They actually had TWO independent proofs from one another. Calcea Johnson's proof (so-called Waffle Proof) is the one you describe. Ne’Kiya Jackson's proof used a right triangle inside a circle bisected by the radius of the circle to create her equations.
It does my heart good to see more kids getting into maths. I couldn’t even figure out the basic geometric proof until halfway through college and I was the straight A student in my math classes.
I'm glad to see this. The AMS and journalists have diverted attention from the real value in what those students did, and this helps bring attention back to that. Thank you.
It’s always the same script too: “I was terrible at math” or “i don’t know how to calculate a tip” etc. Sometimes I wonder if they are truly bad, or if they feel that they need to act dumb to connect with the audience
Each one actually came up with their own proof. There was also a mathematician who came up with 1 in 2009 too, but they are the only three to have done so.
I appreciate your quick, lively, and detailed overview of the context: history, math, and modern journalism - ouch, those were painful behaviors to watch. This is so far my favorite video on the topic to give to students. I'd like to note, in case you would want to know, that the music was difficult for me, maybe due to the sensory issues frequent among math people. I'd love to have a no-music version of this (or very brief intro and ending).
"thinking outside the box" twice was the solution to connect pythagoreas theorem with trigonometry, a lot of proofs and trigonometric properties are limited with one rectangular triangle.
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.
I was scoffing at people thinking that any proof using trigonometry is using the theorem to prove itself. Just now I read an article that used the theorem to prove another trigonometric identity, and said that proof is "the standard one that you see in most textbooks." Maybe that's the approach most often used in textbooks, first proving the Pythagorean theorem, then using that to prove the other identities. Maybe that's why so many people think that there's no way to prove them without it.
I get that analysis can be done without drawing any pictures, but I still think that calculus, and therefore trigonometry, is a consequence of geometry. That said, this is an impressive paper for a high schooler. There's nothing wrong with proving the if in an iff.
Thank you for a great video explaining what these young ladies did. You’re correct it was very impressive given their age. And you were spot on that even some basic problems lend themselves to a new perspective. Kudos to their teacher for giving them the challenge and perhaps maybe not expecting them to come up with a new proof but getting them to think. Education is not the feeling of a bucket it should be the lighting of a fire. I forget who said that but it wasn’t me. Again thank you very much.
The nice thing about identities is that you are free to select whichever side you want to start with. For this one, I'd start with the right hand side and expand both cosine and sine into their exponential representations. From there, it should just be a matter of algebra to reduce to cos(2x)'s exponential representation. This only uses the analytic form of sine and cosine that we got from the differential equation definition.
I love construction proofs, especially when it adds new elements!! Also love that they combined trigonometry with infinite series. Sooo cool!!! I wonder how they arrived at it? I guess it must've started with adding one triangle and then a second and then a third and then BOOM! They mustve noticed that the triangles are scaling down and will converge. This is soooooo cool!!! It's making me happy just thinking about it.
It was the Sumerians, not the Babylonians who wrote the formula. Hundreds of year’s before the Babylonians. The Egyptians also wrote it as they built the pyramids.
The egyptians just used pythagorean triples, they didn't know the law, they exhaustively hand calculated and experimentally found right triangle triples.
9:05 How do you know that sin γ = sin(90^-2α).....? I need to see the proof without using the Pythagorean theorem. In other words, 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?
@@JoelRosenfeld ruclips.net/video/9XFBw8Hfk5k/видео.html Why is cos being called cos(t) and not apple(t)? The term "cosine" is derived from the Latin phrase "complementi sinus" or "complementum sinus." So, we cannot simply call things by arbitrary names. There is a reason behind the naming conventions in mathematics. I'm not here to point out a mistake because it seems there is a plaintext explanation of mathematics emerging, and I'm trying to learn. This is interesting to me because I often have a hard time communicating with mathematicians. They may assume certain things and then use those assumptions as facts later on, similar to the argument presented here. It is important to note that in trigonometry, the names given to mathematical functions, such as sine and cosine, have historical and contextual significance. They are not arbitrary labels. The term "sinus" was used to describe the curve or shape of the sine function due to its sinusoidal graph when plotted against angles. Similarly, "cosine" refers to the complementary relationship it has with the sine function. Understanding these historical origins and the mathematical context behind the naming conventions can help in communicating and comprehending mathematical concepts effectively.
@@DisgruntledSpam Well, that is exactly what Ne'Kita Jackson and Calcea Johnson are denying. They claim that we can get to trigonometry(sin and cos) without Pythagoras.
Nifty. I had heard of this approach a few months ago, but I was not aware of the "punch line," in a manner of speaking. Good for them. Pretty darned clever, I'd say. I wonder if they might have anything else of their sleeves? Staying tuned. Sounds like they've got their Ph.Ds. nailed down too, just write up their findings.
First of all, just the fact that these two students independently created 2 different "proofs" themselves are fantastic, regardless of the claim about being independent from Geometry. I didn't understand how one can say that the Law of Sines from Trigonometry is not based on Geometry, so I will check out that video that explains how it can be derived from calculus.
@@marcfruchtman9473 that’s more or less my own stance. Even if you don’t buy the “without geometry” claim, it’s still two new proofs by high school kids. Remarkable either way.
Great work to Jackson and Johnson!! I love to hear stories of discoveries being made by high school students! I’m surprised this proof by geometric series was overlooked. Perhaps it’s because geometers have bound themselves to a rule of finitely many constructions by ruler and compass and ruled it out, no pun intended. But another proof of this ancient theorem goes to show there are still many gems to be uncovered, even today
Another variation would be to make two smaller new similar right triangles inside our larger new right triangle by dropping a perpendicular from its right angle vertex to its hypotenuse. Then the sum of the areas of the two new smaller right triangles is equal to the area of the new larger right triangle.
What is a good mathematical journal to present a new proof of the Pythagorean Theorem? I proof it differently and have checked with more than 400 solutions and did not find my solution, so I want to submit it for publication. Can you suggest a journal?
It wouldn’t go to a big journal, but probably would fit into an undergraduate journal of some kind. Search for journals dedicated to undergraduate research.
In the 60 Mintes video, it was stated that they only used trigonometry to creat this proof. So, they used only trigonometry to create this so-called new proof? What did they initially start with?, a right triangle, which is geometry. I applaud their effort, but you cannot get to the moon w/o going up. One cannot get to 3 w/o going through 2. 1+1.9 does not equal 3, only 1+2=3.
This is how math curriculum in high school goes: algebra 1 < geometry < algebra 2 < trigonometry. The 60-Minutes video cites the highest level of mathematics that the students used which implies that they have taken geometry and 2 courses of algebra. In the same way, a student who knows algebra 1 knows how to do fractions, integers, decimals, etc., from arithmetic in grade school.
@@williammajor6768 when I made this video there was very little information out there. But I appreciate you all coming into the comments to share what you know
Sure, but our perspective is different. We can see that there was a strong qualitative difference between what the Greeks did and the Egyptians. The Egyptians really didn’t introduce logical proof, which for us is what mathematics is.
@@JoelRosenfeld you should check more about Egyptians. You will find that most of what Greeks did were taken from Egypt, including names of all those Gods! There is just a king of brainwashing state to make all greek.
I loved your proof, but struggle with the math replication of the hypotenuses. If the second triangle in the series moving down the waffle cone has hypotenuse (2a^2/b^2)*c, that is 2a^2/b^2 bigger than the original triangle with hypotenuse of length c. So, why are all the triangles moving down only a^2/b^2 bigger and not 2a^2/b^2 bigger? I know from a different proof I saw that a^2/b^2 is correct, I just cannot figure out in your proof how you go from knowing (2a^2/b^2)*c for the hypotenuse that the one below that has a hypotenuse of length (2a^4/b^4)*c.
Try to recreate the algebra. It's a lot of work but that's how I understood it. It employs proportionality that is derived from similar triangles. My proof in the end is a lot simpler than what was presented in the video.
@@omysoulblessthelord4578 Thanks. I did the math, but at first I had to do more calculations than you did. Then, I realized the triangle below the mirrored isosceles triangle is 2a/b the size of the original triangle and the triangle to the right of that triangle is a/b the size of that triangle. That is why the first right triangle to the right is (2a/b)*(a/b) the size of the original triangle. From there, the triangle below that is a/b the size of that and the triangle to the right of that is a/b the size of that. So, each triangle moving down is (a/b) * (a/b) = a^2/b^2 the size of the previous one above it.
I think I'm still kinda lost here. How is that "just trigonometry" when they clearly used geometric arguments? What makes this proof any more 'trigonometric' than the first proof you gave?
The whole idea is to leverage ideas from trigonometry, specially, the law of sines. Loomis' argument that trigonometry rests on the Pythagorean theorem really isn't true, and so, purportedly, his list doesn't include methods that leverage trigonometry.
I don't think that those students have ever said that their proof is "just trigonometry." I have some ideas about why people are saying that, but it might be a long story. Mayve that's the only way that they can explain to themselves why it's getting so much attention. One possible reason that I see for all the attention is the false claim that mathematicians have always thought that a trigonometric proof was impossible. Another possible reason is people thinking that the story is uplifting three categories of oppressed people.
@@jimhabegger3712 the trigonometry emphasis is coming from their abstract for their talk at the AMS sectional meeting. There they give loomis’ statement about trigonometry, and that caught on and probably got inflated.
Farkle, I'm thinking that the only reason it got into the news is because it fits into three diversity/inclusion categories. If it had been two old white men, they never would have been invited to an AMS meeting in the first place. They never would have made such a ridiculous and easily disproven claim, and even if they had, the AMS would have rejected it.
@@JoelRosenfeld Yes, I understand that, but the more that people talk about it being "trigonometric," the worse it might be for them, if and when the falseness and foolishness of their "considered impossible for 2000 years" claim becomes public knowledge. I think it's better to turn people's attention to the real value of what they did: that they tried to do something they thought was considered impossible; and didn't give up until they succeeded; and what might actually be new and interesting in their proof, which has nothing to do with it being "trigonometric."
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!
They read it in a book that collected all the proofs of the pythagorean theorem. This book claimed that there was no proof that was purely trigonometric, since trigonometry relies on the pythagorean theorem. However, this was in the 1940s or 50s when it was written. Since then, there have been several well documented definitions of trigonometric functions that don't rely on geometry.
The new element is the sum of the convergent geometric series. I think the law of sines is a distraction. We just need to use the ordinary geometrical similarity of right triangles. We get similar right triangles in two ways. 'One', by taking the doubled original shorter side as hypotenuse. 'Two', by taking a copy of our original right triangle. 'One' is a scaled version of 'Two'. From there on, it's repetition.
@@omysoulblessthelord4578 Thank you for your response. I have a prejudice against relying on trigonometric functions when one hasn't derived a way of calculating their values for a general angle. In the present context, the law of sines is a fancy way of talking about similar triangles; the actual values of the sine function are neither used nor calculated in this context. So I would say that the law of sines is a practically irrelevant complication. Important is that the present proof relies on the meeting of the two lines at the convergence point; that is one of Euclid's axioms.
I suppose you must consider that the students are high schoolers and could be exposed to formal mathematics at a very novice level, if at all. They used what they know. I learned the formality of the Euclidean Axioms at the graduate level, not as a high school student.
@@omysoulblessthelord4578 It is an admirable achievement by the students to invent these new proofs. I too didn't understand the Euclidean system of axioms till later in life. I was just reflecting on the proof itself. I think that the summing of a geometric progression is unusual in this context, and therefore notable.
It is thought tjhat Pythagoras's theorem has been known for 4000 years (i.e. before Pythagoras). As I understand it, the recent discovery is trigonometric proof of Pythagoras's theorem Until very recently (1950?) It was thought that this was not possible. Sine and Cosine did not exist, as I understand it until the 12th century A.D.
The real novelty here is the discovery of a new proof of the Pythagorean theorem. There are honestly hundreds that we know, but the fact that a new proof was discovered by high school students is really the icing on the cake. Sine and cosine were known to Ptolemy in antiquity. In the 12th century, Indian mathematicians discovered the power series representations.
6:00 But don’t you need to use pythagoras to know that tan(45) = 1/√2 EDIT: Just realised you can use double angle formula, 2sin45 cos45 = sin90 = 1 with sin45=cos45 (since the ratio of lengths will be the same due bc isosceles)
Min 6:02. sin 45 = 1/√2 is calculated with Pythagoras. The case a=b is given very easily, only with the definition of cos. If a=b => sin=a/c, sin=c/2a => c² = 2a² => c² = a² + a²
@@JoelRosenfeld If you claim that you have a proof for the case of the isoscel triangle, then that proof needs to be air tight. And in the form presented in the video, it is not. Even the verification you make at the end is based on the Pythagorean theorem - circular much? If the length of the video does not allow this, a reference to a complete detailed proof should be provided.
@@silviadoandes7930 this isn’t an academic paper, it’s a RUclips video. If you want a video to be successful, you have to get to the point, and diversions like that will absolutely kill retention and RUclips won’t show it to hardly anyone. I’ve done a lot of experimenting with this, and I’ve learned a lot about retention and communication. That said, even in an academic paper, you are free to assume well known concepts and otherwise provide citations. RUclips has the added benefit of allowing for discussions like this, which can be really helpful. As for the half angle formula, this follows from the sum of angles formulas for trigonometry, which themselves follow from the power series representations. These representations can be derived directly from the exponential function or from the differential equation y’’ = -y. All of this follows from straight analysis, which doesn’t rely on the Pythagorean theorem. You can find a discussion in Rudin’ Principles of Mathematical Analysis or in the RUclips video I linked at the end of this video.
6:03 The fact that sin(45)=1/sqrt(2) derived using Pyphagorean theorem. So you're proving Pyphagorean theorem holds by assuming Pyphagorean theorem holds.
No, it follows from the half angle identity from sin(90)=1. And the Pythagorean identity can be established from calculus and differential equations, with no reference to geometry and triangles
@@bubblegumgun3292 why the woosh? All of trigonometry is a consequence of calculus and differential equations. The dependencies go Calculus -> Trigonometry -> Geometry here.
I'm here after the 60 Minutes story finally arrived in my content algorithm... an entire year after-the-fact. Apparently I've been focused on old-white-male-politics for the past few years to be rewarded with content I actually like.
Well, to prove the equal side case you need to know what sine of 45 degrees is without using Pythagorean theorem which seems tricky. I think it would be better to prove it from the general case by a limit/continuity argument.
From the power series representation of sine and cosine, you can show that sine(pi/2) is equal to 1. Then from there you can use the half angle formula sin(45) = sqrt( 1 - cos(90)) / sqrt(2) = 1/sqrt(2).
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.
They are probably going to publish in an undergrad journal, which has a low bar for entry. The reviewers will chime in and make suggestions, and some things will get hammered out. The novelty of the proof will carry the paper.
So, I never took Trigonometry because they didn't have that in my high school and I didn't need it for anything I took in college. I have to ask - how is this not purely geometry and algebra combined? That's what it appears to be like to me. You just increase the notations (2, then 4, then 6, and so on) as you go down towards the smallest angle. That's still algebra, right? You're just using smaller and smaller versions of the same triangle, oriented differently than the original triangle, so that's just geometry, right?
Trigonometry could be built up in different ways. Classically, it arose from geometric problems associated with tracking the stars. Modern approaches to trigonometry build it up from calculus, without any reference to geometry. The changing side lengths form a geometric series, which is also a tool from calculus. Now you aren’t going to be able to completely remove geometry from geometry, like the existence of similar triangles. But here, the pivotal piece of the proof is the law of sines, which is a trigonometric identity.
That’s the Pythagorean identity, not the theorem. This identity actually doesn’t require geometry at all. Just a touch of calculus. Check out my video on trigonometry without geometry.
@@JoelRosenfeld Geometry is about measuring lengths and angles. Calculus is about finding lengths and angles by adding up a large number of very small lengths and angles. In this sense geometry is a special case of calculus. If you are allowed to use calculus then these kids arguments and Pythogrean geometry are subsumed into this whole.
I’m afraid you don’t have a clear view on the logical structure of Calculus and analysis. What you describe is roughly a 19th century view of the subject. Since then we have gutted the entire subject and built it all up from notions of set theory
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.
The proof which I know for the theorem is the following: Imagine a triangle in a 2d-coordinate system with the right angle at the origin. The first side with length a lies on the x-axis, the second side with length b lies on the y-axis. So the vectors pointing to the two corners are (a,0) and (0, b). The length of the hypotenuse c is the distance between these two points, or the length of (a, 0) - (0, b). In euclidean space this yields (by axiomatic definition of the norm) c = sqrt(a^2 + (-b)^2) or c^2 = a^2 + b^2. That way the theorem directly follows with the assumption that a 2d euclidean vectorspace exists. Why would this proof count as circular, or do I somehow miss the point of Jackson and Johnson's proof?
Johnson and Jackson’s proof just uses trigonometry. There are a lot of different routes to the same place, and it really just depends on the axioms you start with. Your proof seems to just use the definition and axioms of the real numbers, which is another starting point.
@@JoelRosenfeld I don't understand what people mean by saying that it "just uses trigonometry." It uses a trigonometric identity, but that's only a small part of the proof, and a needless one at that. Most of the proof is using geometry, algebra, and infinite series, so how it "just trigonometry"?
@@jimhabegger3712 you are right, this does use a bunch of geometry and calculus. I think the emphasis on the proof with “just” trigonometry really comes in opposition to Loomis’ quote, where he said that a proof using trigonometry is impossible. So not quite true that it’s just trig but I guess that the marketing that’s being thrown around. One way or another, it’s a fresh proof and done by some very talented high school students
@@JoelRosenfeld The abstract doesn't say that the proof uses *only* trigonometry, and I haven't seen those students saying that in any interviews. I have an idea how that started, but it's a long story. However that may be, saying that it uses only trigonometry diverts attention from the real value of what they did, and is one more reason for people discounting, mocking and ridiculing the whole story.
@@jimhabegger3712 the real important piece of the proof is the use of the law of sines, which is a theorem in trigonometry independent of the Pythagorean theorem. Geometric series and all the other bits are from geometry, but the interface with the law of sines is what is new here. At least from my understanding. Hence the emphasis on a trigonometric proof.
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I can't find the reference, but sometime, about 50 years ago, I saw an article in a Swiss (?) anthropology journal that discussed a Sumerian or Babylonian cuneiform tablet that had a picture of a right triangle subdivided into what appeared to be an infinite sequence of similar sub-triangles, all in the same consecutive ratio. Summing the lengths of the sides of these sub-triangles lying along the hypotenuse (or the base) of the original triangle, as a geometric series, gave a proof of Pythagoras' theorem. This seems to be equivalent to (or easily related to) the proof that these high school students thought up. Does anyone know the reference? The following You Tube video gives an elementary explanation of this method of proof: ruclips.net/video/7642iBEOjCk/видео.html
@@TheJackTheLion 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.
You can use the Law of Sines with ANY and ALL proofs of the Pythagorean Theorems. Sine = Opposite/hypotenuse. Law of Sines: a/sin(a) = b/sin(b) = c/sin(c) For any Right triangle: a/sin(a) = a/(a/c) = c b/sin(b) = b/(b/c) = c c/sin(c) = c/(c/c) = c For any Right triangle: a/sin(a) = b/sin(b) = c/sin(c) = c Which means for any Right Triangle: (a/sin(a))^2 = (b/sin(b))^2 = (c/sin(c))^2 = c^2 THEREFORE for any right triangle: a^2 + b^2 = (a/sin(a))^2 = (b/sin(b))^2 = (c/sin(c))^2 = c^2
It's a cool proof, but contrary to the news reports it doesn't tell us anything new about what assumptions are needed to prove the Pythagorean theorem. Forget the law of sines; the proof uses the fact that the interior angles of a triangle add to 180 degrees, and this is already known to be equivalent to the Pythagorean theorem!
@@JoelRosenfeld That's true. I never said the proof was wrong or unoriginal, or that finding the proof was not an impressive feat. But, as you discuss in the video, this is hyped to be more than just proof #272 of the Pythagorean theorem; it supposedly overturns two millennia of conventional wisdom about what assumptions are required to prove the theorem. That is not the case.
I looked at the trigonometry lessons in two textbooks, and I see them using the Pythagorean identity to prove some of the other ones. Maybe that’s one reason for people thinking that any proof using trigonometry is using the theorem to prove itself.
That is absolutely the reason for this, and it is exactly what Loomis says in his text concerning the Pythagorean theorem. The girls quote him in their abstract. Loomis wrote his book in the 1940s, and papers that started to establish Trigonometry as independent of Geometry started to get published in the late 60s. At the end of the day, what makes this significant is that this is a new proof and that it was found by people so young. This is exactly the sort of work professional mathematicians engage in, but they did it with so little training. Calling this a “trigonometric” proof or whatever hasn’t really mattered in this conversation. There is certainly still a lot of geometry in the proof. Proofs are abstract objects and we just put names and descriptors on them to help us understand and distinguish them.
@@jimhabegger3712 I honestly think you are being too conspiratorial about a story about two talented high school girls coming up with a new proof of the Pythagorean theorem. This is just a story where we can just be happy about their success and celebrate them.
This story is completely misreported. To actually understand the math theory read this article: "Is a Trigonometric Proof Possible for the Theorem of Pythagoras? " Michael de Villiers RUMEUS, University of Stellenbosch
@JoelRosenfeld You didn't misreport it. The journalists did. The original Articles state that they solved an "impossible" 2,000 year old problem. However, the "impossible problem" still hasn't been solved by them or Jason Zimba. In the late 1800s a mathematician named Elias Loomis said it was impossible to make a proof of the Pythagorean theorem using trigonometry because that would be circular reasoning. He said this because he defined trigonometry using the Unit Circle definition of Trigonometry. If you define trigonometry using the Unit Circle then he is right and it is impossible to make a valid trigonometric proof of the Pythagorean theorem. However, you can arguably define trigonometry using symmetry. Not all mathematicians would agree, but if you accept the symmetry definition of Trigonometry as valid then you could say Jason Zimba's proof and these girl's proof is a valid Trigonometric proof. The problem with that is many other proofs could also claim to be Trigonometric under the symmetry definition of Trigonometry and then these girl's can't claim to have solved an unsolved problem or a problem that has only been solved once before. Their proof is one of many and the impossible Trigonometric proof that is valid with the Unit circle still doesn't exist.
To be fair, just because you put law of sines in one part of your proof, it doesn't make it a trigonometric proof. It is still a nice thing they did and very creative. News channels should just stop broadcasting it as if it is a major breakthrough in math
It’s not a major breakthrough, but it is remarkable for high schoolers. It’s also a math topic the general public can get behind. So it’s the perfect storm for media fodder. And it is a trigonometric proof because it only presupposes trigonometry. What would it take for you to consider this proof trigonometric?
I cannot understand the interest of this boys proof… it uses convergency in infinite series and geometry obviously, what do you mean by “trigonometric” proof? Not only is long but is requiring an infinite sum that is way less intuitive as the classical proofs, and use same tools that the one at the beginning of the video, adding angles to have a right angle, parallels fifth axiom of Euclides (one cannot avoid it because Pythagoras theorem doesn’t hold in non-Euclidian geometries), and algebraic equalities.
It’s not meant to be an efficient proof. Just a new one. Trigonometry comes in through the law of sines. The geometric series is also classical and was known to the Greeks. The interest here is less mathematical and more social. The achievement is remarkable for high school students. And from a media standpoint, it’s easy to advertise, since everyone has seen the Pythagorean theorem before.
@@JoelRosenfeld I personally don’t appreciate, I find it so mathematical as opposed as you ☺️ What I found strange is the claim that is simpler than others, I remember 3 or 4 of them but this one is the only that requires an infinite series when is not needed. I’m mot saying that use infinite sums are not “greek style” (in fact there is), just seems to me little non esthetic. P.S: I have the impression if was a professional mathematician that came to the proof, we would say “oh, another one”. The charm of this one is age related IMHO
@@miguelalonsoperez5609 i am a professional mathematician. Personally, another proof of the Pythagorean theorem itself is ok. Not something ground breaking. But what is remarkable is the inventiveness of these high school students who actually discovered it. It’s about them more than the proof itself.
@@JoelRosenfeld yeah, they’re cool! 😎 Probably the most prone to reinvent ancient theorems are scholars, no other often tries to go around them! Some years ago I learned general relativity and was confused about Levi-Civita connection (never learned Riemann geometry) and I “invented” a proof of Christoffel symbols formula for covaraint derivative without using anything but basic calculus: I was thinking about publishing but after I thought it was no interesting at all except for me. When one ignores something is the best moment to fresh ideas! Hope the boys continue with maths or something else interesting.
Even, 1 to 1 Even Johnson and Jackson as per simple gematria and be-ginning letter J is 1/1! WHAT!!!!!!!!!!!!!!!! Fluke? In 9 years of individual observation at least 9 out 10 times connection. Not my A The first one "A" was silent. AlefBet.
Are we all just pretending not to see what's going on here? There are over 371 novel proofs to the Pythagorean theorem (and that's just the ones that are found to NOT be derivative of former submissions). My brother and I came up with several when we were in the 5th and 6th grade... but of course, none of them were new. This particular proof IF NEW is a cool thing... and it SHOULD look good on a college application. Good for them. BUT... this story has been BLOWN UP. Why? We all know why. I don't even have to say it. But the media knew that these girls weren't the first to prove the Pythagorean Theorem. After all... it's a Theorem ffs. But if it's not first, they have to blow it up some other way... so what do they say? They say this is the first TRIGONOMETRIC proof. But what they fail to realize is that Trigonometric proofs are CATEGORICALLY disqualified (because all trigonometric functions are derivative of the Pythagorean Theorem. It IS circular reasoning, and IF this theorem gets adopted, it will be AFTER they strip the trigonometric terms from it. This proof can be solved with only geometry and calculus. IN FACT, one COULD say that depending on how much these girls relied on the Law of Sines to solve this, they might not even have to understand their proof geometrically in order to solve it trigonometrically (meaning that this could be viewed as an incomplete proof) You need nothing more than Euklidian geometry to calculate the legs, and you need calculus to solve the series. That's it. Any use of any trigonometric function is circular. Nothing will EVER be proven by trigonometry. Sorry... but trigonometry is just a fairly brief list of triangle geometry properties that have been proven by euclidian geometry and a convention for signifying angle and length relationships. Trigonometry is useful, but it's not foundational. It will always be circular reasoning to use trig in proofs. Trig RELIES on the assumption of the pythagorean theorem. Don't believe me? Try this. *** a² + b² = c² (Where c2 is the hypotenuse)*** Now look at the relationship of sin cos and tan as it pertains to something called the "unit circle" Look at all the relationships. Every point on this circle correlates to sin cos and tan in some way. What an amazing coincidence, right? Do you know what the FORMULA for a unit circle is? It's not technically a function... it's not "f(x)=" so it's technically an equation. ***that equation for charting a unit circle... x² + y² = 1*** 1= 1² btw and 1 is the hypotenuse in a unit circle This proof is a nice, elegant GEOMETRIC/CALCULUS proof. I'm sick of hearing that it's "The first trigonometric proof." That is nonsense. There's no such thing BY DEFINITION. The coverage on this thing has been so inaccurate and sensationalized that I wouldn't be even a little surprised if they find this proof isn't even novel. Everyone just smiles and nods when there's a feel-good story in front of them.
Trigonometry can be demonstrated to be a consequence of calculus and differential equations, without the use of geometry. I made a video on that topic, if you want to check it out.
Sounds a lot like you are hating. This would have gotten coverage, regardless of the ethnicity of who came up with the theory. The only reason this is an issue for you, is because these ladies aren't white or Asian, and thus "didn't earn it" to you. Just because you don't understand it, doesn't mean that the logic is circular reasoning.
Um... correct me if I'm wrong... But isn't an hypotenuse suppose to be a straight line? If so how can a circle have an hypotenuse? But if you mean triangles inside the "Unit circle" with one corner at the centre/origin and a second point on the circle itself then perhaps, just maybe *you should say that*
@@EarlJohn61 I REFERENCED the "Unit Circle". There's a reason they saw fit to NAME the "Unit Circle". It's so that if you say "Unit Circle" and someone who doesn't know what the "Unit Circle" is can look up the "Unit Circle" in Google and find out what the "Unit Circle" is. I'm not going to explain what the "Unit Circle" is in a youtube comment. If it went over your head, just look up the "Unit Circle". That is actually the purpose of giving the "Unit Circle" a name in the first place. So that anyone who's confused can learn about it under the name " Unit Circle". Comprende? Similarly, if you don't "Comprende" what "Comprende" means... don't come b!tching to me. Look it up. It's a word. It means something. If you don't know what it means and you can't be bothered with researching that which you don't know... that's on you. Oh, and btw... you know that anyone who cares to check knows that you liked your own comment, right? LOL!
Remember when I capitalized "IF NEW" referring to this so-called "novel proof"? Well guess what... it isn't even new. It has come to my attention that this is nothing more than a redressing of John Arioni's proof "Pythagorean Theorem via Geometric Progression" using trigonometric terms (which on its own invalidates the proof). So this proof is not only invalid (circular reasoning) because it used trigonometric terms It's also invalid because it's extraneous (ie, not simplified) which is another reason we don't use trigonometric terms in pythagorean proofs. They are inherently not simplified. And after all that... this proof isn't even original/novel. Look for yourself. Look up "Pythagorean Theorem via Geometric Progression by John arioni" and you will find the very same proof complete with "waffle cone"... except it was completed using geometric identities (because DUH)... and it was done in 1996 NOW what do y'all have to say? You still simping for these 2 girls? It's so sad that the media these days has absolutely no sense of reality or integrity. You can pretty well set your watch to the idea that nothing the media tells you is 100% accurate. If ever the media encountered a VERIFIED MIRACLE, they would exaggerate it anyway.
You raise the specter of circular reasoning between geometry and trig at :52-:56, but wait another 10 minutes or so to acknowledge that it may not be true about this proof. A proof that you have not seen. Also you make sure everyone knows right up front that the math community does actually have plenty of proofs of this theorem, in case the uninitiated might think that these 2 young people have done something math teachers and others have been unable to do for millennia. Allowing anyone to think THAT for 10 minutes would be just terrible, but leading your audience to believe that this proof, which you have not read, by these 2 young women is based on circular reasoning....well, it's OK to let them think THAT for 10 minutes. And maybe think it forever since many people don't watch RUclips videos to their conclusion. Their proof could be all trig by way of calculus or something else and not be based on geometry. If this is the case, the circular reasoning you talk about up front may not apply at all. Since you haven't seen it, you don't know, but somehow you feel free to cast aspersions on their achievement before having the read the proof. I'm thinking that these students and their teacher also know what circular reasoning is and understand the difference between geometry and trig, and so wouldn't come forward in this very public way if those things were part of this proof. I could be wrong, but I could be right. Which is why this video is so distasteful to me. It is all about you and not about whatever their achievement may have been. Why are you talking if you haven't actually read the thing you are talking about?
Their proof has not been published, but has been publicized. This video is here to clear up the details of their proof. That picture is enough to figure out the proof. There is a concept in mathematics of a “proof by picture.” It was popularized back in the 80s or so, and that is what this becomes without the writing. It’s a game mathematicians play, where they craft a picture that can present a proof on its own. What I am doing here is giving context. That’s my job as a professor. And then I’m talking about what they accomplished. The question of circular reasoning can’t be determined until you’ve been through the proof and examined it, which is exactly what we did. The media brought up the issue of circular reasoning, and so I addressed it in this video. This video is just me reporting on what they did. I just have more experience than a typical news person, so I can give a lot of details that they cannot. It’s not about me, it’s about exploring their reasoning based on what was available when I made the video. Do you think news organizations are making it about themselves too?
I should say that the math you described was interesting and woke up the unused geometry and trig stored in my brain. (Much has rusted away.) I followed most of it, which was mostly due to how you explained it. So thank you for that. Yes, sometimes the news reports things about research (about any topic, doesn’t even have to be about things that confuse the average person) that are not true just because it’s more sensational or gets them more attention. That is them making it about them. The choices around how to construct a piece, how to deliver the content, are what I take issue with. You are not alone in making these types of choices. But I also am not alone in taking issue with it.
The Babylonians didn’t engage in mathematics in the same way that the Greeks did. There was a lot of intuition but not proof. Babylonians were obsessive data collectors, and had developed some nice results in their own way. I mentioned that they knew of the Pythagorean theorem in this video, but they also knew the quadratic equation. I wouldn’t say “European” here as much as “Ancient Greeks.”
Why does anyone care about anything they didn’t do themselves? That is an interesting philosophical question. People are interested, because a couple of very young people found something new about a topic that everyone has familiarity with. That’s a natural thing that people are interested in. The real question is, why did you end up here on a video about a math topic, if YOU don’t care?
@@ishmaelbenn4002 Ancient Greeks aren't European in the way we think of it today. They were part of a cluster of civilizations right next to Egypt and Babylon.
I didn't realize that I needed a history lesson for the first half of your video but then your mansplaining was almost too much after you said in the beginning that their proofs were f-ed up. When you conclude that both of them did this proof, you're failing to acknowledge that one of them came up with a separate entirely different approach, and hers was much simpler. Your video comes across as jealous and frankly, your explanation was unnecessary. Now you're getting views on the backs of someone else's work...especially since these are young high school African American girls. Time for self- reflection....
The whole point to the video is to give contextual background to their accomplishment. I’m sorry that you felt I was mansplaining, but I’m a professor and my job is to make this sort of content digestible to the general public. I’m proud of them for their accomplishments. I never said their proofs were bad. Maybe you feel the explanation was unnecessary, but others do not.
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/1.gif)
They actually had TWO independent proofs from one another. Calcea Johnson's proof (so-called Waffle Proof) is the one you describe. Ne’Kiya Jackson's proof used a right triangle inside a circle bisected by the radius of the circle to create her equations.
They currently have 10 proofs and passed peer review this week.
It does my heart good to see more kids getting into maths. I couldn’t even figure out the basic geometric proof until halfway through college and I was the straight A student in my math classes.
Math wasn’t even on my radar in high school to be honest. I’m thoroughly impressed
I'm glad to see this. The AMS and journalists have diverted attention from the real value in what those students did, and this helps bring attention back to that. Thank you.
I’m really happy you think so. Thanks for taking the time to leave a comment!
Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!🎉
The 2 students submitted their Proofs and they are working on additional Proofs as well. Thanks for covering this topic.
fake
I am a retired teacher of secondary school maths in the UK. I couldn't have done this. Well done, ladies😍
News casters are almost never serious when covering mathematics stories.
It’s always the same script too: “I was terrible at math” or “i don’t know how to calculate a tip” etc.
Sometimes I wonder if they are truly bad, or if they feel that they need to act dumb to connect with the audience
They think playing ignorant will make them look cool
@@JoelRosenfeld I believe them though. So many people hate math.
@@godzgag I don’t think they are playing.
@@JoelRosenfeld Exactly. Because media/newscasters are totally uncreative unoriginal and lazy.
Each one actually came up with their own proof. There was also a mathematician who came up with 1 in 2009 too, but they are the only three to have done so.
Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!🎉
I know I'm not the only one who watched the entire video completely lost.
Congratulations Nekita and Calcea
Me too ! 🤣🤣🤣🤣🤣❤️
Got lost at 6:50
I appreciate your quick, lively, and detailed overview of the context: history, math, and modern journalism - ouch, those were painful behaviors to watch. This is so far my favorite video on the topic to give to students. I'd like to note, in case you would want to know, that the music was difficult for me, maybe due to the sensory issues frequent among math people. I'd love to have a no-music version of this (or very brief intro and ending).
"thinking outside the box" twice was the solution to connect pythagoreas theorem with trigonometry, a lot of proofs and trigonometric properties are limited with one rectangular triangle.
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.
Well done Ne’Kiya Jackson and Calcea Johnson!!!
Really appreciate the summary!
Glad to see you here! And I’m happy you liked it!
I was scoffing at people thinking that any proof using trigonometry is using the theorem to prove itself. Just now I read an article that used the theorem to prove another trigonometric identity, and said that proof is "the standard one that you see in most textbooks." Maybe that's the approach most often used in textbooks, first proving the Pythagorean theorem, then using that to prove the other identities. Maybe that's why so many people think that there's no way to prove them without it.
I get that analysis can be done without drawing any pictures, but I still think that calculus, and therefore trigonometry, is a consequence of geometry. That said, this is an impressive paper for a high schooler. There's nothing wrong with proving the if in an iff.
Thank you for a great video explaining what these young ladies did. You’re correct it was very impressive given their age. And you were spot on that even some basic problems lend themselves to a new perspective. Kudos to their teacher for giving them the challenge and perhaps maybe not expecting them to come up with a new proof but getting them to think. Education is not the feeling of a bucket it should be the lighting of a fire. I forget who said that but it wasn’t me. Again thank you very much.
As a math simpleton I still don't completely understand how Calcea's proof works, but this video helped a little bit.
Reading and understanding proofs is an art. Sometimes it takes me a few tries and a few days for something to click.
I have 1 question. How can we prove cos(2x) = cos^2(x) - sin^2(x) without using Pitagore theorem in the formula sin^2(x) + cos^2(x) = 1
The nice thing about identities is that you are free to select whichever side you want to start with. For this one, I'd start with the right hand side and expand both cosine and sine into their exponential representations. From there, it should just be a matter of algebra to reduce to cos(2x)'s exponential representation.
This only uses the analytic form of sine and cosine that we got from the differential equation definition.
I love construction proofs, especially when it adds new elements!! Also love that they combined trigonometry with infinite series. Sooo cool!!! I wonder how they arrived at it? I guess it must've started with adding one triangle and then a second and then a third and then BOOM! They mustve noticed that the triangles are scaling down and will converge. This is soooooo cool!!! It's making me happy just thinking about it.
It was the Sumerians, not the Babylonians who wrote the formula. Hundreds of year’s before the Babylonians. The Egyptians also wrote it as they built the pyramids.
Corrrect😎👍💯💯💯
It's all aliens
@@vectoralphaSec More Proof That The Balcks Are Not Off This World !!!
Repent
The egyptians just used pythagorean triples, they didn't know the law, they exhaustively hand calculated and experimentally found right triangle triples.
@@freeshavaacadooo1095 are you certain that's all the Egyptians used and what about the Sudanese??
9:05 How do you know that sin γ = sin(90^-2α).....? I need to see the proof without using the Pythagorean theorem. In other words, 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?
I have a link at the end of the video that does just that
@@JoelRosenfeld ruclips.net/video/9XFBw8Hfk5k/видео.html Why is cos being called cos(t) and not apple(t)?
The term "cosine" is derived from the Latin phrase "complementi sinus" or "complementum sinus." So, we cannot simply call things by arbitrary names. There is a reason behind the naming conventions in mathematics. I'm not here to point out a mistake because it seems there is a plaintext explanation of mathematics emerging, and I'm trying to learn. This is interesting to me because I often have a hard time communicating with mathematicians. They may assume certain things and then use those assumptions as facts later on, similar to the argument presented here.
It is important to note that in trigonometry, the names given to mathematical functions, such as sine and cosine, have historical and contextual significance. They are not arbitrary labels. The term "sinus" was used to describe the curve or shape of the sine function due to its sinusoidal graph when plotted against angles. Similarly, "cosine" refers to the complementary relationship it has with the sine function.
Understanding these historical origins and the mathematical context behind the naming conventions can help in communicating and comprehending mathematical concepts effectively.
No one is denying that we came to trigonometry via geometry and Pythagoras.
@@DisgruntledSpam Well, that is exactly what Ne'Kita Jackson and Calcea Johnson are denying. They claim that we can get to trigonometry(sin and cos) without Pythagoras.
Nifty. I had heard of this approach a few months ago, but I was not aware of the "punch line," in a manner of speaking. Good for them. Pretty darned clever, I'd say. I wonder if they might have anything else of their sleeves? Staying tuned. Sounds like they've got their Ph.Ds. nailed down too, just write up their findings.
First of all, just the fact that these two students independently created 2 different "proofs" themselves are fantastic, regardless of the claim about being independent from Geometry.
I didn't understand how one can say that the Law of Sines from Trigonometry is not based on Geometry, so I will check out that video that explains how it can be derived from calculus.
@@marcfruchtman9473 that’s more or less my own stance. Even if you don’t buy the “without geometry” claim, it’s still two new proofs by high school kids. Remarkable either way.
Great work to Jackson and Johnson!! I love to hear stories of discoveries being made by high school students! I’m surprised this proof by geometric series was overlooked. Perhaps it’s because geometers have bound themselves to a rule of finitely many constructions by ruler and compass and ruled it out, no pun intended. But another proof of this ancient theorem goes to show there are still many gems to be uncovered, even today
Another variation would be to make two smaller new similar right triangles inside our larger new right triangle by dropping a perpendicular from its right angle vertex to its hypotenuse. Then the sum of the areas of the two new smaller right triangles is equal to the area of the new larger right triangle.
Couldn't they extend this to Ptolemy's theorem?
I don’t know! That’s an interesting idea though.
I was thinking like you. Once I saw the picture. You clearly have something that looks like an integral with limits.
What is a good mathematical journal to present a new proof of the Pythagorean Theorem? I proof it differently and have checked with more than 400 solutions and did not find my solution, so I want to submit it for publication. Can you suggest a journal?
It wouldn’t go to a big journal, but probably would fit into an undergraduate journal of some kind. Search for journals dedicated to undergraduate research.
What is this proof?
Try submitting it to The Mathematics Teacher or American Mathematical Association of Two-Year Colleges
in the last part you would have done the cos2x directly instead of the law of sines xd good video.
I also have a trigonometric proof of the pythagorean theorem
But where should I publish it??
@@khalilbsfic give it to me bro
In the 60 Mintes video, it was stated that they only used trigonometry to creat this proof. So, they used only trigonometry to create this so-called new proof? What did they initially start with?, a right triangle, which is geometry. I applaud their effort, but you cannot get to the moon w/o going up. One cannot get to 3 w/o going through 2. 1+1.9 does not equal 3, only 1+2=3.
This is how math curriculum in high school goes: algebra 1 < geometry < algebra 2 < trigonometry. The 60-Minutes video cites the highest level of mathematics that the students used which implies that they have taken geometry and 2 courses of algebra. In the same way, a student who knows algebra 1 knows how to do fractions, integers, decimals, etc., from arithmetic in grade school.
I watched over and over as I did when I listened to metallica's kill'm all for the first time 40 years ago ...
You 2 are an inspiration to us all. ! Thanks 🙏 so much !!
That was one of two seperate proofs.
Read the notes B4 posting!
@@williammajor6768 when I made this video there was very little information out there. But I appreciate you all coming into the comments to share what you know
Greeks may have been credited for the creation of proof and logic but they credited the Egyptian for everything they learned. Including Pythagoras.
Sure, but our perspective is different. We can see that there was a strong qualitative difference between what the Greeks did and the Egyptians. The Egyptians really didn’t introduce logical proof, which for us is what mathematics is.
@@JoelRosenfeld you should check more about Egyptians. You will find that most of what Greeks did were taken from Egypt, including names of all those Gods!
There is just a king of brainwashing state to make all greek.
I loved your proof, but struggle with the math replication of the hypotenuses. If the second triangle in the series moving down the waffle cone has hypotenuse (2a^2/b^2)*c, that is 2a^2/b^2 bigger than the original triangle with hypotenuse of length c. So, why are all the triangles moving down only a^2/b^2 bigger and not 2a^2/b^2 bigger? I know from a different proof I saw that a^2/b^2 is correct, I just cannot figure out in your proof how you go from knowing (2a^2/b^2)*c for the hypotenuse that the one below that has a hypotenuse of length (2a^4/b^4)*c.
Try to recreate the algebra. It's a lot of work but that's how I understood it. It employs proportionality that is derived from similar triangles. My proof in the end is a lot simpler than what was presented in the video.
@@omysoulblessthelord4578 Thanks. I did the math, but at first I had to do more calculations than you did. Then, I realized the triangle below the mirrored isosceles triangle is 2a/b the size of the original triangle and the triangle to the right of that triangle is a/b the size of that triangle. That is why the first right triangle to the right is (2a/b)*(a/b) the size of the original triangle. From there, the triangle below that is a/b the size of that and the triangle to the right of that is a/b the size of that. So, each triangle moving down is (a/b) * (a/b) = a^2/b^2 the size of the previous one above it.
The peer review process finished this week. They submitted 10 proofs.
@@rayhs1984 that is quite impressive!
I think I'm still kinda lost here. How is that "just trigonometry" when they clearly used geometric arguments? What makes this proof any more 'trigonometric' than the first proof you gave?
The whole idea is to leverage ideas from trigonometry, specially, the law of sines. Loomis' argument that trigonometry rests on the Pythagorean theorem really isn't true, and so, purportedly, his list doesn't include methods that leverage trigonometry.
I don't think that those students have ever said that their proof is "just trigonometry." I have some ideas about why people are saying that, but it might be a long story. Mayve that's the only way that they can explain to themselves why it's getting so much attention. One possible reason that I see for all the attention is the false claim that mathematicians have always thought that a trigonometric proof was impossible. Another possible reason is people thinking that the story is uplifting three categories of oppressed people.
@@jimhabegger3712 the trigonometry emphasis is coming from their abstract for their talk at the AMS sectional meeting. There they give loomis’ statement about trigonometry, and that caught on and probably got inflated.
Farkle, I'm thinking that the only reason it got into the news is because it fits into three diversity/inclusion categories. If it had been two old white men, they never would have been invited to an AMS meeting in the first place. They never would have made such a ridiculous and easily disproven claim, and even if they had, the AMS would have rejected it.
@@JoelRosenfeld Yes, I understand that, but the more that people talk about it being "trigonometric," the worse it might be for them, if and when the falseness and foolishness of their "considered impossible for 2000 years" claim becomes public knowledge. I think it's better to turn people's attention to the real value of what they did: that they tried to do something they thought was considered impossible; and didn't give up until they succeeded; and what might actually be new and interesting in their proof, which has nothing to do with it being "trigonometric."
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!
They read it in a book that collected all the proofs of the pythagorean theorem. This book claimed that there was no proof that was purely trigonometric, since trigonometry relies on the pythagorean theorem. However, this was in the 1940s or 50s when it was written. Since then, there have been several well documented definitions of trigonometric functions that don't rely on geometry.
The new element is the sum of the convergent geometric series. I think the law of sines is a distraction. We just need to use the ordinary geometrical similarity of right triangles. We get similar right triangles in two ways. 'One', by taking the doubled original shorter side as hypotenuse. 'Two', by taking a copy of our original right triangle. 'One' is a scaled version of 'Two'. From there on, it's repetition.
Applying the Law of Sines makes the trigonometric claim legit.
@@omysoulblessthelord4578 Thank you for your response. I have a prejudice against relying on trigonometric functions when one hasn't derived a way of calculating their values for a general angle. In the present context, the law of sines is a fancy way of talking about similar triangles; the actual values of the sine function are neither used nor calculated in this context. So I would say that the law of sines is a practically irrelevant complication. Important is that the present proof relies on the meeting of the two lines at the convergence point; that is one of Euclid's axioms.
I suppose you must consider that the students are high schoolers and could be exposed to formal mathematics at a very novice level, if at all. They used what they know. I learned the formality of the Euclidean Axioms at the graduate level, not as a high school student.
@@omysoulblessthelord4578 It is an admirable achievement by the students to invent these new proofs. I too didn't understand the Euclidean system of axioms till later in life. I was just reflecting on the proof itself. I think that the summing of a geometric progression is unusual in this context, and therefore notable.
It is thought tjhat Pythagoras's theorem has been known for 4000 years (i.e. before Pythagoras). As I understand it, the recent discovery is trigonometric proof of Pythagoras's theorem Until very recently (1950?) It was thought that this was not possible. Sine and Cosine did not exist, as I understand it until the 12th century A.D.
The real novelty here is the discovery of a new proof of the Pythagorean theorem. There are honestly hundreds that we know, but the fact that a new proof was discovered by high school students is really the icing on the cake.
Sine and cosine were known to Ptolemy in antiquity. In the 12th century, Indian mathematicians discovered the power series representations.
6:00
But don’t you need to use pythagoras to know that tan(45) = 1/√2
EDIT: Just realised you can use double angle formula, 2sin45 cos45 = sin90 = 1 with sin45=cos45 (since the ratio of lengths will be the same due bc isosceles)
Min 6:02.
sin 45 = 1/√2 is calculated with Pythagoras.
The case a=b is given very easily, only with the definition of cos.
If a=b => sin=a/c, sin=c/2a => c² = 2a² => c² = a² + a²
Sin 45 = 1/sqrt 2 comes from the half angle formula, which follows from the power series definition.
@@JoelRosenfeld The above argument is missing from the video clip. You also need to prove the half angle formula without using Pitagora's theorem.
@@silviadoandes7930 I didn’t want the video to go too long, and it’s not to bad to prove on its own. But I appreciate your eagle eyes.
@@JoelRosenfeld If you claim that you have a proof for the case of the isoscel triangle, then that proof needs to be air tight.
And in the form presented in the video, it is not.
Even the verification you make at the end is based on the Pythagorean theorem - circular much?
If the length of the video does not allow this, a reference to a complete detailed proof should be provided.
@@silviadoandes7930 this isn’t an academic paper, it’s a RUclips video. If you want a video to be successful, you have to get to the point, and diversions like that will absolutely kill retention and RUclips won’t show it to hardly anyone. I’ve done a lot of experimenting with this, and I’ve learned a lot about retention and communication.
That said, even in an academic paper, you are free to assume well known concepts and otherwise provide citations. RUclips has the added benefit of allowing for discussions like this, which can be really helpful.
As for the half angle formula, this follows from the sum of angles formulas for trigonometry, which themselves follow from the power series representations. These representations can be derived directly from the exponential function or from the differential equation y’’ = -y.
All of this follows from straight analysis, which doesn’t rely on the Pythagorean theorem. You can find a discussion in Rudin’ Principles of Mathematical Analysis or in the RUclips video I linked at the end of this video.
6:03 The fact that sin(45)=1/sqrt(2) derived using Pyphagorean theorem. So you're proving Pyphagorean theorem holds by assuming Pyphagorean theorem holds.
No, it follows from the half angle identity from sin(90)=1. And the Pythagorean identity can be established from calculus and differential equations, with no reference to geometry and triangles
@@JoelRosenfeld >>>>>"can be established from calculus" 😂😂XD LMAO WOOOSH
@@bubblegumgun3292 why the woosh? All of trigonometry is a consequence of calculus and differential equations. The dependencies go Calculus -> Trigonometry -> Geometry here.
I'm here after the 60 Minutes story finally arrived in my content algorithm... an entire year after-the-fact. Apparently I've been focused on old-white-male-politics for the past few years to be rewarded with content I actually like.
Welcome! Happy to have you here!
Well, to prove the equal side case you need to know what sine of 45 degrees is without using Pythagorean theorem which seems tricky. I think it would be better to prove it from the general case by a limit/continuity argument.
From the power series representation of sine and cosine, you can show that sine(pi/2) is equal to 1. Then from there you can use the half angle formula sin(45) = sqrt( 1 - cos(90)) / sqrt(2) = 1/sqrt(2).
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.
They are probably going to publish in an undergrad journal, which has a low bar for entry. The reviewers will chime in and make suggestions, and some things will get hammered out. The novelty of the proof will carry the paper.
So, I never took Trigonometry because they didn't have that in my high school and I didn't need it for anything I took in college. I have to ask - how is this not purely geometry and algebra combined? That's what it appears to be like to me. You just increase the notations (2, then 4, then 6, and so on) as you go down towards the smallest angle. That's still algebra, right? You're just using smaller and smaller versions of the same triangle, oriented differently than the original triangle, so that's just geometry, right?
Trigonometry could be built up in different ways. Classically, it arose from geometric problems associated with tracking the stars. Modern approaches to trigonometry build it up from calculus, without any reference to geometry.
The changing side lengths form a geometric series, which is also a tool from calculus.
Now you aren’t going to be able to completely remove geometry from geometry, like the existence of similar triangles. But here, the pivotal piece of the proof is the law of sines, which is a trigonometric identity.
This is circular because trignometry implies pythogrean theorem sin^2(x) + cos^2(x) = 1
That’s the Pythagorean identity, not the theorem. This identity actually doesn’t require geometry at all. Just a touch of calculus. Check out my video on trigonometry without geometry.
@@JoelRosenfeld Geometry is about measuring lengths and angles. Calculus is about finding lengths and angles by adding up a large number of very small lengths and angles. In this sense geometry is a special case of calculus. If you are allowed to use calculus then these kids arguments and Pythogrean geometry are subsumed into this whole.
I’m afraid you don’t have a clear view on the logical structure of Calculus and analysis. What you describe is roughly a 19th century view of the subject. Since then we have gutted the entire subject and built it all up from notions of set theory
This makes me think of those videos where college students can't answer questions like what is 3x3x3 or 9+9+9 or 25+85 or 15% of 100 or 4x7.
Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!🎉
The answers my friend (from the top of my head, no calculator, not even pen and paper, are respectively: *27, 27, 110, 15 & 28*
@@EarlJohn61 Good job. 🙂
the square within a square is a more elegant proof.
Certainly, but this isn’t about making the most elegant proof. Just a new one.
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.
The proof which I know for the theorem is the following: Imagine a triangle in a 2d-coordinate system with the right angle at the origin. The first side with length a lies on the x-axis, the second side with length b lies on the y-axis. So the vectors pointing to the two corners are (a,0) and (0, b). The length of the hypotenuse c is the distance between these two points, or the length of (a, 0) - (0, b). In euclidean space this yields (by axiomatic definition of the norm) c = sqrt(a^2 + (-b)^2) or c^2 = a^2 + b^2. That way the theorem directly follows with the assumption that a 2d euclidean vectorspace exists.
Why would this proof count as circular, or do I somehow miss the point of Jackson and Johnson's proof?
Johnson and Jackson’s proof just uses trigonometry. There are a lot of different routes to the same place, and it really just depends on the axioms you start with.
Your proof seems to just use the definition and axioms of the real numbers, which is another starting point.
@@JoelRosenfeld I don't understand what people mean by saying that it "just uses trigonometry." It uses a trigonometric identity, but that's only a small part of the proof, and a needless one at that. Most of the proof is using geometry, algebra, and infinite series, so how it "just trigonometry"?
@@jimhabegger3712 you are right, this does use a bunch of geometry and calculus. I think the emphasis on the proof with “just” trigonometry really comes in opposition to Loomis’ quote, where he said that a proof using trigonometry is impossible. So not quite true that it’s just trig but I guess that the marketing that’s being thrown around. One way or another, it’s a fresh proof and done by some very talented high school students
@@JoelRosenfeld The abstract doesn't say that the proof uses *only* trigonometry, and I haven't seen those students saying that in any interviews. I have an idea how that started, but it's a long story. However that may be, saying that it uses only trigonometry diverts attention from the real value of what they did, and is one more reason for people discounting, mocking and ridiculing the whole story.
@@jimhabegger3712 the real important piece of the proof is the use of the law of sines, which is a theorem in trigonometry independent of the Pythagorean theorem. Geometric series and all the other bits are from geometry, but the interface with the law of sines is what is new here. At least from my understanding. Hence the emphasis on a trigonometric proof.
Ya but its no different than asking equal distances of each line of a triangle that can then form a perfect square
Do you think no one else made an attempt before them as opposed to couldn't figure it out. There was a guy in 2009 that also found the answer.
If you bounce light in a straight line should it not take an expanding angle.
crazy thing is it wasn't even new
See the steps! Steps "Divine" "Divine" steps Every stair has a starting place. A place that can be defined as a step. A place to start or a place to "step" from.
The "next step" in terms of human advancement is at a be-ginning non-material. time, times, and half time. Material even-ing always followed by material morn (mourn) Day portion is light the part void of light is night. In this reality the light portion is materially 1/2 1 and 2 and every relationship between this variation as per first math constant now seen through first offering as per sacred be-ginning. yet the first pair-ing has no variation 1/1
Not about the proof, but Pythagoras lived approximately 2500 years ago.
I found out another trigonometry proof. Video with it is on my channel.
Indians invented the Pythagorean theorem centuries before the Egyptian or anyone else, it is is mentioned in the Sulba Sutra
How come all these grown men couldn’t find this? Amazing for these girls
because grown men aren’t the answer for everything.
It’s named after a a man, and one guy did it in 2009. These young women found two more.
A dude did back in 2009, it's already published using trig
cause they weren't looking. it's not that interesting.
I want to watch their video...
🤔
And to infinity.......
peace and love peace and love, don't have a light pointing directly into the camera.
I can't find the reference, but sometime, about 50 years ago, I saw an article in a Swiss (?) anthropology journal that discussed a Sumerian or Babylonian cuneiform tablet that had a picture of a right triangle subdivided into what appeared to be an infinite sequence of similar sub-triangles, all in the same consecutive ratio. Summing the lengths of the sides of these sub-triangles lying along the hypotenuse (or the base) of the original triangle, as a geometric series, gave a proof of Pythagoras' theorem. This seems to be equivalent to (or easily related to) the proof that these high school students thought up. Does anyone know the reference?
The following You Tube video gives an elementary explanation of this method of proof: ruclips.net/video/7642iBEOjCk/видео.html
Lol...Is this not the exact same thing as described in their "new" proof? The exception being that this is 3700 years ago.
@@TheJackTheLion 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.
You can use the Law of Sines with ANY and ALL proofs of the Pythagorean Theorems.
Sine = Opposite/hypotenuse.
Law of Sines: a/sin(a) = b/sin(b) = c/sin(c)
For any Right triangle:
a/sin(a) = a/(a/c) = c
b/sin(b) = b/(b/c) = c
c/sin(c) = c/(c/c) = c
For any Right triangle: a/sin(a) = b/sin(b) = c/sin(c) = c
Which means for any Right Triangle: (a/sin(a))^2 = (b/sin(b))^2 = (c/sin(c))^2 = c^2
THEREFORE for any right triangle: a^2 + b^2 = (a/sin(a))^2 = (b/sin(b))^2 = (c/sin(c))^2 = c^2
fresh eyes! congrats!! goes further back to Egypt
Wow
It's a cool proof, but contrary to the news reports it doesn't tell us anything new about what assumptions are needed to prove the Pythagorean theorem. Forget the law of sines; the proof uses the fact that the interior angles of a triangle add to 180 degrees, and this is already known to be equivalent to the Pythagorean theorem!
A proof doesn’t have to be efficient to be correct or new
@@JoelRosenfeld That's true. I never said the proof was wrong or unoriginal, or that finding the proof was not an impressive feat. But, as you discuss in the video, this is hyped to be more than just proof #272 of the Pythagorean theorem; it supposedly overturns two millennia of conventional wisdom about what assumptions are required to prove the theorem. That is not the case.
Ya, it was proofed using trig back in 2009 by Jason zima
Why did yoU climb the mountain? because it is there. in doing so what has been bettered?
I looked at the trigonometry lessons in two textbooks, and I see them using the Pythagorean identity to prove some of the other ones. Maybe that’s one reason for people thinking that any proof using trigonometry is using the theorem to prove itself.
That is absolutely the reason for this, and it is exactly what Loomis says in his text concerning the Pythagorean theorem. The girls quote him in their abstract. Loomis wrote his book in the 1940s, and papers that started to establish Trigonometry as independent of Geometry started to get published in the late 60s.
At the end of the day, what makes this significant is that this is a new proof and that it was found by people so young. This is exactly the sort of work professional mathematicians engage in, but they did it with so little training.
Calling this a “trigonometric” proof or whatever hasn’t really mattered in this conversation. There is certainly still a lot of geometry in the proof. Proofs are abstract objects and we just put names and descriptors on them to help us understand and distinguish them.
@@JoelRosenfeld … and to make the story say what we want it to say. 😀
@@jimhabegger3712 I honestly think you are being too conspiratorial about a story about two talented high school girls coming up with a new proof of the Pythagorean theorem. This is just a story where we can just be happy about their success and celebrate them.
You missed. . . Denise Feder and the MATH CLUB, baby!!!!!
I’m not sure I understand
Nice 😎👍💯💯💯
Introduction to introduction to introduction to ....
This story is completely misreported.
To actually understand the math theory read this article:
"Is a Trigonometric Proof Possible for the Theorem of Pythagoras? "
Michael de Villiers
RUMEUS, University of Stellenbosch
@@kirkb2665 what do you think I misreported?
@JoelRosenfeld You didn't misreport it. The journalists did. The original Articles state that they solved an "impossible" 2,000 year old problem. However, the "impossible problem" still hasn't been solved by them or Jason Zimba. In the late 1800s a mathematician named Elias Loomis said it was impossible to make a proof of the Pythagorean theorem using trigonometry because that would be circular reasoning. He said this because he defined trigonometry using the Unit Circle definition of Trigonometry. If you define trigonometry using the Unit Circle then he is right and it is impossible to make a valid trigonometric proof of the Pythagorean theorem. However, you can arguably define trigonometry using symmetry. Not all mathematicians would agree, but if you accept the symmetry definition of Trigonometry as valid then you could say Jason Zimba's proof and these girl's proof is a valid Trigonometric proof. The problem with that is many other proofs could also claim to be Trigonometric under the symmetry definition of Trigonometry and then these girl's can't claim to have solved an unsolved problem or a problem that has only been solved once before. Their proof is one of many and the impossible Trigonometric proof that is valid with the Unit circle still doesn't exist.
To be fair, just because you put law of sines in one part of your proof, it doesn't make it a trigonometric proof.
It is still a nice thing they did and very creative. News channels should just stop broadcasting it as if it is a major breakthrough in math
It’s not a major breakthrough, but it is remarkable for high schoolers. It’s also a math topic the general public can get behind. So it’s the perfect storm for media fodder.
And it is a trigonometric proof because it only presupposes trigonometry. What would it take for you to consider this proof trigonometric?
@JoelRosenfeld it uses calculus. If we say this is a trigonometric proof, then proofs that only use calculus would also be trigonometric proof
I cannot understand the interest of this boys proof… it uses convergency in infinite series and geometry obviously, what do you mean by “trigonometric” proof?
Not only is long but is requiring an infinite sum that is way less intuitive as the classical proofs, and use same tools that the one at the beginning of the video, adding angles to have a right angle, parallels fifth axiom of Euclides (one cannot avoid it because Pythagoras theorem doesn’t hold in non-Euclidian geometries), and algebraic equalities.
It’s not meant to be an efficient proof. Just a new one. Trigonometry comes in through the law of sines. The geometric series is also classical and was known to the Greeks.
The interest here is less mathematical and more social. The achievement is remarkable for high school students. And from a media standpoint, it’s easy to advertise, since everyone has seen the Pythagorean theorem before.
@@JoelRosenfeld I personally don’t appreciate, I find it so mathematical as opposed as you ☺️
What I found strange is the claim that is simpler than others, I remember 3 or 4 of them but this one is the only that requires an infinite series when is not needed. I’m mot saying that use infinite sums are not “greek style” (in fact there is), just seems to me little non esthetic.
P.S: I have the impression if was a professional mathematician that came to the proof, we would say “oh, another one”. The charm of this one is age related IMHO
@@miguelalonsoperez5609 i am a professional mathematician. Personally, another proof of the Pythagorean theorem itself is ok. Not something ground breaking. But what is remarkable is the inventiveness of these high school students who actually discovered it. It’s about them more than the proof itself.
@@JoelRosenfeld yeah, they’re cool! 😎
Probably the most prone to reinvent ancient theorems are scholars, no other often tries to go around them!
Some years ago I learned general relativity and was confused about Levi-Civita connection (never learned Riemann geometry) and I “invented” a proof of Christoffel symbols formula for covaraint derivative without using anything but basic calculus: I was thinking about publishing but after I thought it was no interesting at all except for me.
When one ignores something is the best moment to fresh ideas! Hope the boys continue with maths or something else interesting.
So, they are real😊
Even, 1 to 1 Even Johnson and Jackson as per simple gematria and be-ginning letter J is 1/1! WHAT!!!!!!!!!!!!!!!! Fluke? In 9 years of individual observation at least 9 out 10 times connection. Not my A The first one "A" was silent. AlefBet.
circular proof
no, it isn't
Are we all just pretending not to see what's going on here?
There are over 371 novel proofs to the Pythagorean theorem (and that's just the ones that are found to NOT be derivative of former submissions). My brother and I came up with several when we were in the 5th and 6th grade... but of course, none of them were new.
This particular proof IF NEW is a cool thing... and it SHOULD look good on a college application. Good for them.
BUT... this story has been BLOWN UP. Why? We all know why. I don't even have to say it.
But the media knew that these girls weren't the first to prove the Pythagorean Theorem. After all... it's a Theorem ffs.
But if it's not first, they have to blow it up some other way... so what do they say? They say this is the first TRIGONOMETRIC proof.
But what they fail to realize is that Trigonometric proofs are CATEGORICALLY disqualified (because all trigonometric functions are derivative of the Pythagorean Theorem. It IS circular reasoning, and IF this theorem gets adopted, it will be AFTER they strip the trigonometric terms from it.
This proof can be solved with only geometry and calculus. IN FACT, one COULD say that depending on how much these girls relied on the Law of Sines to solve this, they might not even have to understand their proof geometrically in order to solve it trigonometrically (meaning that this could be viewed as an incomplete proof)
You need nothing more than Euklidian geometry to calculate the legs, and you need calculus to solve the series. That's it. Any use of any trigonometric function is circular.
Nothing will EVER be proven by trigonometry. Sorry... but trigonometry is just a fairly brief list of triangle geometry properties that have been proven by euclidian geometry and a convention for signifying angle and length relationships. Trigonometry is useful, but it's not foundational. It will always be circular reasoning to use trig in proofs. Trig RELIES on the assumption of the pythagorean theorem.
Don't believe me? Try this.
*** a² + b² = c² (Where c2 is the hypotenuse)***
Now look at the relationship of sin cos and tan as it pertains to something called the "unit circle"
Look at all the relationships. Every point on this circle correlates to sin cos and tan in some way. What an amazing coincidence, right?
Do you know what the FORMULA for a unit circle is? It's not technically a function... it's not "f(x)=" so it's technically an equation.
***that equation for charting a unit circle... x² + y² = 1***
1= 1² btw
and 1 is the hypotenuse in a unit circle
This proof is a nice, elegant GEOMETRIC/CALCULUS proof. I'm sick of hearing that it's "The first trigonometric proof." That is nonsense. There's no such thing BY DEFINITION.
The coverage on this thing has been so inaccurate and sensationalized that I wouldn't be even a little surprised if they find this proof isn't even novel. Everyone just smiles and nods when there's a feel-good story in front of them.
Trigonometry can be demonstrated to be a consequence of calculus and differential equations, without the use of geometry. I made a video on that topic, if you want to check it out.
Sounds a lot like you are hating. This would have gotten coverage, regardless of the ethnicity of who came up with the theory. The only reason this is an issue for you, is because these ladies aren't white or Asian, and thus "didn't earn it" to you. Just because you don't understand it, doesn't mean that the logic is circular reasoning.
Um... correct me if I'm wrong...
But isn't an hypotenuse suppose to be a straight line?
If so how can a circle have an hypotenuse?
But if you mean triangles inside the "Unit circle" with one corner at the centre/origin and a second point on the circle itself then perhaps, just maybe *you should say that*
@@EarlJohn61
I REFERENCED the "Unit Circle". There's a reason they saw fit to NAME the "Unit Circle". It's so that if you say "Unit Circle" and someone who doesn't know what the "Unit Circle" is can look up the "Unit Circle" in Google and find out what the "Unit Circle" is. I'm not going to explain what the "Unit Circle" is in a youtube comment. If it went over your head, just look up the "Unit Circle". That is actually the purpose of giving the "Unit Circle" a name in the first place. So that anyone who's confused can learn about it under the name " Unit Circle".
Comprende?
Similarly, if you don't "Comprende" what "Comprende" means... don't come b!tching to me. Look it up. It's a word. It means something. If you don't know what it means and you can't be bothered with researching that which you don't know... that's on you.
Oh, and btw... you know that anyone who cares to check knows that you liked your own comment, right? LOL!
Remember when I capitalized "IF NEW" referring to this so-called "novel proof"?
Well guess what... it isn't even new.
It has come to my attention that this is nothing more than a redressing of John Arioni's proof "Pythagorean Theorem via Geometric Progression" using trigonometric terms (which on its own invalidates the proof).
So this proof is not only invalid (circular reasoning) because it used trigonometric terms
It's also invalid because it's extraneous (ie, not simplified) which is another reason we don't use trigonometric terms in pythagorean proofs. They are inherently not simplified.
And after all that... this proof isn't even original/novel.
Look for yourself. Look up "Pythagorean Theorem via Geometric Progression by John arioni" and you will find the very same proof complete with "waffle cone"... except it was completed using geometric identities (because DUH)... and it was done in 1996
NOW what do y'all have to say? You still simping for these 2 girls?
It's so sad that the media these days has absolutely no sense of reality or integrity. You can pretty well set your watch to the idea that nothing the media tells you is 100% accurate.
If ever the media encountered a VERIFIED MIRACLE, they would exaggerate it anyway.
Oohhh ... go home folks ... go home ...
Egyptians are the authors of it all.
Never cancel TikTok. I found out about this after a year from TikTok
Cancel TikTok. You can find out information from other social media.
Wow 😂
What?
WHAT?!
You raise the specter of circular reasoning between geometry and trig at :52-:56, but wait another 10 minutes or so to acknowledge that it may not be true about this proof. A proof that you have not seen. Also you make sure everyone knows right up front that the math community does actually have plenty of proofs of this theorem, in case the uninitiated might think that these 2 young people have done something math teachers and others have been unable to do for millennia. Allowing anyone to think THAT for 10 minutes would be just terrible, but leading your audience to believe that this proof, which you have not read, by these 2 young women is based on circular reasoning....well, it's OK to let them think THAT for 10 minutes. And maybe think it forever since many people don't watch RUclips videos to their conclusion. Their proof could be all trig by way of calculus or something else and not be based on geometry. If this is the case, the circular reasoning you talk about up front may not apply at all. Since you haven't seen it, you don't know, but somehow you feel free to cast aspersions on their achievement before having the read the proof. I'm thinking that these students and their teacher also know what circular reasoning is and understand the difference between geometry and trig, and so wouldn't come forward in this very public way if those things were part of this proof. I could be wrong, but I could be right. Which is why this video is so distasteful to me. It is all about you and not about whatever their achievement may have been. Why are you talking if you haven't actually read the thing you are talking about?
Their proof has not been published, but has been publicized. This video is here to clear up the details of their proof.
That picture is enough to figure out the proof. There is a concept in mathematics of a “proof by picture.” It was popularized back in the 80s or so, and that is what this becomes without the writing. It’s a game mathematicians play, where they craft a picture that can present a proof on its own.
What I am doing here is giving context. That’s my job as a professor. And then I’m talking about what they accomplished.
The question of circular reasoning can’t be determined until you’ve been through the proof and examined it, which is exactly what we did. The media brought up the issue of circular reasoning, and so I addressed it in this video.
This video is just me reporting on what they did. I just have more experience than a typical news person, so I can give a lot of details that they cannot. It’s not about me, it’s about exploring their reasoning based on what was available when I made the video.
Do you think news organizations are making it about themselves too?
I should say that the math you described was interesting and woke up the unused geometry and trig stored in my brain. (Much has rusted away.) I followed most of it, which was mostly due to how you explained it. So thank you for that. Yes, sometimes the news reports things about research (about any topic, doesn’t even have to be about things that confuse the average person) that are not true just because it’s more sensational or gets them more attention. That is them making it about them. The choices around how to construct a piece, how to deliver the content, are what I take issue with. You are not alone in making these types of choices. But I also am not alone in taking issue with it.
Really the Europeans got it from the Babylonians.
The Babylonians didn’t engage in mathematics in the same way that the Greeks did. There was a lot of intuition but not proof. Babylonians were obsessive data collectors, and had developed some nice results in their own way. I mentioned that they knew of the Pythagorean theorem in this video, but they also knew the quadratic equation.
I wouldn’t say “European” here as much as “Ancient Greeks.”
@@JoelRosenfeldAnd the Greeks got it from the Ancient Egyptians.😎👍
I’m looking for someone who cares about the harm that has been done to those two students and might continue to be done.
Please elaborate!
@@TheJackTheLion That was a year ago, and I should have said that I was looking for someone who *sees* the harm and cares about it.
I have no clue what this is all for. Why dose anyone care.
Why does anyone care about anything they didn’t do themselves? That is an interesting philosophical question. People are interested, because a couple of very young people found something new about a topic that everyone has familiarity with. That’s a natural thing that people are interested in. The real question is, why did you end up here on a video about a math topic, if YOU don’t care?
Your explanation is too long. Theirs was shorter.
We don’t actually know what their argument was yet, since they haven’t published it. As long as mine is correct, I’m ok with this.
@@JoelRosenfeld well we know how Europeans are good with intellectual colonization ( patents). Just little cautious.
@@ishmaelbenn4002 Ancient Greeks aren't European in the way we think of it today. They were part of a cluster of civilizations right next to Egypt and Babylon.
@@JoelRosenfeld cluster nah. Humm go on
@@ishmaelbenn4002 oh sorry, I thought we were in your other comment thread. Maybe I’m not understanding what you are getting at in this one.
I didn't realize that I needed a history lesson for the first half of your video but then your mansplaining was almost too much after you said in the beginning that their proofs were f-ed up. When you conclude that both of them did this proof, you're failing to acknowledge that one of them came up with a separate entirely different approach, and hers was much simpler. Your video comes across as jealous and frankly, your explanation was unnecessary. Now you're getting views on the backs of someone else's work...especially since these are young high school African American girls. Time for self- reflection....
The whole point to the video is to give contextual background to their accomplishment. I’m sorry that you felt I was mansplaining, but I’m a professor and my job is to make this sort of content digestible to the general public.
I’m proud of them for their accomplishments.
I never said their proofs were bad.
Maybe you feel the explanation was unnecessary, but others do not.
nothing special this method
One got a triangle, and the other one a circle, a right angle and circle doesn’t fit, I’m confused!!!😂😅🤣
Hidden Figures,..high school edition.📐🧐🤔👏👏