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Posted on 04/21/2023 2:30:05 PM PDT by nickcarraway
At an American Mathematical Society meeting, high school students presented a proof of the Pythagorean theorem that used trigonometry—an approach that some once considered impossible
Two high school students have proved the Pythagorean theorem in a way that one early 20th-century mathematician thought was impossible: using trigonometry.
Calcea Johnson and Ne’Kiya Jackson, both at St. Mary’s Academy in New Orleans, announced their achievement last month at an American Mathematical Society meeting. “It’s an unparalleled feeling, honestly, because there’s just nothing like it, being able to do something that ... people don’t think that young people can do,” Johnson told WWL-TV, a New Orleans CBS affiliate.
If verified, Johnson and Jackson’s proof would contradict mathematician and educator Elisha Loomis, who stated in his 1927 book The Pythagorean Proposition that no trigonometric proof of the Pythagorean theorem could be correct. Their work joins a handful of other trigonometric proofs that were added to the mathematical archives over the years. Each sidestepped “circular logic” to prove the pivotal theorem. So what exactly is a trigonometric proof of the Pythagorean theorem, and why was Loomis so closed off to the idea?
The Pythagorean theorem provides an equation to calculate the longer side of a right triangle by summing the squares of the other two sides. It is often phrased as a2 + b2 = c2. In this equation, a, b and c represent the lengths of the three sides of a right triangle, a triangle with a 90-degree angle between two of its sides. The quantity c is the length of the longest side, called the hypotenuse. Though the theorem is named for the ancient Greek philosopher Pythagoras, some historians believe it was known in Babylon around 1,000 years earlier.
The theorem “connects algebra and geometry,” says Stuart Anderson, a professor emeritus of mathematics at Texas A&M University–Commerce. “The statement a2 + b2 = c2, that’s an algebraic statement. But the figure that it comes from is a geometric one.”
Meanwhile trigonometry focuses on functions that depend on angles. These functions, such as the sine and cosine, are defined using right triangles. Imagine a right triangle with one side that lies flat against a table and another that shoots straight up from where it meets the first side at a right angle. The hypotenuse will reach diagonally between these two sides.
Now measure the angle between the hypotenuse and the table. Mathematicians define the sine of this angle as the height of the vertical side divided by the length of the hypotenuse. The cosine of this angle is the length of the horizontal side divided by the hypotenuse. The Pythagorean theorem is therefore equivalent to the equation sin2 x + cos2 x = 1. “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem,” explains Anderson, referring to equations that describe relationships among different trigonometric functions.
Loomis believed that if you used these functions in a proof of the Pythagorean theorem, you would have assumed the theorem to begin with—a circular argument and thus an unforgivable mathematical error.
But that’s not always true. In their talk at the American Mathematical Society meeting, Jackson and Johnson said a trigonometric identity called the law of sines didn’t depend on the Pythagorean theorem and that they could use it to prove the theorem.
Anderson hopes that Jackson and Johnson’s proof will raise interest in mathematics among students. “It kind of makes me wish I still had a class so I could talk about it,” he says.
The other trigonometric proofs of the theorem that have appeared in the past include a few that are described on mathematician Alexander Bogomolny’s website. One of these was crafted by Jason Zimba, then a physicist and mathematician at Bennington College, and published in Forum Geometricorum in 2009. This proof used a trigonometric identity that allows you to calculate the cosine and sine of an angle x – y without using the Pythagorean theorem—if you know the cosines and sines of x and y on their own.
On October 26, 2009, Bogomolny added Zimba’s proof to his website, writing “Elisha Loomis, myself and no doubt many others believed and still believe that no trigonometric proof of the Pythagorean theorem is possible.... I happily admit to being in the wrong.” Over time, Bogomolny added more trigonometric proofs to the site: one such proof could be written in just four lines.
The saga shows how even the simplest mathematics can surprise us. “Mathematicians, I think, have learned to not make a bold claim that something is impossible because we’ve been embarrassed over the years too many times by doing that,” Anderson says.
The American Mathematical Society has encouraged the New Orleans students to submit their proof for publication in a peer-reviewed scientific journal.
Leila Sloman is a math writer based in Princeton, New Jersey. She is a contributor to Quanta Magazine, and she creates and edits outreach content for the American Mathematical Society. She holds a Ph.D. in mathematics from Stanford University.
Recent Articles by Leila Sloman
I agre with most of what you wrote except for the above. There is an area between open skepticism and unthinking affirmation that is simple open-mindedness, without coming to a conclusion either way. If I'm in a camp at all, (because usually I'm an original thinker), that would be the one. Nothing that I wrote claimed belief about their ostensible discovery; just that the possiblity couldn't be dismissed until resoundingly disproven.
would it even have mede the news if they were not black?
I programmed on the IBM 360 through the 1970s At my first job I programmed on one of the first commercially available minicomputers - SEL32. Minicomputers is not even a distinction any more. My main objection with the movie was the implication the missions would have been impossible without those women mathematicians. They were talented and good at their jobs but not the only people there capable of understanding orbital mechanics and programming it into the main frame or minicomputer. Orbital mechanics basically started with Newton, then a host of European mathematicians improved on it. Every physics and aerospace engineering student studies it, even back then. Might have even programmed the equations up on an analog computer, maybe even a mainframe digital computer if available. It was a team effort something g barely acknowledged by the movie.
The movie was essentially a biopic about those three particular mathematicians, and not focused on the engineering process except to acknowledge that its upper echeolon was a rarified atmosphere to which blacks and women were not typically admitted.
I personally experienced the struggle for equal treatment during my entire corporate career; and I am a white conservative. Not having worked on the engineering side of tech but rather the thought leadership R&D and marketing sides, and also having grown up post-WW2 immersed in the stereotypes about blacks and the South, I came away from the movie just astonished at what these women had been able to do. Today's social justice ingrates would be out on their asses in no time, or else dragging the entire effort down because nowadays they can't be fired.
OK, Langley Virginia is not the Deep South, but even in the DC area, that degree of daily interaction, not even to mention the acceptance of women as team members in any previously male-dominated domains, was quite unusual in the early 60s.
To your point that many others could have done the job: while that is true, the picture was about what actually happened.
It's possible that someone higher-up could have wanted to make a stand for civil rights at the time by seeking to hire black women (there were also white women in those same Langley positions); but it's also possible that these women were only available to Langley because they were shunned from comparable opportunities in most employment arenas. They could also have been seen as a bargain; at that time, not only were women generally paid less for the same work, but also blacks of the era had grown up conditioned to be submissive to the power structures, and therefore hiring black women may have been seen as less scary to male management than hiring white female or male "human computers", as they were called.
I'm not endorsing any of these attitudes; just relating the kinds of things I witnessed.
A lot of snake oil salesmen make a living off people who go by that belief, just sayin.
ping
So, you mischaracterized what I said, and now you think you won? Okay. Have a good weekend.
That is beautiful, ansel12.
Huh? I wasn’t conversing to ‘win’. Sorry you feel that way.
It is isn’t it, it is at Notre Dame University.
Maybe it’s just a habit.
Looks like that'll be harped on at every grade level, as basic education is itself dismantled. Thanks GOPJ.
P2+I2+N2=G2!........................
Their work joins a handful of other trigonometric proofs that were added to the mathematical archives over the years. Each sidestepped “circular logic” to prove the pivotal theorem.
...I might as well fly with it. 🕊️
The entire universe is Hebrew school.
Directional indicator:
Vector quantities have a direction and a magnitude. However, sometimes one is interested in only the direction of the vector and not the magnitude. In such cases, for convenience, vectors are often "normalized" to be of unit length. These unit vectors are commonly used to indicate direction, with a scalar coefficient providing the magnitude.
yechidah ("unity")
Unit vector: "vector yechidah" [וקטור יחידה] = 358
A Comprehensive Soul
1The soul of Mashiach comprises the souls of the entire Jewish people. This is what enables him to redeem all of Israel from exile.
Mashiach, as is known,2 is the all-embracing yechidah of the Jewish people.
https://www.chabad.org/library/article_cdo/aid/2326847/jewish/Chapter-1-The-King-Mashiach.htm
Mashiach [משיח] = 358
As the Maverick solution to the take-a-number and wait business-as-usual model, he's the very opposite of a bot [בוט], because flipped to the reverse it's tov טוב, "good".
The unexpected one. A literal mashal:
Hebrew: מש"ל
(What needs to be proven or what we wanted to prove or what was to be proven.)
It's a black tombstone, or an empty one,
a victory yechidah in black and white, to indicate the direction (up 1). There's a higher way, outside of the dogmatic/settled/bot/paralysis-is-king interpretations that *don't* come from God, evident just on general principle.
"High School Students Prove Pythagorean Theorem. Here’s What That Means":
↑
Outside the box *is* where the abstract is the literal meaning. People just need to be shown the way up and out.
Sunshine, fresh air... it's not complicated. It all begins with a little Tender Loving Care.
🎁 ❤️ 🎈🔴 ♂️ ☘️
Johnny Cash - Keep On the Sunny Side
Isaiah 40:3
The voice of him that crieth in the wilderness, Prepare ye the way of the LORD, make straight in the desert a highway for our God.
🪙🪙🕊️
It always looks like I wander. Priorities.
This topic was posted , thanks nickcarraway.
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