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MathTrain
Добавлен 16 апр 2009
How High Schoolers Proved Pythagoras Using Just Trig! (and some other stuff)
How did two high school students prove the Pythagorean Theorem using only trigonometry?
This is a summary of the proof described in the article by The Guardian: "US teens say they have new proof for 2,000-year-old mathematical theorem: New Orleans students Calcea Johnson and Ne’Kiya Jackson recently presented their findings on the Pythagorean theorem"
0:00 Math News
1:14 Proof Preview
2:00 The Proof
5:54 Final Thoughts
Guardian article: www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove
WWLTV report: ruclips.net/video/Ka1k4i1ueNU/видео.html
AMS abstract: meetings.ams.org/math/spring2023se/meetingapp.cgi/Paper/23621
This is a summary of the proof described in the article by The Guardian: "US teens say they have new proof for 2,000-year-old mathematical theorem: New Orleans students Calcea Johnson and Ne’Kiya Jackson recently presented their findings on the Pythagorean theorem"
0:00 Math News
1:14 Proof Preview
2:00 The Proof
5:54 Final Thoughts
Guardian article: www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove
WWLTV report: ruclips.net/video/Ka1k4i1ueNU/видео.html
AMS abstract: meetings.ams.org/math/spring2023se/meetingapp.cgi/Paper/23621
Просмотров: 206 045
Видео
A New Way to Measure Sets! (How to build a strictly monotone measure) #SoME2 #3b1b
Просмотров 32 тыс.2 года назад
*List of corrections/clarifications (including "clopen", "topologist", and "measure") below* This is a summary of about a year or two of my own personal math research. It is not complete overview, nor does it constitute peer-reviewed research. I am a teacher by trade, not a professional mathematician. This is just a fun intro to my research to see if people are interested. I also do claim to ha...
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.
You misused the terminology. “Clopen” doesn’t mean what you indicated.
My understanding is that the young ladies each did separate proofs on their own. They did not come up with one proof in collaboration. The methodologies were completely different. Only one of the proofs is illustrated in this video.
She like hypasus and godel i think
Jackson's proof is an exact copy of a proof originally made by B. F. Yanney and J. A. Calderhead in 1896 which can be found at cut-the-knot, proof 60. Johnsons's proof can be found in many Calculus textbooks and is a "geometric series" that was already a proof of the pythagorean theorem. She just added an extra triangle.
One thing I wonder: is it a problem that this proof fails in the special case where alpha=45°? In that case 2*alpha is a right angle and the sides X and Z never intersect.
This is the "waffle cone" part of their proof: (I can't post a link.) Search: "math.stackexchange Is this series representation of the hypotenuse symmetric with respect to the sides of a right triangle?"
Jaykee the Wolf: No. They used calculus to find out the limit results for u and v. Luis: true. but good work, still. Jaykee the Wolf: not really. just like einstein ripped off maxwell's theory to shape the theory of special relativity, a little of the same is going on here.
I cant believe that nobody thought about this before. It´s amazing!
To summarize in three words - ingenious, simple and elegant - the highest level in mathematics.
.Yay! Thanks for doing this. It took me two passes through, but I get it. These gals claim not to be math whizzes, but clearly they are.
FYI: They each came up with 2 separate trigonometric proofs on their own.
what a beautiful proof, so elegant
The 60 minute coverage said that they each discovered an independent proof and showed a drawing of some of the second proof. Please look at that one. Interested how that one works.
nothing special
Why hasn’t their paper been released yet?
How this uses trigonometry? It basically uses similarity of triangles and "scaling factors". Also a cocept of infinite constructions such as this does not really fit Euclidean geometry.
This is an interesting proof, but no more interesting than any other Pythagorean Theorem proof. OK, it uses the Law of Sines (big whoop), but it also uses the theorem of infinite geometric series, so it's not "just by trigonometry". But of course, the whole reason that the media is going so bonkers over it is because of the DEI agenda; if a white male had figured this out, would the media cover it?
Jackson's proof is an exact copy of a proof originally made by B. F. Yanney and J. A. Calderhead in 1896 which can be found at cut-the-knot, proof 60. Johnson's proof can be found in many Calculus textbooks and is a "geometric series" that was already a proof of the Pythagorean theorem. She just added an extra triangle.
Questa cosa non solo è fichissima, e non (almeno apparentemente) tautologica. In matematica e geometria si dice anche "elegante". Ma soprattutto è molto greca, molto pitagorica, molto egizia per come immaginiamo quel mondo, che ancora non amava scrivere tutto, e la cui vita media era molto breve. In altre parole, è possibile che Johnson e Jackson, più brave delle sorelle Williams, abbiano trovato l'origine del Teorema di Pitagora, la sua dimostrazione che in realtà costituisce la sua intuizione logico matematica.
Cool! 😎 #'s dude! (. . . and dude-ettes! :)
Nice💯💯💯
interesting
That’s pure genius ❤
If the pathagorean theorm is proved, does it then become the law of Pythagoras?
This proof only works if the triangle is not isosceles. If a=b, then r=1 and the geometric series doesn’t converge.
You don't need the staircase at all if a=b. As 2α = 90°, by the definition of sine, sin(2α) = 1. Either way, the formula sin(2α) = 2ab/(a²+b²) holds. The rest of the proof after the staircase remains unchanged. I'm sure in the original paper this was explained thoroughly.
I looked in a endless circle, lol, for this. Thank you for finding it.
have you ever heard of the name ZIMBA? Go do some research!
There is absolutely no need of trigonometry in this proof. The orthogonal rectangle with one side being 2a is homotethic to the original one with scale 2a/b so no need to use any trigonometry. Then once you have X and Z you can see Z^2 = X^2+c^2 proving the Pythagorean theorem without using the sin(2 alpha).
Jackson's proof is an exact copy of a proof originally made by B. F. Yanney and J. A. Calderhead in 1896 which can be found at cut-the-knot, proof 60. Johnsons's proof can be found in many Calculus textbooks and is a "geometric series" that was already proof of the pythagorean theorem. She just added an extra triangle.
can you add ‘nonstandard analysis’ to the description?
Very clever proof... But, I'm not sure the usage of the geometric series makes it a purely trigonometric solution.
Limit theory...sum to infinity...."using only trig!" Hardly.
Where's this guy? Did he stop making videos?
Haha I still exist but life has gotten in the way of video making
@@MathTrain1 owun. That's ... little sad. Well, I hope you are ok. Big hug from Brasil.
@@MathTrain1 best of luck!
The idea and visual is intresting, but in the end of the day it feels frustrating, that the subject of the topic came out to be just specific subsets of plane. I think 3rd question should be the 1st one. Only after dealing with it other questions has meaning. The definition is pretty poor. Feels like it useful only in the cases when there is no special intrigue about sets. If the set contains countable number of points it's already fails. But it's just my aesthetic of feeling math. Considering the title of the video.
I'm gonna say what no one will. I see a lot of assholes in this comment section who are trying to pretend to be soooooo smart and are so bothered by the fact two teenage black girls have discovered something so beautiful and great so some of y'all just have to try and disprove it or belittle it in any way u can and it's really pathetic!!😂😂 kudos to these young queens and may they go far far far beyond the stars in life and ignore haters trying to pull them down!! I myself cannot wait for that official peer reviewed paper to come out published to shut the lot of you tiny dick incels down!!!
If they had claimed that they made "a proof for the Pythagorean theorem with an exception for isosceles triangles" that would be an acceptable claim. There are literally hundreds of proofs of the Pythagorean theory. Stating that they made a "proof using just trigonometry" is false. So you can call people assholes if you like, but their claim is false.
Too many flaws. Will update this comment once I finish my exam this week
You done yet?
@@MathTrain1 before 06:00 02:56 - A contains all points of B does not necessarily mean A is larger than B. One example is B = (0, 1) and A = (-inf, +inf), these two are homeomorphic then equal. Even if B = (0, 1), A = [0, 1], there is also a bijection between A and B 05:33 - f(x) must be continuous to conclude that f(x_0) = lim_{x -> x_0} f(x) 05:40 - you have not proved that area is preserved under translation/rotation, and area is additive (area(A cup B) = area(A) + area(B) with A B disjoint)
@@MathTrain1 Just watched the whole video, and the idea is very interesting. The first term corresponding to omega^2 recovers the usual Lebesgue measure. I think this idea will be useful when we need to subtract infinity from infinity (in the usual analysis sense)
@@MathTrain1 fyi, this construction I believe is a special case of Hausdorff outer measure.
I was hoping you'd mention fractional dimensions. I've pondered over the concepts you've presented here before, though I never took it anywhere.
just noticed that the constant term of your polynomial is exactly baez's so-called "euler-schanuel characteristic"! fascinating stuff!!
Wonderful video. No one is talking about the excellent pencil animations. Amazing work.
Thank you very much!
Right before you showed your first take on μ, I immediately started thinking about the same idea. However, instead of using the letter ω to differentiate between different dimensional parts, I simply thought of an infinitely long vector where the nth element is for the nth dimension. (With the count starting at zero, of course.) So what you call 20ω²+5ω+1, I would call (1, 5, 20, 0, 0, 0, ...). To compare two vectors, you just use a lexicographic ordering. The multiplication of two polynomials is the only non-obvious thing here. Two vectors can be multiplied by calculating the outer product and then summing up elements along the minor diagonals. I wonder if there is an established operation for that? Other than that, this vector representation does not open up any further questions about derivates, integrals, fractal dimensions, etc. I can't really see how those would work or what they would mean anyway, tbh.
Well, polynomials are vectors! Really the only difference beyond how they're usually visualised is that polynomials have specific multiplication rules that don't apply to other types of vectors, though they don't conflict with them being vectors either.
Math is all about creativity and this video does just that! Keep it up, my friend!
I was a little confused with the example of circle at 17:48 (might have to re-watch this a few times and experiment myself a little bit) with 2 points not included... because I understand that is we are including a circle with it's circumference, the circumference when opened up into a line segment, it would have a open hole on one end and closed one on other but when closed it would have no hole since the closed end would overlap with closed one hence completing the circumference, but I still cannot wrap my head around how would I do it if I was not including the circumference like you did at 17:48. But you did intrigue my brain and at the end I do agree with the way this formula works... I might come back to this video again and spend a couple of hours sketching and scribbling things around on a pad.... Though the most important thing I might have taken from this is my brain at this point at least works like a topologist, cos it feels like when we are trying to add area and circumference and the points together what we are doing is we are removing any additional points or like that we might have counted twice... like we do when we are counting items in a Venn diagram, we add the number of elements in both sets and subtract the elements common in both to get the count of a union of both sets.... And this approach of sets the my brain prefers to take is what makes me think that I might be thinking like a topologist... Though I am pretty sure I am no way as good an intellectual as an actual topologist.... But thanks a tonne again for this video.
It's most simple to use the law of cosines (LoC) directly to prove the Pythagorean theorem, if someone wants a purely trigonometric way. LoC itself has a trigonometric proof itself, check the Wikipedia article for it: the proof does not use the sin²x+cos²x=1 identity anywhere, so it is not circular. Why they thought then trigonometric proofs were impossible? Or am I missing something? Someone please enlighten me!
law of cosines is basically a generalization of the Pythagorean theorem, so i don't think it counts
what about weird and undefined behavior like the se of all rational point between (0,0) and (1,1) or lines from infinityor you have the line from 0 to 1 and remove the points which are 1/2^n
Very interesting questions... I was just thinking, how does this affect the sizes of infinities. The finite number of points is clearly enumerable, while the omega's are not. How do we write w*w as a line with w length, vs w*w as a square with length 1. Same for squares, w*w*w as an infinite plane vs a cube of sides 1. And can we even compare the half plane x>0 with the whole plane? I'd argue that it's impossible since translations shouldn't affect area but translation will inevitably affect any half plane "area".
Good question about the size and proprieties of infinities. The set of all rationals Q in countably infinite, while ω is uncountably infinite (it is equivalent to the real numbers R), so ω > Q. In that sense, the set of all rationals are all points and no 1D line ω. Say C is a countable infinity, then the polynomial for Q would be p(ω) = C. (countably infinite points) Also, ω represent a line segment of size 1, so an infinite line could be measured with a countably infinite number of these: p(ω) = Cω. It is also important to distinguish between a measure of length (here ω) and an amount of something (C or U maybe for a uncountable infinity). There are case where they could be considered equal (or equivalent) but that would not always be the case. You could for example take an infinite line defined by p(ω) = Uω, then warp the line into a square of side length 1 so that every point is covered by the line. But you could just as well warp it into a rectangle of 1 x 2, or even a square. As long as U is not defined in term of the measure ω, then Uω has arguably no other measure the Uω and its dimension is undefined. (ω was created specifically to define the size and dimensionality.)
This proof is purely trigonometric which is impressive.
It isn't purely trigonometric as it relies on series which isn't trig.
the way i think about it, if they are the same, why need to specify if the boundries are closed or open? when it doesnt change anything? thats like saying A is blue, and B is yellow, tell me which is larger? neither, they are the same, the colour doesnt affect it. This is why i say A is larger than B
My question is, how. How could it be that that some of the most brilliant minds of the past could not figure it out, while 2 brilliant high schoolers could?
Because it's not what it claims to be. It's not "solely trig" and is really a geometric solution and limit theory with the veneer of trig.
What about bijection? If there are continuously many points in A and B, then we could find a bijection from A to B...
Suppose angle A and angle B are not known, then angle A = tan^-1(a/b) and angle B = tan^-1(b/a). So, then, c^2 = 2ab/sin2A = 2ab/sin2B.