The Guardian reports –

https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove

US teens say they have new proof for 2,000-year-old mathematical theorem

New Orleans students Calcea Johnson and Ne’Kiya Jackson recently presented their findings on the Pythagorean theorem.

Ne’Kiya Jackson, left, and Calcea Johnson recently presented their findings at the American Mathematical Society’s south-eastern chapter’s semi-annual meeting. Photograph: WWL-TV

Two New Orleans high school seniors who say they have proven Pythagoras’s theorem by using trigonometry – which academics for two millennia have thought to be impossible – are being encouraged by a prominent US mathematical research organization to submit their work to a peer-reviewed journal.

Calcea Johnson and Ne’Kiya Jackson, who are students of St Mary’s Academy, recently gave a presentation of their findings at the American Mathematical Society south-eastern chapter’s semi-annual meeting in Georgia.

They were reportedly the only two high schoolers to give presentations at the meeting attended by math researchers from institutions including the universities of Alabama, Georgia, Louisiana State, Ohio State, Oklahoma and Texas Tech. And they spoke about how they had discovered a new proof for the Pythagorean theorem.

The 2,000-year-old theorem established that the sum of the squares of a right triangle’s two shorter sides equals the square of the hypotenuse – the third, longest side opposite the shape’s right angle. Legions of schoolchildren have learned the notation summarizing the theorem in their geometry classes: a2+b2=c2.

As mentioned in the abstract of Johnson and Jackson’s 18 March mathematical society presentation, trigonometry – the study of triangles – depends on the theorem. And since that particular field of study was discovered, mathematicians have maintained that any alleged proof of the Pythagorean theorem which uses trigonometry constitutes a logical fallacy known as circular reasoning, a term used when someone tries to validate an idea with the idea itself.

Johnson and Jackson’s abstract adds that the book with the largest known collection of proofs for the theorem – Elisha Loomis’s The Pythagorean Proposition – “flatly states that ‘there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem’.”

But, the abstract counters, “that isn’t quite true”. The pair asserts: “We present a new proof of Pythagoras’s Theorem which is based on a fundamental result in trigonometry – the Law of Sines – and we show that the proof is independent of the Pythagorean trig identity sin2x+cos2x=1.” In short, they could prove the theorem using trigonometry and without resorting to circular reasoning.

Johnson told the New Orleans television news station WWL it was an “unparalleled feeling” to present her and Jackson’s work alongside university researchers.

Calcea Johnson and Ne’Kiya Jackson are being encouraged to submit their work to a peer-reviewed journal after being

“There’s nothing like it – being able to do something that people don’t think that young people can do,” Johnson said to the station. “You don’t see kids like us doing this – it’s usually, like, you have to be an adult to do this.”

Alluding to how St Mary’s slogan is “No excellence without hard labor,” the two students credited their teachers at the all-girls school in New Orleans’s Plum Orchard neighborhood for challenging them to accomplish something which mathematicians thought was not possible.

“We have really great teachers,” Jackson said to WWL during an interview published Thursday.

WWL reported that Jackson and Johnson are on pace to graduate this spring, and they intend to pursue careers in environmental engineering as well as biochemistry.

St Mary’s Academy administrators did not immediately respond to a request for comment on Friday. Prominent alumnae of the school include judge Dana Douglas, who is the first Black woman to serve on the bench of the federal fifth circuit court of appeals, and renowned restaurateur Leah Chase.

Catherine Roberts, executive director for the American Mathematical Society, said she encouraged the St Mary’s students to see about getting their work examined by a peer-reviewed journal, even at their relatively young age.

“Members of our community can examine their results to determine whether their proof is a correct contribution to the mathematics literature,” said Roberts, whose group hosts scientific meetings and publishes research journals.

Roberts also said American Mathematical Society members “celebrate these early career mathematicians for sharing their work with the wider mathematics community”.

“We encourage them to continue their studies in mathematics,” Roberts added.

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