In December 2022, a high school in Louisiana challenged its students with a bonus math problem that came with a $500 cash prize: Using trigonometry, they had to create a new proof of the Pythagorean theorem; This is the 2,000 year old rule. The relationship between side lengths in a right triangle. Nobody told the young people He said mathematicians had long considered this task “impossible.”

When Ne’Kiya Jackson and Calcea Johnson found a solution, it sent shockwaves through the mathematical world. Them They presented their evidence at the American Mathematical Society a few months later in 2023, but it still needed further review. This week, two students now at the university published their findings, along with additional proof of the theorem, in a new study.

“Perhaps no topic in mathematics creates as much confusion and anxiety in high school students as trigonometry,” Jackson and Johnson write in their article. “There were times when we both wanted to give up on this project, but we decided to insist on finishing what we started.”

Research published on Sunday American Mathematics Quarterly It outlines “five or ten” new ways to prove the Pythagorean theorem using trigonometry. In other words, all their evidence shows that the square of the longest side of a right triangle is equal to the sum of the squares of its two short sides.

Jackson and Johnson spent months, even working weekends and holidays, to find their original evidence. In the new article, they describe this success and the four new proofs they worked on afterwards. They also outline a new method that could result in five additional proofs.

“Some people have the impression that you have to be in academia for years before you can actually produce new mathematics,” he says Álvaro Lozano-Robledoa mathematician at the University of Connecticut Science NewsNikk Ogasa. But Jackson and Johnson’s study shows that “even as a high school student, you can make a splash,” he adds.

Although the Pythagorean theorem has been proven with algebra and geometry, mathematicians previously thought that it could not be proven using trigonometry. This is because the foundations in trigonometry are based on the same theorem; therefore experts have suggested that any proof of this would have to use principles that assume that the theorem is already true, which would lead to a logical fallacy called “circular reasoning”.

Instead, the pair of students used the trigonometric rule. Law of SinesNot based on the Pythagorean theorem. Two people have previously proven the theorem using trigonometry: Jason Zimba in 2009 and Nuno Luzia in 2015.

“I didn’t think it would go this far,” Jackson said. expression. “I was pretty surprised it got published.”

Jackson and Johnson first worked on rehearsals at St. Paul’s in New Orleans, one of the oldest Catholic schools for Black women in the country. They studied at St. Mary’s Academy. After presenting their work, the duo set out to publish their findings in a peer-reviewed journal and worked on past proofs of the theorem. This “triggered a creative process,” Jackson says Science NewsHe added that they developed additional evidence from there.

Currently, Jackson is pursuing a doctorate in pharmacy at Xavier University of Louisiana and Johnson is studying environmental engineering at Louisiana State University.

The duo’s success demonstrated how a simple bonus question in high school could lead to surprising mathematical discoveries, even discoveries previously thought “impossible.”

“I think mathematicians have learned not to make bold claims that something is impossible, because we have been embarrassed to do so many times over the years.” stuart andersonA mathematician from Texas A&M University-Commerce said: Scientific American‘s Leila Sloman last year after Jackson and Johnson first presented their work.

Anderson, a retired professor, hopes this new evidence will increase interest in mathematics among students. “I wish I still had a class so I could talk about it,” he says. Scientific American.

This enthusiasm is echoed by Johnson, who emphasizes the importance of representation on the field. “I am so proud that we have both been able to make such a positive impact in showing that young women and women of color can do these things,” she says in the statement.