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New Orleans teenagers found a new proof of the Pythagorean Theorem
251–260 of 287 posts
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#252I am puzzled by the article's spin on this, which really centers on the fact that this original proof was authored by two teenage African-American girls from the South , as if the interesting thing here was not so much the proof itself than the idea that there are gifted mathematicians from underrepresented backgrounds and skin colors. In my experience, pretty much in any place and social stratus you might visit, the…
Skewed representation is the end product of multiple filters working in tandem. You're correct that there are still further filters and that college is one of them. But the filters don't start in college, it just continues. The girls in question have already passed through several.
That's a good point
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#253I am puzzled by the article's spin on this, which really centers on the fact that this original proof was authored by two teenage African-American girls from the South , as if the interesting thing here was not so much the proof itself than the idea that there are gifted mathematicians from underrepresented backgrounds and skin colors. In my experience, pretty much in any place and social stratus you might visit, the…
It's telling that your comment is currently 2nd ranked. It comes across generous "there are bright math kids everywhere" but really boils down to "don't talk about how they're black" and "don't talk about how they're women." And finished with "my male friend was disadvantaged, the conversation should be about that." Obviously a large number of the HN crowd agrees with you because these types of comments always land a…
well said
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#254Earlier quoted context omitted.
> What am I missing? Nearly all the nontrivial results of trigonometry do in fact rest on the pythagorean theorem. The trig identities you learned in high school, as well as more advanced results like power series, etc. These results would be inadmissible. So the “uses trigonometry” part of this story feels like an attempt to manufacture mystery and hype. Which is a shame, because the geometric series construction is…
Apparently (found this while reading an article on the girls' accomplishments) somebody proved sin^2x + cos^2x = 1 without using the Pythagorean theorem in 2009: https://forumgeom.fau.edu/FG2009volume9/FG200925index.html . I don't think the "uses trigonometry" part is hype. They do use the definitions and law of sines, they just cleverly avoid the parts of trigonometry that depend on the Pythagorean theorem.
This theorem is essentially the Pythagorean theorem, so … a proof of one is a proof of the other.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#255Earlier quoted context omitted.
Your explanation is very clear, but I still think that the article is confusing about that. Note how the author sets the expecatations at the beginning of the article: "...the proof these young trailblazers have proposed might make a few established mathematicians eat their words. This is because their proof uses trigonometry." This wording implies to me that Jackson and Johnson are actually the first who came up wit…
I see your confusion. The thing is that “trigonometric proof” is not a well-defined mathematical object (even in this HN thread, you can see a lot of discussion about whether the trigonometry is really essential to the proof, ways to remove it, etc). So the “establishment” idea that “There are no trigonometric proofs” (Loomis, 1927) is not a mathematical conjecture, which can be demolished by the first counterexample…
Thanks, this has been very explanatory. It’s unusual that maths, the language known for precise definitions, has no unambiguous definition of trigonometry. It’s what it is though.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#256I am puzzled by the article's spin on this, which really centers on the fact that this original proof was authored by two teenage African-American girls from the South , as if the interesting thing here was not so much the proof itself than the idea that there are gifted mathematicians from underrepresented backgrounds and skin colors. In my experience, pretty much in any place and social stratus you might visit, the…
It's telling that your comment is currently 2nd ranked. It comes across generous "there are bright math kids everywhere" but really boils down to "don't talk about how they're black" and "don't talk about how they're women." And finished with "my male friend was disadvantaged, the conversation should be about that." Obviously a large number of the HN crowd agrees with you because these types of comments always land a…
I read through the article as well and was put off mostly by how it was presented that the students are black females:
> They are female, they are African-American, and they come from an area which is not particularly renowned for producing high academic achievers. This is just an awesome turn of events and one which should inspire anyone — no matter what their gender, ethnic or socio-demographic background — that excellence in your chosen field of study is always attainable if you have enough joy and passion for what you do.
I'm a person of color myself (not black) and seeing this statement (and the fact that the author is white) made it come across as "Look, even a black female can excel in math if they have enough joy and passion in what they do." On the surface, it seems like an innocuous statement, but what it really reads is "the only thing holding you back as an underrepresented person in society, especially being black and female, is your joy and passion, so keep working at it and you too can excel at math". It just reads as tone deaf to me.
My point is there's a way to present the fact that they're black and female, but you have to be careful how you word it because it can otherwise come across as almost condescending.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#257I have read about this proof for a bit and this is the first write-up that gives the slightest details. The phrase "using trigonometry" is confusing. What they do is assume functions sine and cosine exist, as normally defined, as ratios of triangle values, without assuming these have the various Pythagorean-theorem derived properties. They then construct an infinite series of nested triangles and use the formula for…
I don't see any Cosine being mentioned in the proof. I don't see any circular logic here.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#258Earlier 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?
Note that when a = b we have an equilateral triangle, with area a^2/2.
Draw a line from the 90 degree angle to side c, bisecting the 90 degree angle into two 45 degree angles. This divides the original triangle into two smaller triangles.
From the fact that the sum of the interior angles of a triangle is 180 degrees, it is not hard to see that the two smaller triangles are both equilateral, with sides of a, c/2, and c/2, and the angle between their two c/2 sides is 90 degrees.
That gives c^2/8 for the area of each of the smaller triangles, or c^2/4 for the area of the original triangle which we know to be a^2/2. So c^2/4 = a^2/2 or c^2 = 2 a^2 = a^ + a^2.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#259for completeness, here is a reference to the AMS presentation. https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Pa... [my post updated to include abstract] Abstract In the 2000 years since trigonometry was discovered it's always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythag…
Note that in math "flatly" means "asserted without proof", not "incontrovertible". "Trigonometry" is not formally defined, so Loomis's statement is merely tautological.
Maybe this is my mathematical background, but is that not the common meaning of “flatly” stating something?
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#260I have read about this proof for a bit and this is the first write-up that gives the slightest details. The phrase "using trigonometry" is confusing. What they do is assume functions sine and cosine exist, as normally defined, as ratios of triangle values, without assuming these have the various Pythagorean-theorem derived properties. They then construct an infinite series of nested triangles and use the formula for…
Here is a proof of Pythagoras using a geometric series:
https://www.cut-the-knot.org/pythagoras/Proof100.shtml
The proof is by John Arioni and also features an infinite number of similar triangles.