211–220 of 287 posts
Cool proof, though it doesn't consider the case where a=b. If so, the geometric series is non-converging since the ratio isn't less than 1. Geometrically, the construction wouldn't work because the sides A and C of the large "triangle" would be parallel to each other.
When a = b then the proof of the Pythagorean theorem is trivial so this is the sort of nit-picking that actual mathematicians don't care about.
Earlier quoted context omitted.
It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…
> It's a very clever proof. Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.
'Aesthetic' appearance in math is important in helping drive mathematical innovation and help new human beings derive pleasure from that wonderful field.
Programming is kind of an applied mathematics where efficiency does matter because it's a tool, a means to an end.
Not to say that people can't find aesthetics in programming, nor that they shouldn't, rather in math at least the pleasure of discovering a new way of doing/proving something is the end in of itself.
It's pleasure for me to see another way to do or prove something; I can only imagine the feelings this teenager got from actually making a discovery.
Earlier quoted context omitted.
It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…
> It's a very clever proof. Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.
but related, i wonder what a mathematics demo party would look like.
Earlier quoted context omitted.
It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…
> It's a very clever proof. Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.
Earlier quoted context omitted.
When a = b then the proof of the Pythagorean theorem is trivial so this is the sort of nit-picking that actual mathematicians don't care about.
How is it trivial?
Okay so take the triangle made by taking the diagonal of the unit square. This has side lengths 1, 1, and c and has area 1/2.
Now, take four of these and arrange them in a square with the side length being c. It would be easier to draw this... basically you stick the right angles in the center. If this isn't clear I can draw a diagram.
Anyway, you just made a square with side length c but since its made of four of those original triangles we know that the area of it is 4 * (1/2) = c^2 so c^2 = 2.
EDIT: made an excalidraw to explain this construction - maybe helpful https://excalidraw.com/#room=2298a8fd232d5f58e8ca,HmUwSqOt6J...
Earlier quoted context omitted.
It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…
> It's a very clever proof. Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.
while programming and math overlaps in so many places, i think mathemathicians are quite a separate species and KISS is often not on the menu.
a fun, non math heavy book with a window into this i enjoyed is https://en.wikipedia.org/wiki/Uncle_Petros_and_Goldbach%27s_...
"...a few trigonometric proofs of Pythagoras have made the rounds since then. Claims in the media that Johnson and Jackson’s proof is the first trigonometric proof of Pythagoras are overblown..." I don't like how the author drops the bomb "this has been done before" and tries to compensate that with the "simple and lively" language of the proof, or the background of the girls. Why didn't author reference any of the p…
Doesn't seem like a serious article about math so much as a feel-good story.
I mean, Wikipedia shows a better proof in [this section titled "proof using similar triangles"](https://en.wikipedia.org/wiki/Pythagorean_theorem#Proof_usin...), which kinda starts the same way, but it's a lot shorter and doesn't require the Law-of-Sines nor infinite-series.
If anyone wants to design further "trigonometric" proofs, they can just start from what's on Wikipedia and add in steps that rely on trigonometry.
Earlier quoted context omitted.
The nice thing about mathematics is that for every true statement there are infinitely many proofs. Granted, some are just variations of others, but there many ways to reach the same point.
> for every true statement there are infinitely many proofs No. There are true statements which cannot be proven. For example: "This statement cannot be proven." (Technically it's truth value is neither true nor false. It is an imaginary Boolean value.)
[1] https://eneim.notion.site/On-a-new-proof-of-the-Pythagorean-....