High Schoolers’ NEW PROOF of Pythagoras’ Theorem
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- Опубликовано: 23 окт 2024
- High school students Calcea Johnson and Ne'Kiya Jackson proved the Pythaogras' theorem using their waffle cone observation, coupled with various facts in trigonometry, astounding the math community.
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![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](http://i.ytimg.com/vi/p6j2nZKwf20/mqdefault.jpg)
If those students want to have their proof published, one way to make it more interesting and useful might be to expand it by including some research and discussion about how widespread has been the false view, possibly even among mathematicians, that there couldn’t be any valid proof using trigonometry, and how that could have happened. One reason might be because of textbooks using the Pythagorean identity to prove some other ones. Another might be people not knowing that the ways of thinking and doing things that they learned in school are not the only ways.
Totally agree with you! On both the clarity of their potential publication as well as the surprisingly flexible (yet logically tight) ways to solve problems.
But it's not a false view. You still cannot prove the theorem using only trigonometry. Using some trigonometry was never prohibited lol.
They are so badass! My favorite quote was something about how they wanted to prove it just because no one expected it
The best kind of mathematical flexing there is out there. :P
And they failed.
Very elegant!🎉
My teacher didn't show interest when I told him I had a similar proof in 2022.😢 It's been almost 1 year now. I'm out of high school now.
damn that sucks man...
Actually, there is a somewhat simpler proof of the Pythagorian theorem, along similar lines, pictured in an ancient Sumerian (or Babylonian?) cuneiform clay tablet of about 4000 years ago. What it depicts is the subdivision of a right triangle into an infinite sequence of similar sub-triangles related sequentially in size by a common scaling factor r
The real question is: did the ancient Babylonians/ Sumerians really understand the concept of summing an infinite series, and, in particular, the formula for summing a geometric series? This was all about 1500 years before Zeno's paradox! The ancient Greeks did certainly use the concept of infinite limits, since they knew the formulae for the circumference and area of a circle, and other areas and volumes. But algebraic manipulations of arbitrary "unknowns" were not yet really developed, and the formula for an infinite geometric sum may perhaps not have been known. So attributing the use of the geometric sum formula in proving Pythagoras' theorem may just be a bit too optimistic.
Actually, although the proofs are similar, with both involving an infinite sequence of similar triangles of decreasing size, and evaluation of the sum of a geometric series, they are not, in fact, the same.
In the proof described in this video, the triangles are attached to the outside of the original right triangle, sequentially, and converge to a similar one of larger size, whereas in the proof explained at: ruclips.net/video/7642iBEOjCk/видео.html, there is a sequence of sub-triangles within the original one. Moreover, the zig-zag path described by following the edges of the sequence of added triangles in this video involves turns by 90 degrees, in alternating directions, with only the hypotenuses combining to form both the hypotenuse and base of the resulting large triangle. In the proof explained at: ruclips.net/video/7642iBEOjCk/видео.html, the zig-zag path traced by the edges of the sub-triangles involves turns at angles alternating between the two acute angles of the original triangle, and both the hypotenuse and base of the original triangle consist of intervals combining, alternately, the bases and sides of the smaller ones.
Really interesting video, and what a way to prove this fabulous theorem.
Yeah! And as a teacher what I find amazing is the students' ingenuity to apply their knowledge. :)
I've seen some crazy riddles with infinite geometric series, but this is by far the most impressive!
@kindiakmath Don't younthink a lot of us could've come ypmwotb this too..Inxould never admit I'm not a math genius and couldn't have come up with this myself..it would be too depressing and discouraging
hell yeah, that shit is way over my pay grade but good for them
How is this not a copy of John Arioni's "Pythagorean Theorem via Geometric Progression" proof (proof 100) at the cut-the-knot?
Nice proof
we can prove cosine law without any circular reasoning or without Pythagoras theorem
Wow, it's interesting Can you explain how to define the sin function without relying on the Pythagorean theorem? What is the definition of sin not using the Pythagorean theorem?
Hmm I would still define sine and cosine in terms of a right triangle (ratios of sides), the catch being that it’s not obvious what the hypotenuse of the triangle might be (even if it has a length, represented by a line segment).
Is a Trigonometric Proof Possible for the Theorem of Pythagoras?
Michael de Villiers
RUMEUS, University of Stellenbosch
CONCLUDING COMMENTS
To get back to the original question of whether a trigonometric proof for the theorem of Pythagoras is possible, the answer is unfortunately twofold: yes and no.
1) Yes, if we restrict the domain to positive acute angles, any valid similarity proof can
be translated into a corresponding trigonometric one, or alternatively, we could use
an approach like that of Zimba (2009) or Luzia (2015).
2) No, if we strictly adhere to the unit circle definitions of the trigonometric ratios as
analytic functions, since that would lead to a circularity.
there's lots of proofs but the best are the simplest like Einstein's proof, pure genius
Yea! Has a similar flavor (pun intended).
@kindiakmath I'm not sure why they or.anyone else would.think to draw the right angle at 1:46 like yiu said. And isn't the other line supposed to be annextensuon ofnthe hypotension c..but you don't necessarily know when rhar additional line on the lower left will intersect tje extended hypotenuse..
Isn’t the simple area of triangle proof much much simpler?
I made various proofs of less cool things in high school. I should have tried proving things we already knew. Darn
Pour moi, le raisonnement est bien circulaire !
Bro the existing one is way more easy and clr... 😂
This is an awkward proof compared to simple geometric proofs like ruclips.net/video/yfGtbNgcrQ8/видео.html
It uses stuff normally learnt after Pythagoras' Theorem like trig and geometric series summation.
But most of all it's just not as much fun as the simple geometric proof 🤽♂
I’m thinking that it might be better for those students, instead of submitting their proof by itself, to make it part of a study of how widespread was the view that there couldn’t be any non-circular “trigonometric proof” or proof “based on trigonometry,” and how that happened.
But they did not prove it using only trigonometry, so the hype is false.
@@sergeyromanov2116 There have been some people widely considered as “mathematicians” who were saying that using trigonometry in any part of the proof is using the theorem to prove itself, which is false. Also, there actually is a proof using only trigonometry, defining the sine and cosine functions with differential equations and initial conditions.
I learned this one as an undergraduate!
@@sergeyromanov2116 If you mean the hype about doing something that has been considered impossible for 2000 years, or about this proof being new and unusual because it’s “trigonometric” or “based on trigonometry,” then I agree that the hype is false. It is contrary though to what some people thought was possible, including some people called “mathematicians” in media stories and public discussions. I don’t know if any of them have math degrees or not. I think that, and the reasons for it, could make an interesting study in sociology and psychology. If anyone does that study, it will be a consequence of this proof and the hype around it, and those students will deserve some credit for it. Even more if they do that study themselves.
@@kindiakmath Do you mean, the one that defines sine and cosine with differential equations?
The Egyptians had these proofs thousands of years ago. Why do we have to keep trying to appear like we are smarter than everyone else. All this is is a feel good proof.
Any PROOF? Get it? Lol.
Well the proof that these two girls use is best cause they didn't use thing which involved Pythagoras to proove
Yes, their ancestors were also very smart.
I’m looking for someone who cares about the harm that has been done to those two students and might continue to be done.
Oh, harm in what manner? If the harm lies in the rigidity in the education system I'm with you on that.
@@kindiakmath Is there any to discuss it privately? Some way to PM on RUclips, or email, or Facebook Messenger for example?
Feel free to contact me! I have some contact points on the main channel page.
@@kindiakmath Okay, thanks.
@@kindiakmath I couldn’t find any contact information for you. I’ll try again on my computer when I get home from my trip.
Far to complicated proof compared to other proofs