2 High School Students Prove Pythagorean Theorem. Here’s What That Means

Scientific American ^ | April 10, 2023 | Leila Sloman

[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

[freedomjusticeruleoflaw]

I bet they have a mom and a dad, which should be, but won’t be the story.

[ChessExpert]

“Calcea Johnson and Ne’Kiya Jackson, both at St. Mary’s Academy in New Orleans.”

I bet if you went to the public schools, no one would have a clue as to what was attempted, probably not even the math teachers.

[Scrambler Bob]

Mathematics is the study of answers to which there are no questions.

= = =

“5”

Oh, you mean 2+2

[Varda]

I don’t know but I’d guess you’re right. The story I read spoke about what a top school it was. ( kids living with mom and dad are ubiquitous at those schools) The girls said that their teachers encouraged them to try it.
I like that. Nobody is telling them they can’t try something hard.

[Magnum44]

I saw the movie and have no doubt the characters were smart and able to do math. That was real physics. They were also a part of a huge team that made the space program possible.

Thats not the same as a high schooler claiming she can prove something with circular logic that the greatest minds in history have failed to prove over centuries.

[Apparatchik]

I had always thought that trigonometry was based on the Pythagorean Theorem. Who knew?

Good for these kids.

[Reily]

Computers existed!

https://astronomy.com/news/2019/05/apollo-computers-when-ibm-engineers-gave-rockets-a-brain

[freedomjusticeruleoflaw]

They will be heroes (deservedly) up until the moment the parents make a statement about the need for nuclear families.

[Honest Nigerian]

I am not racist.
Math is racist.
Further, I highly doubt the “proof” which they claim was actually achieved by the rigorous rules on math.
But it doesn’t matter because the exercise is irrelevant to anything practical or necessary.

[FatherofFive]

Mathematics is the study of answers to which there are no questions.

42

[Organic Panic]

“IF VERIFIED”

In other words the story matches the desperate narrative like the 14 year old “boy genius” who proved you can get unlimited free energy.

Next up... I can trisect an angle using geometric construction, produce monopole magnets, and finally bring my AC battery to market... I just need a few billion tax dollars...

[jimfr]

But math is racist!

[Nifster]

Good for them

[Mr. Jeeves]

We’ve seen too many fakes and exaggerations in recent years to immediately accept the story at face value. That’s the true dividend of affirmative action..

[Albion Wilde]

Computers existed!
https://astronomy.com/news/2019/05/apollo-computers-when-ibm-engineers-gave-rockets-a-brain

Your article is great, but it is about the Apollo missions starting in the late 1960s, which was several years after the mission depicted in Hidden Figures—the U.S.'s first manned orbital flight in 1962 with John Glenn. It llasted only 5 hours, but had required years of NASA development starting in the 1950s. The leading woman depicted in the film, Katherine Johnson, had gone to work for NASA in 1953. (She died two years ago at age 101.)

Part of the film version of the story depicted another of the black women mathematicians urging the use of a large computer for the Manned Space Center; but certainly no one had desktops yet in that era.

Don't know how old you are, but I remember when a publishing company I did business with in the 1970's installed a mainframe computer. The hardware was the size of a U-Haul delivery van, and they had to build a temperature-controlled room and extensive air-conditioning to house it—just to do word processing for typography, not terabytes of math. Today we have more computing power in our phones.

The entire trajectory of tech development is "more reach, smaller size, cheaper per unit of measurement."

There's a great moment in the 1995 film Apollo 13 about NASA's first lunar mission in 1970. The Houston control center was depicted with two or three rows of dozens of desktop computer stations where engineers in short-sleeved white dress shirts and neckties were tensely monitoring the voice communications with the astronauts as they approached the moon. Some glitch came up, and after a moment of extreme panic, they all suddenly whipped out their slide rules! (It's been almost 30 years since I saw the movie, so I hope my recollection is correct, but that should give you an idea how powerful a moment it was—and I'm a writer, not a STEM person.)

[MD Expat in PA]

Part of the film version of the story depicted another of the black women mathematicians urging the use of a large computer for the Manned Space Center; but certainly no one had desktops yet in that era.

Dorothy Vaughn. A remarkable woman as well.

https://en.wikipedia.org/wiki/Dorothy_Vaughan

[Albion Wilde]

The posts that inspired my pushback implied skepticism due to the current social mania that elevates fashionably intersectional but otherwise unqualified persons. So I’m keeping an open mind, not being adept at higher math myself.

Until it is reliably shown that the girls’ reasoning is invalid, I don’t wish to dismiss their claim simply because of their sex and race.

[Albion Wilde]

Dorothy Vaughn. A remarkable woman as well.
https://en.wikipedia.org/wiki/Dorothy_Vaughan

Totally agree; and wonderfully well played by Olivia Spencer. Your link says she had been Katherine Johnson's supervisor at first! I salute them. This is Dorothy Vaughan through her ages:

[Magnum44]

I wasn’t dismissing the claim based on sex or race, but some sure seem to jump right to that conclusion. The fact that many in todays work environment want to artificially elevate peoples of no particular accomplishment to places of stature may have had something to do with it. But more so, is giving children (cause that’s what they still are) the impression they can change the world in some incredible way when the fact is, the only way to do that is through teams where you play small incremental parts. No one does it by themselves. We should teach kids that if they want to change the world, be part of something great. Not that you can be an army of one.

Anyway, I don’t have any ill will towards these girls, but I ain’t blowing smoke up their butts to make them feel like they are smarter than all the other true math greats who have laid the foundations for our mathematical and physics based accomplishments.

I fall in a different camp than you. I’m skeptical until you prove it, as opposed to believing a claim until it’s disproven. My only exception to that is faith in The Creator.

[Albion Wilde]

Some glitch came up

PS
This restrained but very dramatic moment in the film popularized the now-familiar saying, "Houston, we have a problem."