itself, the two young women overcame a failure of logic known as circular reasoning.
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:
The reader may be surprised to learn that the catalyst for us to start this project was a bonus question of a high school math contest. The bonus question was to create a new proof of the Pythagorean theorem. Motivated by the $500 prize, we independently decided to take on this task. It proved to be much harder than we first imagined, and we each spent many long nights trying and failing to create a proof. After roughly a month of mental labor, we each completed and submitted our work. Mr. Rich, a math volunteer at our high school, believed our proofs were novel enough to be presented at a mathematical conference. Neither of us had such confidence in our work at that point, but we decided to go along with it anyway. This is when we began to work together.
For the next two to three months, we spent all of our free time perfecting and polishing our work. We worked both independently and together after school, on weekends, and even during holidays. In the process, with Mr. Rich as our faculty advisor, we created additional proofs. We did all of this not knowing if we would even be allowed to present at the conference, which is usually only done by professional mathematicians, and occasionally college students. To our surprise, our high school work was taken seriously, and we were approved to present at the American Mathematical Society’s Southeastern Sectional conference in March of 2023. Being the youngest people in the room and the youngest presenters was terrifying, but knowing that this was the culmination of all of our previous efforts gave us the confidence to present.
We were then encouraged by the AMS to submit our findings to an academic journal. This proved to be the most daunting task of all, since we had absolutely no experience writing for an academic journal. We were both also dealing with the stressors that come with adjusting to the college environment. Learning how to code in LaTeX is not so simple when you’re also trying to write a 5 page essay with a group, and submit a data analysis for a lab. With the guidance and wisdom of our mentors, and a lot of personal dedication, we were able to craft this paper. The support of our family and later our community helped us to persevere. Our journey to this point was by no means simple or straightforward. There was no road map laid out for us, and there certainly was no guarantee that any of our work would go further than our own heads. There were many times when both of us wanted to abandon this project, but we decided to persevere to finish what we started.
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’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…
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.
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!
Does this mean it’s going to henceforth be referred to as the Pythagorean Principle or something or does it still remain a theorem after proving.
A conjecture is a problem that is expected to be true but has not been proven. So it remains a theorem because it has already been proven. What they did is to prove it in a new way. New proofs could just be interesting or they could provide a new way to think about other un proven problems.
There are hundreds of proofs, this is the first one to use trigonometry. Not sure about the wording though 🤔
Jackson and Johnson are only the third and fourth people known to have proven the Pythagorean theorem using trigonometry and without resorting to circular reasoning. The two other people were professional mathematicians, according to the statement.
Oh so they aren’t even the first to do it. Just the first to do it this specific way.
Not to downplay their achievement because not only are they joint third in proving it using trigonometry (without any circular logic) but they then put forward another bundle of proofs that smacks of them saying “it ain’t that hard!”
I hope this is just the first time we hear their names like this!
HELL YEAH
That’s awesome. How guaranteed are they for college now?
Is this an example of common core math actually working since forces them to prove the same thing in ten different ways?
