18 year old students just discovered a proof of Pythagoras that mathematicians said was impossible
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- Опубліковано 18 тра 2024
- This is a truly remarkable story. A textbook said that it was impossible to prove the Pythagorean Theorem from trigonometry alone. Two US students questioned this claim and developed a new trigonometric proof. Congratulations to Ne'Kiya Jackson and Calcea Johnson for developing this wonderful proof.
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2:00 Outline
3:26 Preliminaries
5:58 Proof
14:20 More Proofs
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![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](http://i.ytimg.com/vi/p6j2nZKwf20/mqdefault.jpg)
It took these 18 year olds 2000 years to solve this math problem. That’s what I call dedication
wut? Why 2000 years?
@@shadesmarerik4112It is quite an obvious joke, and like the first thing said in the video
@@sovietmogus3263 and no, its not nearly what was said in the video
@@shadesmarerik4112 copium just admit you missed the joke
@@shadowsavage3693 ow its kinda revealing, that u project onto me that this "joke" appeared intellectually challenging to u. i feel sorry for u
Very clever high school students. This would have made a great undergraduate senior project in mathematics.
I think this is already known
hoeever this isnt really trigonometric, but rather graphical. Its just that you call a/c=sin(α) but at that point this is just a name. So it has nothing to do at all with trigonometry itselve.
@@neutronenstern. I think the point is that if you draw triangles with the same angles, the lengths of the sides may vary but the ration between the side lengths stays the same. This is the whole basis behind trigonometry, and is why we can say that there is such a thing as sin(α).
It's a little like how Ohm's Law in physics actually says that in a conductor the current that flows is (all things being the same) proportional to the voltage. This means that you can define resistance as being this ratio, and can say that V = IR, which is what many people (including some electronics experts) think is Ohm's Law.
@@pauln7869 yes, but with sin(x) its special, i think. Cause in this proove no property of sin was used,without deriving it geometrically first. no actual value of sin was used.
So its kind of a trick to say, this is trigonometry. This proove works exactly the same, without needing the help of any sine.
"This would have made a great undergraduate senior project..." Why do you need it to be college students for it to be "great"? It is high school students, it is a great project, they deserve all credit accorded mathematicians of any age or school level, for having developed this proof.
Good for them. Even if this proof may not be commonly used in everyday-life solving the Pythagoras theorem, it is still a spectacular discovery.
Applications for their discovery may be discovered in the future.
This proof has many indirect applications and it reminds me of Archimedes' Method of Exhaustion.
Já tem inúmeras aplicações. Pq o método delas se relaciona diretamente com limites e cálculo infinitesimal que é o que se faz em matemática na prática. Adaptando esse método já deve ter surgido vários métodos baseados em pós-graduação associados a matemática computacional e muito mais.
has it been verified that no one else came up with this proof first?
These proofs will take years for them to be put into action.
Thanks so much for the shout out! You didn't have to be so forthright with your citation but you were and i greatly appreciate it.
You are the high school student ??
@@kiloperson5680 no, this channel was presh talwalker’s inspiration for the video, as seen at 2:05
The most fascinating thing that no one is talking about is that we human beings uses shared knowledges across time periods and always figure out a way to make things better, and discoveries aren't always made by people that society deems genius, but people that are curious enough to put in the time and effort.
But every person who society deems a genius has discovered something or made something new, because otherwise they wouldn't be considered a genius.
yeah bro. no one has ever talked about scaffolding.
And then published it in a closed form so it’s (again) not commonly available until someone reverse engineers it - at least without solving a riddle or poem in this case
@@-AxisA-That's just completely untrue.
@@jasonnelson9141 Name 1 or a few generally recognized geniuses then who didn't invent or discover something.
To be fair, trigonometry is based on the parallel postulate, which is equivalent to the Pythagoras theorem. The students theorem is based on the assumption the extended lines must intersect.
1:19
Elegantly stated. :)
True, and that is dependent on whether alpha and beta are equal. If you take the limit as they approach the same angle, you would get the answer for when those two are equal.
I might have missed something, but doesn’t using an infinite series avoid having to rely on Euclid’s fifth postulate?
@@frogmorely Nope.
The fifth postulate presupposes Euclidian (flat) space. Euclidian space may be more formally described as a space in which distances between points in the coordinate system are calculated using pythagoras. (There are other possibilities.)
Euclidian space has the feature where non-parallel lines intersect in exactly one point. This feature is required for the construction used.
IOW, Pythagoras is actually smuggled into the construction when the lines are extended to the intersection.
Nevertheless, still a clever way of tackling the problem.
These two are geniuses for figuring this out.
One thing that I was interested in hearing is your opinions on Animation vs. Math.
yesss
absolutely
why do you say they are geniuses? don't you think you couldve done the same thing??
@@leif1075 It's called a compliment
I called them geniuses because I would've never been able to guess that.
@@michaelgum97so if someone figures something out you couldn't then that means they are geniuses? I am not saying they may not be very smart or their accomplishment shouldn't be acknowledged.
I can see how all this mathematics is accessible to high schools (albeit advanced high schoolers), but it also took the creativity and grit of two young mathematicians to put it all together. Bravo!
Man, his voice by the end of the second proof sounds so happy and excited, it shows genuine love to mathematics and the topic in hand.
it's a computer voice.
@@sb3nder proof? It's better to voice the video than do an AI voice
I would assume this proof was overlooked until now because it relied on taking the sum of an inifinite series, which is a relatively modern concept. By the time that idea had been well established, the dogma that there were no trigonometric proofs was well established, so no one went looking until now. Bravo!
This.
Infinite series were around since the time of Euler and earlier, before the 19th century
How to reduce someone’s accomplishment on one hand while complimenting on the other. Well done
A trigonometric proof was published in 2009 by Jason Zima.. still cool cuz they're new proofs, good on them for sure
The sum of all angles in a triangle is 180 degrees. With the obtuse angle you will never have 180 degrees. And when the lines don't intersect and it looks like they don't intersect you don't have a closed triangle. The lines that don't inersect go into infinity. You have an infinite series which diverges the cone shaped pseudo-triangle because it is NOT a triangle. Thereore, even in the first part of placing the next trinagles can't work because the side U (adjecent to the right angle and hypotenuse go on forever not touching each other never forming the right triangle. Thus, any movement of any point on these lines changes as a variable but should be constant with well defined lengths of solid triangle. The only constant side of that elongated pseudo-tiangle is the side c. When you begin drawing triangles in the cone the more you move downwards the more you have to correct the angles of the added triangles to have right triangles and correct lenghts of these added triangles. The worst thing that they did in the video was that the physicically did not measure the sides of the added triangles and if you did that you would notice that the Pythagorean rule would not stand. a^2+b^2 would not eaqual to c^2 because the lines depicting the cone are further aparat from each other even right in the begining at points of angles of Alpha and Beta. Also, if you want to be very picky then calculate the area of the right triangle ( the formula of A=1/2(bh) and draw a triangle with two lines never intersecting and you will notice that (it is a bit strange to calculate area of a triangle that is not complete) the area of the triangle that is not closed is larger than the completely drawn right triangle or any triangle for that matter. So, even areas would not match. They did not prove the theorem at all. It is a bit of an illusion of what they are doing but in physical measurements the math would not even up.
I don't know... To me Pyth and Trig were always different formulations of the same qualities of triangles, expressed differently, but NOT from different principles. It always felt like they both started from the way that right triangles can be inscribed into circles. But I never devoted much thought into this. So it's interesting but ultimately I don't think a proof from Trig is independent at all.
Absolutely, no way are Pythagoras, Trigonometry separate. I don't remember ever being taught one without using the other.
You lost me but i trust you
It depends a lot on what we mean by *only* trigonometry. In particular, it means that there is something used in the common proofs of the theorem that has to be excluded. If not one could just apply a thin varnish of trigonometry on another proof.
An obvious candidate would be to exclude geometry (which would then exclude this new proof) but then how do we define the trigonometric functions?
We should do it axiomatically. If we give differential equations as axioms, it might be possible to prove the theorem using analysis but would not really fit the notion of what trigonometry *is*. If we want to use some of the common trigonometric formulas as axioms, we would need to not include sin²x+cos²x=1 (as that is essentially what we want to prove) but even from other formulas (like the formulas for sum of angles), we can quite trivially 'prove' the sum of squares (and so decide that we kind of put the result in the axioms).
Nevertheless it is an interesting proof with non-trivial trigonometry, geometry and limits in the long tradition of alternative proves of this most famous theorem. Congrats to them!
does einstein's proof on Wikipedia of pythagoras theorem not constitute as using trigonometry without circular logic?
the Taylor series was published in 1715, over 300 years ago
For some reason adding a character different from a space right before or right after your *bold text* breaks the formatting (my guess is that they didn't want people censoring words to have their 3 asterisks reduced to one bold one)
At least on mobile it does
the sine rule is based on right angle triangle drawn by taking one of the heights. so even though law of sine extends to any triangle the law inherently is based on basic definition of sin in relation to a right angled triangle. so I don't think the paragraph is wrong, these students found nothing new but just a longer proof of proving something which can be itself written as sin²a + cos²a=1
yes you're right, but all the idiots can't understand that. the greeks already discovered it 2000 years ago, and their whole point was to make it simple, not unnecessarily convoluted
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
I think the most beautiful proof of all is the following. You have a right triangle. Divide the triangle into two triangles by a line that is perpendicular to the hypotenuse and connects to the vertex of the right angle. Now you have three similar triangles: the original one and the two pieces. Because of the fact that the triangles are similar, each occupies the same fraction 'x' of the square on its hypothenuse. If the hypotenuses and surface areas of the three triangles are a, b, c and A, B, C, respectively, you therefore have A = x*a^2, B = x*b^2 and C = x*c^2. Because of the fact that A + B = C, then x*a^2 + x*b^2 = x*c^2 and therefore a^2 + b^2 = c^2.
just a fractal simple yet full of beauty i think about this when in traffic
"occupies the same fraction 'x' of the square on its hypotenuse" -- I think you mean "of the square _of_ its hypotenuse".
But beautiful proof! Thanks for posting this, very educational and much appreciated.
@@yurenchu You are right, that might have been a better wording. I originally found out about this proof when I was watching a video by James Grime that is called "A Pythagorean Theorem for Pentagons + Einstein's Proof." Definitely one on my favorite math videos on UA-cam so check that out if you're interested.
@@jimi02468 Thanks, I went to watch it. It's been quite some time since I've watched one of "Singing Banana" 's videos.
But to be honest, I found the video not so surprising anymore, after having read the proof that you posted here. :-)
And I was expecting to see some equivalent theorem that's valid for pentagons instead of for (right) triangles, but there wasn't such a thing.
:-(
But still interesting to learn how Einstein looked at it, so thanks anyway!
The most beautiful proofs of this are picture proofs with no words needed.
I'm an old PhD in mathematics & I never heard the assertion anywhere at any time that it was "impossible to prove the Pythagorean theorem using trigonometry alone". Lol! Perhaps you can send me a link to someone making that assertion? ;)
Elisha scott loomis did make that asseration in a book titled "The Pythagorean Proposition" however it was disproven by Jason Zimba way before the students did.and most likely someone probably disproved it before zimba
@@tank2256 It's not even that Elisha Scott was wrong, very often the chosen analytic _definitions_ of the trigonometric functions are based on the Pythagorean Theorem. If your trig functions are defined that way, you can't prove Pythagorean Theorem with trig. Elisha was probably referring to that.
Indeed that is crap. You have to tell what axioms you use before claiming such a thing. I don't know what "using trigonometry alone" means. The Pythagorean theorem is very elementary. Trigonometry is a very sophisticated tool. This amounts to boasting you can crack a nut with a hydrogen bomb.
@@tank22561914 J. Versluys as cited at the bottom of Zimba’s paper.
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) It's almost an exact copy of an existing proof. Look up John Arioni "Pythagorean Theorem via Geometric Progression"
I like the history you are providing in recent days. is this related to your earlier poll and will you discuss more mathematical history? It's definitely interesting to know, especially considering how despite different cultures and different times, mathematics connects us all. I remember seeing depictions of 'Pascal's Triangle' from a Chinese mathematician and Ancient Indian Mathematician. Even without knowing the language, I can understand it as Pascal's triangle.
The "gameplay vs lore" Meme absolutely sums up maths perfectly
While doing maths in general is.... "Fun", the history that maths itself has is insanely interesting and intricate. Throughout hundreds to thousands of years, humans have advanced in this language, the language of truth itself
My go at it;
The angle addition formula can be prooven without pythagoras theorem
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
let b = -a
cos(a-a) = cos(a)cos(-a) - sin(a)sin(-a)
= cos(a)cos(a) + sin(a)sin(a)
= cos²(a) + sin²(a)
We also have
cos(a-a) = cos(0) = 1
cos²(a) + sin²(a) = cos(a-a) = 1
q.e.d.
The identities cos(-x) = cos(x) and sin(-x) = -sin(x) can also be prooven with the angle addition formula.
Assuming cos and sin are well defined on the interval 0≤x≤0.5π, I am going to extended their definition to all real numbers, in a way that preserves the angle addition formula.
First, I evaluate sin and cos at π, 1.5π and 2π, by evaluating their angle addition formulas at 0.5π + 0.5π, π + 0.5π and π + π respectively.
Then I can proove that they have a period of 2π, by evaluating their angle addition formulas at x + 2π
cos(x + 2π) = cos(x)cos(2π) - sin(x)sin(2π) = cos(x)•1 - sin(x)•0 = cos(x)
Similar proof for sin(x+2π) = sin(x)
The identities cos(0.5π-x) = sin(x) and sin(0.5π-x) = cos(x) in the interval 0≤x≤0.5π can be directly observed in a right triangle.
Final step, evaluate cos and sin at -x, by evaluating them at 1.5π + (0.5π-x). This is the same as evaluating them at -x, due to the period of 2π.
cos(1.5π + (0.5π - x)) = cos(1.5π)cos(0.5π - x) - sin(1.5π)sin(0.5π - x) = 0•sin(x) - (-1)•cos(x) = cos(x)
Similarly sin(-x)=sin(2π-x)=sin(1.5π+(0.5π-x)) evaluates to -sin(x)
Insert pythagoras theorem proof here
Things like cos(π-x)=-cos(x) can all be prooven in a similar manner, extending the definition of cos and sin to all real numbers
In that last proof, after drawing in the rest of the chord that made up sin(theta), you can use the intersecting chord theorem to get sin^2+cos^2=1
The last proof involving just the circle is (at 16:21s) the best and elegant proof of fundamental trig identity w/o using "pythagoras theorem"
When it comes to lost and/or missing proofs, I want you to reflect on how much knowledge has been lost to time because our own destructive tendencies.
There are well over 371 Pythagorean Theorem proofs, originally collected and put into a book in 1927, which includes those by a 12-year-old Einstein (who uses the theorem two decades later for something about relatively), Leonardo da Vinci and President of the United States James A. Garfield.
Exactly. The premise of the video is bogus.
@@hoochygucci9432 Not to mention that there are countless proofs that also claimed to be fully trigonometric, which is supposedly the thing that makes this proof interesting. The most recent example was a proof by Zimba that was made in the early 2000s. So clearly the video is bogus, and whatever these students are being rewarded for is clearly not what the video claims, because we've already had fully trigonometric proofs of Pythagoras for a looooong time. And when I say fully trigonometric I really mean it. Like, really really mean it.
@@tiranito2834why is this your perspective on life. There’s no need to be so negative. It’s impressive that high schoolers whose highest math is probably basic calculus were able to come up with a new and interesting proof for this theorem. Also the video isn’t the one making a big deal about it it’s just explaining the proof. Why can’t you just appreciate math, why does it have to be a contest that requires putting passionate people down.
@@joshuafrank1246 The problem is that they didn't make a new proof, look it up, they copied parts of multiple proofs, most of it straight out of Zimba's proof.
It is precisely because I appreciate math that this situation infuriates me. Because the fact that this is being treated as news is nothing but proof of lost knowledge. Things that used to be common knowledge just about a decade ago are now cutting edge and people act as if they had never heard of fully trigonometric proofs for pythagoras before.
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
Media blowing up with nonsense: Zimba already did the "impossible" (itself a questionable statement by Elisha Loomis) nearly 15 years ago.
Honestly, I still don't really get what is so special about this proof. Of course, it's a very unique proof and those teenagers are apparantly brilliant. But the 'never done before' part of it is not entirely clear to me. A statement in a book that some kind of proof is impossible is not the same as a real mathematical conjecture.
So I can't really tell if the media is blowing things up, but I'm suspicious.
The media but also other UA-camrs are clueless on this.
The majority of this "proof" is showing the trigonometry identity:
sin(2 alpha) = 2*tan(alpha) / ( 1 + tan(alpha)^2 )
But this is already a known fact and has nothing to do with proving the Pythagorean theorem.
Replacing tan(alpha) with a/b gets you pretty close to the end result.
Nevertheless, these students discovered this by themselves and that is quite an achievement.
its because its a bunch of kids. Why can't you just let them feel like they did something cool? Imagine if you were a highschooler who discovered something and online everyone is just saying how you are a fraud and your work isn't important.
You all are obviously jealous because instead of being positive you are going out of your way to find something negative to say. Its so sad.
@@Brad-qw1te I agree that it's cool, but I don't see how it merits media coverage to this extent.
@@Brad-qw1telol don't waste your energy on toxic people they are always there in our lives . I mean if the scientific community acknowledges that this is something to celebrate then some nonentities comes and dispute that what can you say? You should just let them swallow their toxicity alone
the need for a < b was for the similarity between the triangles ?
It's possible to prove the law of cosines without using the Pythagorean theorem. The proof can be found on Wikipedia for example. Now using the law of cosines on a right triangle just gives the Pythagorean theorem as a special case.
Well... Actually this proof uses not only trigonometry. It uses also limits and similarity. And you can easily prove the theorem using similarity only. :-)
Premislao, ok then provide us with the actual alternate solution.
@@PHlophe there was a proof published in 2009 using trig (Jason Zima) look it up
Okay, that was really impressive!👍🏻
Put a small square inside a larger square with their centers coinciding. Rotate the smaller square until its 4 corners touch the sides of the large square. The large square will be comprised of 4 identical right triangles and the small square. Write an equation stating that the area of the large square is equal to the area of the small square + the area of the 4 identical triangles. Rearrange the terms in the equation and voila!
Of course it is required that the area of the smaller square is at least half of the area of the larger square (otherwise the corners of the smaller square would never touch the sides of the larger square, and we wouldn't have any right triangles in the diagram). But yes, this is an excellent proof!
If the right triangles have right sides a and b (with a ≤ b) and hypotenuse c, then the larger square has side (a+b) while the smaller square has side c. The area of each triangle equals ½ab . Since the area of the larger square equals the area of the smaller square + the area of the four triangles, the equation is (a+b)² = c² + 4*(½ab) . After expanding, we get a² + b² + 2ab = c² + 2ab , and hence a² + b² = c² .
Additionally, to physically demonstrate the result, the four right triangles can also be paired to each other so that they form two separate, identical rectangles (with their diagonal equal to the triangle's hypotenuse), and these two rectangles can then be placed back inside the (emptied) large square: one rectangle horizontally in the bottom left corner, and the other rectangle vertically into the top right corner. The area of the large square that is not covered by the two rectangles consists of exactly one square in the top left corner with side b and area b² , and one square in the bottom right corner with side a and area a² ; and of course, it follows that these two squares together have the same area as the rotated smaller square (with side c and area c² ) that we had at the beginning.
@@yurenchu Yes, U got it. I never calc'd exactly the minimum size of the small square such that its corners eventually contact the sides of the big square when rotated but your 1st statement does precisely define it. The length of the diagonal from the small square's center to its corners must be greater than half the length of the big square's side, which works out to what you said.
The sum of all angles in a triangle is 180 degrees. With the obtuse angle you will never have 180 degrees. And when the lines don't intersect and it looks like they don't intersect you don't have a closed triangle. The lines that don't inersect go into infinity. You have an infinite series which diverges the cone shaped pseudo-triangle because it is NOT a triangle. Thereore, even in the first part of placing the next trinagles can't work because the side U (adjecent to the right angle and hypotenuse go on forever not touching each other never forming the right triangle. Thus, any movement of any point on these lines changes as a variable but should be constant with well defined lengths of solid triangle. The only constant side of that elongated pseudo-tiangle is the side c. When you begin drawing triangles in the cone the more you move downwards the more you have to correct the angles of the added triangles to have right triangles and correct lenghts of these added triangles. The worst thing that they did in the video was that the physicically did not measure the sides of the added triangles and if you did that you would notice that the Pythagorean rule would not stand. a^2+b^2 would not eaqual to c^2 because the lines depicting the cone are further aparat from each other even right in the begining at points of angles of Alpha and Beta. Also, if you want to be very picky then calculate the area of the right triangle ( the formula of A=1/2(bh) and draw a triangle with two lines never intersecting and you will notice that (it is a bit strange to calculate area of a triangle that is not complete) the area of the triangle that is not closed is larger than the completely drawn right triangle or any triangle for that matter. So, even areas would not match. They did not prove the theorem at all. It is a bit of an illusion of what they are doing but in physical measurements the math would not even up.
1:27
I wonder, why they came up with this rather strange statement. The sine theorems in (right) triangles can easily be deducted purely from the definition of the sine, just like it's done in the video. Also it seems quite obvious that sin alpha = a/c will hold if and only if gamma=90° .
So anyway I just googled this book... it's from 1940. Not that people couldn't have known better back then, but it is quite imaginable that nowadays such books would be checked much more thouroughly by many more people. So that might explain it in conjection with someone being a bit too excited about the topic.
The funny thing is that neither of them wanted to pursue mathematics in college but instead, if my memory is correct, environmental engineering and biochemistry. Of course both are math heavy but only math. Also, I believe it originated out of school competition and not just a paragraph in a textbook.
Burnham there are lots of hateful phd and math grads in the comments. 2 high school teens born in in the late 2000s just crushed them .
There is another trigonometric proof I have been taught at school. The only thing you need is to come up with a shape of an outer square with the side a+b, inner square with side c and the rest of space filled with 4 equal right triangles rotated 4 ways (0, 90, 180, 270 degrees). Essentially, you put those in the corners of a big square. Then you write the area of the big square two ways and voila - very easy;)
That’s innovative. Never would have thought of this. Very nice
By the way, the last proof that you showed is also one of the fundamental equalities in basic trigonometry, taken nearly straight from the definition of cosinus.
Can someone explain why exactly the other proofs told at the end of the video are not considered trigonometric proofs for the Pythagoras theorem, and why does this new proof special?
I also want to know
More views
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
It's called a geometric series and scaling.
It's infinitely repetitive.
It says a^2 + b^2 = C^2 because a^2 + b^2 = c^2 over and over and over again for infinity.
It's was also already done in a previous proof:
John Arioni "Pythagorean Theorem via Geometric Progression"
Even with everything it's really really hard to follow but hell I had a hell of a time with plain geometry so this was practically impossible for me to follow
There's an infinite limit, so that's a calculus proof disguised as a trigonometry proof.
So just because it has a specific field of math, it overtakes the proof that was also using another field of it, even if the latter field is more prominent in the proof?
The award of the dumbest thing I've seen today goes to you, because on top of being wrong about one field being more important than the other, you didn't talk about the starting point: finding a proof that involves trigonometry without using the Pythagorean theorem.
@@clemente3966 No, he's correct to point this out given the context of the "discovery" of a proof that is supposed to rely on "trigonometry alone". BTW we covered the second proof (which is by trigonometry alone) in maths class in the UK back in the 70s, so it seems the heart of the "story" here is really that the math textbook they use is wrong!
@@georgecurrie4808
Wait, when was it stated that the proof needed to be based on trigonometry *alone* ?
Because it was clear that the proof was made to contradict a paragraph which stated the trigonometry couldn't prove the theorem without the theorem. Nothing in there stated that trigonometry was to be used completely alone aside from the theorem.
As for the second proof, tbh, I have yet to see that part because I got interrupted when I watched the video and had to close it and forgot about it.
@clemente3966
It stated "there is no trigonometric proofs...". If it contains infinite series, the whole point of it change to calculus proof. Period. No matter how you'd try to explain or "hide" it, this series is still there, therefore it's based on calculus and not trigonometry. You can try to hide tofu as a chicken by many more or less effective methods, but that doesn't change the fact that it's still tofu
The award of the dumbest thing anybody will see today goes to@@clemente3966, because on top of using a strawman argument and getting the strawman argument wrong, he doesn't know just how many proofs of the Pythagorean theorem exist that use the Taylor series for the trigonometric functions sin(x) and cos(x).
Congrats to these high students. I would not say they used trigonometry to prove the theorem. Instead, they used calculus plus a clever geometry construction to prove it, which is a great effort. BTW, I never heard such comments from mathematicians that a particular tool is IMPOSSIBLE to solve a specific problem.
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
Right on. Thanks for sharing.
For other non-European mathematics, read "The Crest of the Peacock" by George Joseph.
I love the new proof using infinite series …
I think i recall learning the first alternate/other proof in high school, - a new proof - genius! these students are legends in high school. Thanks everyone for shining light on this!
The thing that is still missing is that it is BY DEFINITION what sin and cos mean, and they are based on right triangles. There is nothing new in the “proof” here, except to take as defined one thing vs another, neither of which are independent (of course not, otherwise Pyth theorem would not be true).
Sin and cos can be defined outside of triangles using Taylor series
1) It's not the first trigonometric proof of the Pythagorean Theorem.
2) It's not pure trigonometry.
3) There is an exception: It doesn't work for isosceles triangles.
I don't know if I missed it in the video; but I was a tad unconvinced, at first, that the similarity factor of all of the waffle-triangles would be the same. They ARE definitely similar, but I think it just needed a short explanation as to how you can tell the similarity factor for all of the triangles would be the same.
Your old side "a" becomes your new side "b" (ie, your short side length from the larger triangle is now your shorter side length of the smaller triangle). Based on the similar triangles, the ratio of the short side to the long side is a/b. That's basically it. Every time you want to calculate the next unknown perpendicular length, you would use this ratio (its also the tangent ratio!), so your similarity scaling factor is a/b.
After typing this out, I think this was touched on, but I hope this extra explanation is helpful for anyone who might have been initially unconvinced.
"With the few screenshots we got, we can predict what they did to do this"
Proceeds to give a detailed explanation for how they did it
Whoa.. that is quite a work! Thank you for making the video so we could understand it easier👍
Perhaps even more amazing than these students coming up with this never-before-discovered proof, is that the proof they came up with is relatively understandable by lay people. Great job!!
Using trig on a triangle with side a, b, c and angle A.
a = c*cosA. b = c*sinA, a^2 + b^2 = c^2*(cosA)^2 + c^2*(sinA)^2 = c^2*((cosA)^2 + (sinA)^2) = c^2. I used only trignometry.
You was using (cosA)^2 + (sinA)^2 = 1, which is one result derived from Pythagorean Theorem. It is mentioned at 1:17.
@@tranhadminthich That can be derived from the expressions for sin and cos in terms of e^iA and e^-iA. What are they if not trig?
At 12:18 you have expressions for all three sides of the big triangle, which is a right triangle. You can show that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. That proves the pythagorean theorem without needing to go back to the original triangle.
I had thought the same
he needs to prove that a^2 + b^2 = c^2, just because the sum of the square of the two shorter sides is equal to the hypotenuse, that doesn't prove a^2 + b^2 = c^2. that would just prove the case for the super large triangle, which could just be a coincidence.
@@rohangeorge712no it wouldn’t because all the triangles are similar are form a right triangle cause of corresponding angles. It wouldn’t be a “coincidence”
@@prod_EYES yea nvm ur right mb
@@prod_EYESTechnically yeah, it would prove the Pythagoras Theorem but I guess they did that because the main objective was to prove specifically 'a²+b²=c²'.
7:46 If alpha + beta is 90 degrees then you're assuming the euclidian parallel postulate, which is equivalent to the pythagorean theorem. Seems to me that the students are implicitly assuming that pythagoras is true in this proof, but I might be wrong.
well he just assming that all angles in a right triangle add up to 180 degrees right? if that implies pythagorean theorem, then idk
The Pythagorean Theorem are derived from Euclid's Postulantes, not the inverse.
This strikes me as more of a geometric proof than a trigonometric one. Most of the steps in Case 2 are using trig functions to add extra steps to simply taking ratios of similar triangles. Nonetheless, it is a very nice and creative proof.
Just draw in any right angled triangle the vertical to the hypotenuse c going through the opposite corner. Name the 2 segments a and b. With similarity follows a^2 + b^2 = c^2
Very interesting!
- You can use same approach in generating triangles right inside base one triangle, but in innerside direction, by making first triangle by line going from 90⁰ to hypotenuse. So it will bring same area you have already. For both of starting triangles.
Another question - is how to prove that square area is a². And proving anything you use something you need to prove, always. It leads you to axioms.
- It can be stop, if it has no changeable base.
I saw this proof when it first came out last April. A very elegant proof and good on the students.
I'm confused... you say the students found the impossible: A trigonometric proof of the Pythagorean Theorem. Then you proceed to show it, and it uses infinite iterations of smaller triangles. That's fine. But then in the "More Proofs" section of this video you show a different, much simpler trigonometric proof. Wasn't that simpler proof already known?
you are right. People like to think about these things as if they're mystical so they fall into the trap of fruitless erroneous imagination (and also fall under the illusion that they are smart and very special)
The infinite geometric series part is a pre-existing proof.
They just slapped a couple of triangles on top of it and it just happened to work.
Makes me wonder if you could do the same with other known proofs.
The simplest is to take the limit as spherical trigonometry approaches plane trignometry. Use the formula cos A = cos B cos C + sin B sin C cos(a - b). A B and C are the sides and a, b, c the vertex angles of a spherical triangle - all dimensionless. In the limit A, B, and C are very small and cos A -> 1 - 1/2 A^2 and sin A -> A to highest order. Let the angle a-b be pi/2, then we get (1 - 1/2 A^2) = (1 - 1/2 B^2) (1 - 1/2 C^2) - working out to highest order, A^2 = B^2 + C^2. This proof does not presuppose the theorem in any way, just the power series expansions for sin and cos and the angular geometry of a sphere. It works in any number of dimensions as well.
No mathematician ever said it was 'impossible'. Nor has any professional mathematician been trying to do a trig proof in the past 2000 years. There is so much nonsense in this story that it makes my brain hurt.
É incrível como na matematica existem coisas que ainda não descobrimos, e ainda não provamos
A matemática sempre me fascina pois ela é um alfabeto completo que podemos descobrir coisas com puro raciocínio e apenas lendo de forma alfabética, é legal resolver problemas nessa elegância de ler matematicamente, e por isso apoio totalmente olimpíadas de matemática, são incríveis
A matemática é cheia de incompletudes e apesar de parecer estressante, só me deixa mais fascinado com essa idéia
A matemática tem muita coisa ainda a ser tratada. Mesmo que demore 300 anos pra provar algum problema proposto por algum matematico maldoso. Kkkkk quem sabe mexer de verdade com a matemática, pode usufruir de tudo dela e dos problemas, mas também pode ser um cara chato que aterroriza todos do ensino médio com sua fórmula que ele cria para provar alguma coisa bem doida. Amo
Reflective! Now to return to the first known constants, 1,2. 1/2. .5. And take a reflective look without limitation.
This proof is awesome. My biggest question is how did these two students specifically collaborate on such a unique proof?
And also, why are they unable to explain it..?
@@CrowsDoMath Are they unable to explain it?
@@CrowsDoMath They can explain it.
I think it was part of a competition
I was taught in grad school the Greeks did not have algebra. The Greeks had a straight edge and a compass, so using trigonometry with algebra seems slightly anachronistic. I would be really blown away to see the proof using just a straight edge and a compass!
On the other hand, my hat goes off to the two high school students. Great work.
Uncle Tony, we should praise the ladies for doing what 99% truly can't . but it is sad that older dudes in the comments are salty because 2 girls not only came up with something new and they did so with 2 separate solutions of their own.
15:17 in our maths textbook similar proof is given only difference is instead of sin it relies on rules of similar triangles
You're quite right. The use of trigonometry in this proof is quite unnecessary. The use of sines is simply a different way of expressing the similar triangles and adds nothing useful to the proof.
i've could have derived that, but I screwed up my knee
Did you take an arrow to the knee
I am surprised no one has thought of this before for 2000 years. It was so satisfying when I realized how they were going to attack the proof. Good job!
I derived this myself 6 or 7 years ago, but I assumed it was already known.
@richardcheney6964 sure you did. Did u also discover that E=mc^2??
Well, the thing is that there are other proofs that use the same law of cosine, are much simpler and do not rely on limits. Kudos to the students for their creativity, but they have just reinvented a crooked wheel.
@@kyriakosphilitas6950 Jason Zimba did 14 years ago though, and you don't even know who he is. Why did the the news of the first to discover it not make it to you but this news did? Most likely because they are black, female, high school students. If they read this paper he wrote beforehand it is less impressive.
forumgeom.fau.edu/FG2009volume9/FG200925.pdf
@@kyriakosphilitas6950 He most likely did considering how the first officially known and approved modern FULLY TRIGONOMETRIC proof of Pythagoras was published in the early 2000s by Jason Zimba so... it is clear that these student's didn't really come up with anything new. As a matter of fact, Zimba's proof was taught in highschool to us and is written in my book, which I still have in my posession to this day.
from reddit: "Their proof itself may be new insofar as involving the law of sines in a new way, but would not be the "first impossible proof" or "first ever non-circular proof". That feat goes to Jason Zimba. I'm sure many folks here can construct their favorite sum or product and squeeze it to a convenient value in the limit like in the second link above to obtain a "new non-circular proof". There is a reason this theorem has seen hundreds of different proofs.
I'm not going to comment further on the claims made or sudden acclaim and media attention - which I don't recall Zimba getting in 2009 - other than to say it's disappointing that no paper or even slides were published. Not doing that allows for all sort of funky stuff to be done post-fact, e.g. adding new references, perhaps Zimba's paper above that the girls might or might not have been aware of - and we'll never know"
I love when a proof that seems to be spiraling away into unintelligible complexity suddenly is resolved into simplicity itself, seemingly out of the blue.
Bravo!
Muito bom! Parabéns e obrigado!
I think you are implicitly assuming the parallel postulate here. The sum angles in a triangle don’t have to be 180 degrees, but if they are, then you basically assuming thing that make the Pythagorean theorem true
What about it
The parallel postule and the Pythagorean theorem are equivalent
So all proofs must assume or imply something equivalent to the postule, else you have a proof that's independant of to it
Could this progression be roughly labelled as a Negative Fractal but with corners.
This proof is implicitly using the flatness of euclidean geometry in that current form of the law of sines (there is a modified form in the other geometries), and is using this flatness in saying sin(alpha)=a/c etc. since this is directly derived from the law of sines. In general, this just simply is not the case. Therefor, this proof is using implicitly the assumption of a flat euclidean geometry, which is the geometry which assumes the parallel postulate, and the parallel postulate is equivalent to the pythagorean theorem. Therefor, the proof is starting with the assumption that the pythagorean theorem is true, albeit subtly and implicitly. It is a smart proof, but it is circular.
Jackson just copied a proof by B. F. Yanney and J. A. Calderhead.
Johnson just copied a proof from a calculus textbook.
Did you know that the Pythagorean Theorem or the Equation to the Unit Circle is embedded within the simplest of all arithmetic equations: 1+1=2? And with that, so are all the definitions, identities and properties of Trigonometry. If you can construct a Unit Circle you already have the Pythagorean Theorem and from that Unit Circle you can construct a Right Triangle and this is why the Trigonometric Functions have a Pythagorean Identity. The equation 1+1 = 2 is the unit circle located with its center (h,k) located at (1,0). Each unit value of 1 is a radii and their sum of 2 is the unit circle's diameter.
In order to understand this proof, you can not think of 1 and 2 as being a scalar value. You have to think of them as being unit vectors with the operation of addition (+) as being a linear transformation of those unit vectors. When you look at them within this context, then it becomes clear as to why the cosine also has a direct relationship to the dot product.
I could even go one step deeper than this. All of these properties are actually embedded not just within this arithmetic or algebraic expression but they are also embedded within the identity properties of both addition and multiplication a+0 = a, a*1 = a, as well as in the equality or assignment operator as in a = a. If you don't think so, then consider the following equation: y = x. This is a specific case of the slope-intercept form of y = mx+b where b is the y-intercept and m is the slope of the line defined as rise over run which can be solved by m = (y2-y1)/(x2-x1) where (x1,y1) and (x2,y2) are any two points on that line. This can be simplified to dy/dx. With the case of y = x, the intercept is 0 as this line crosses through the origin (0,0) and the slope here is 1. The line y = x bisects the first and third quadrants at 45 and 225 degrees or PI/4 and 5*PI/4 radians. At 45 degrees or PI/4 radians, both the sine and cosine of those functions are equal, sqrt(2)/2 . This is also tan(45) or tan(PI/4). Within the linear equation in the slope-intercept form we can substitute m = dy/dx with sin(t)/cos(t) = tan(t) where t, theta is the angle between the line y=mx+b and the +x-axis. And this can be derived or constructed simply from the line y = x which is equivalent to x = x which is equivalent to x+0 = x and/or x*1 = x.
x | y
====
.. | ...
-3 | -3
-2 | -2
-1 | -1
0 | 0
1 | 1
2 | 2
3 | 3
... | ...
And this is just an simple expression or equation which doesn't even involve function composition. The reason this works is because the Pythagorean Theorem A^2 + B^2 = C^2 for all tense and purposes is equivalent to or a simplified form the equation to a circle (x-h)^2 + (y-k)^2 = r^2 where (x,y) is any point on the circumference (h,k) is the center of the circle and r is it's radius. So going back to the expression 1+1 = 2 (full circle) the tail of the first vector of (1,0) starts at (0,0) and it's head ends at (1,0). The addition operator (+) translates this vector in the same direction (+) by convention to the right by its magnitude of (1) since the second operand of the equation is also a unit vector. The result of this linear transformation (translation) now has the 2nd vector starting at (1,0) and its head is at (2,0). The center of the unit circle is (1,0) and both points (0,0) and (2,0) are on the circumference of the circle. We can substitute these in to the equation of the circle (x-h)^2 + (y-k)^2 = r^2 == (2-1)^2 + (0-0)^2 = 1^2 which == 1^2 = 1^2 = 1 which is the radius of the unit circle and is also the unit vector. The diameter of the unit circle is 2r which is 2*1.
Therefore when you see any term in any mathematics of 2x or 2n, you actually have a solution to a given circle where x or n is its radius.This is also why the distance formula between any two points is directly related to the equations of a line as well as the Pythagorean Theorem. You don't need a "Triangle" to have a "Pythagorean Theorem". You only need a Line Segment. With that, everything within mathematics is related from basic arithmetics to algebra to geometry to trigonometry to linear algebra to calculus, and so on...
These are all possible simply just because we can Count or Enumerate. This is why I appreciate numbers and mathematics.
that's great.
I can dig it. You put a wonderful deal of effort into this reply and the relations are mutual. Excellent work!
But.
Trigonometry: tri. You need more than one line segment. In the circle, the perimeter is the so-called "continuous line", drawing not a boundary, but a measurable distance away from some point, for all points swept out by the so-called "full rotation" about the point... another line, here, delineates the distance r to some point p.
I'm an engineering student and I don't know who said no proof existed before. I've proven it with a basic method that takes 2 minutes to explain with 1 square
Is that andrew taint as your pfp?? ewwww
@@endxofxeternity you know it looks good
I think PT is only needed to prove sin²+cos² if they're defined geometrically. It's easy to prove it if they're defined as sin x = (e**ix - e**-ix)/2i, cos x = (e**ix + e**-ix)/2. Just substitute and simplify:
((e**ix - e**-ix)/2i)**2 + ((e**ix + e**-ix)/2)**2 = (e**ix - e**-ix)**2/-4 + (e**ix + e**-ix)**2/4 = ((e**ix + e**-ix)**2 - (e**ix - e**-ix)**2)/4 = (e**2x*2*e**-2x*2)/4 = e**0 = 1
Great work by the students! The part of the proof that involves infinite serries can actually be proved by trigonometry. Consider the triangle made of all of the infinitely many triangles except the first two and apply Sine Law on this triangle. You will be able to find the unknown sides of the big right triangle (the waffle cone) in terms of a,b,c. The rest is the same.
15:06 This is such an elegant, easy to understand proof, I love it.
I once sat a test and I wrote
"therefore by Baudhayan a^2 + b^2 = c^2"
and I got 0/10
The teacher commented,
"It's the Pythagoras theorem, what is wrong with you?"
I don't believe you that the teacher subtracted 10 out of 10 points just because of a name. Maybe 1 point.
@@efi3825 its a joke lmao
Two Extended lines are 10:49 intersect at infinity needs to be proved.
It cannot be proven, it is an assumption of the geometry that you are working in. If you don't assume it, then you might be in spherical geometry where Pythagoras' theorem is not true. The fundamental axiom that makes planar geometry planar is that non-parallel straight lines meet when extended infinitely.
I really appreciated this presentation, thank you.
...................I'm gonna go watch Animation vs Math
@MindYourDecisions I appreciate that you try to be historically sensitive. I really do. But there is a lot of value in naming conventions. If a large portion of people know the thing under the same name, the Pythagorean theorem, then that's an achievement all on its own. And it's not worth giving that up for the chase of "Who did it first".
This makes me want to revisit the idea I had about measuring the one way speed of light
I really wish math instructors would focus more on the details. I get that this is youtube, and there is a certain standard for the length of videos, but I would have loved to learn all the reasoning for example visually of all the connecting ideas within the equations. (and can literally be a second-long, you can throw a picture in with text that explains how you were able to move forward). I loved that I remembered how to simplify complex formulae, but even for youth, I can see it being extremely useful for reviewing material. Also, how much do I bet that length and rate of 'pause' in a video is the single most defining factor in how interested people are in your videos.
omg just start your own youtube channel already
Bravo, students. Brilliant!!!
they are only tricking us into thinking, its a trigonometric proove. However its a graphical one, cause it only gives a name to a/c and call it sin(α). But at that point its just a name,and has nothing to do with trigonometry itselve.
Presh said “Pythagorus”! I’ve been waiting years for this day.
as
Here's another way to prove it with trigonometry:
d/dx (cosx) = -sinx and d/dx (sinx) = cosx is proven without pythagoras.
We say that:
(sinx)^2 + (cosx)^2 = f(x), for some function f
DIFFERENTIATE BY X TO GET:
2sinx(cosx) + 2cosx(-sinx) = f'(x)
0 = f'(x) --> f(x) = C, where C is a constant
(sin0)^2 + (cos0)^2 = 1 --> C = 1
--> (sinx)^2 + (cosx)^2 = 1
This is an excellent proof and fantastic accomplishment for these young students. Their textbook was wrong, there is at least one other proof based on the law of sines and law of cosines.
What makes me fascinated in mathematics is that even an 18 year old can make a groundbreaking discovery.
that’s why no Nobel prize for math 🙂 jk
This is not groundbreaking in math. It is just a nice geometry problem plus a little bit of calculus. Groundbreaking works in mathematics generally shed light on a subject in a way that has never been considered before. This does not really come close to any revelation about Pythagoras' theorem, it is just another proof using limits, in fact you never have to mention sin and the proof stays the same. Groundbreaking work is stuff like the introduction of perfectoid spaces, or the development of the Lebesgue measure, central models, etc etc. These are brand new ideas that fundamentally change their subject.
New Calculus of John Gabriel is groundbreaking
truly amazing!
is there any significance in that the final smallest triangle is the same as the entirety of the waffle sequence excluding the intitial mirrored right triangles as well as those being the only two triangles that are not right triangles?
Your proof at 15:10 is the same as mine when I was a secondary student.
It is not an impossible discovery. All we have is another proof of the Theorem, which is nice for the students, but other than that, not a remarkable discovery that goes into history books. The reason why this is "hyped up" is because the Pythagorean Theorem relates to a lot of people through their high school math. So easier accessible for general audience. The proof of Fermats theorem by Wiles was far more ground breaking than anything else but then...who knows Fermat anyway?
The true master equation of planar geometry is Norman J. Wildberger's Cross Law, which is the cosine law squared (=1-sin²), out of which Pythagora's theorem is a limit case when the spread ("sine squared") between two sides of a triangle is 1. The other limit case is when a spread of a (degenerate) triangle is 0, hence about 3 collinear points, as strong and as fundamental as Pythagora's theorem. This video is much wind for nothing, NJWildberger cracked geometry properly, without the sine qua none need for infinite processes 😊 Nevertheless, I enjoy the channel and content very much!! Keep up the good work 😁🙏🙋
at 2:36 this formula is given by trigonometry or another method ?
The part that I don’t understand is how does the geometric series count under trigonometry? I would’ve considered that to be a more algebra or calculus concept
These 18 year olds took 2000 years to solve this question. That’s some mad dedication
They used trigonometry to prove that x = 2ac/b (see ua-cam.com/video/juFdo2bijic/v-deo.html) , but they could have proven it by the proportionality of sides in similar triangles (in fact, that's what Pythagoras used). In other words, it is really just one more proof of the theorem. Anyway, it is great for high school students.
It is, first, a circular reasoning test... pun subtended.
Using the definition of sin to prove sin is literally circular reasoning. This is a big gut check: drop nuts and call a spade a spade.
The first example of other proof was what we always had to do in math class.
This is really cool. I love the wor of those students! And I wish I had their genius by myself...
I thought this must have been done long time ago… It’s so simple that ordinary high school students should be able to understand it!!
They aren't the first to prove it this way. The video is misinformation
Everything is obvious after you know about it
I would guess because there wasn’t really any knowledge about infinite geomteric series back then, or infinite series in general. Those were concepts that were only really developed much later.
That would me my guess.
@@Ninja20704 But I’m referring to the hundreds of proofs published in the 20th and 21st centuries. And they come in all flavors.
@@wesleysuen4140 ok sorry for my misunderstanding
Trying to cancel Pythagoras for 18 minutes straight