Two years ago, several high school classmates each created a mathematical wonder: a trigonometric proof of the Pythagorean theorem. We are currently publishing 10 more.
For over 2,000 years, such a proof was thought to be impossible. Undeterred, Nekiyah Jackson and Calthea Johnson published it. their new evidence October 28th American Mathematics Monthly Magazine.
“Some people have the impression that you have to be in academia for years to actually create new mathematics,” said mathematician Alvaro Lozano Robredo of the University of Connecticut at Storrs. But Jackson and Johnson are proving that “high schoolers can make a splash,” he said.
Ms. Jackson is currently a pharmacy student at Xavier University of Louisiana in New Orleans, and Ms. Johnson is studying environmental engineering at Louisiana State University in Baton Rouge.
A mathematical proof is a series of statements that proves a claim to be true or false. Pythagorean theorem —2 +b2 =c2relating the length of the hypotenuse of a right triangle to the lengths of the other two sides has been proven many times in algebra and geometry (SN: April 2, 2003).
But in 1927, mathematician Elisha Loomis argued that this feat could not be accomplished using the rules of trigonometry, the part of geometry that deals with the relationship between the angles and side lengths of triangles. He believed that the Pythagorean theorem was the basis of trigonometry, and that if you tried to prove a theorem based on trigonometry, you first had to assume that it was true, and as a result you had to resort to circular logic. was.
Jackson and Johnson devised the first proof based on trigonometry in 2022, along with senior students at St. Mary’s Academy in New Orleans, a Catholic school attended primarily by young black women. At the time, there were only two other trigonometric proofs of the Pythagorean theorem, published by mathematicians Jason Zimba and Nuno Luzia in 2009 and 2015, respectively. Jackson says working on the initial proofs “started a creative process from which we developed additional proofs.”
officially then present their work At the March 2023 meeting of the American Mathematical Society, the two aimed to publish their findings in a peer-reviewed journal. “This turned out to be the most difficult task of all,” they wrote in their paper. While attending college, the two had to learn new skills in addition to writing. “Learn how to code with LaTeX” [a typesetting software] “Writing a five-page essay in a group and trying to submit data analysis to a lab is not that easy,” they write.
Nevertheless, they were motivated to finish what they started. “It was important to me to publish the evidence to ensure that our research is valid and worthy of respect,” Johnson says.
According to Jackson and Johnson, terms in trigonometric functions can be defined in two different ways, which can complicate the task of proving the Pythagorean theorem. Focusing on only one of these methods, we developed four proofs for right triangles with sides of unequal length and one proof for right triangles with two equal sides.
Among them, one piece of evidence stands out for Lozano-Robredo. In it, students fill one large triangle with an infinite number of smaller triangles and use calculus to find the side lengths of the large triangle. “It’s like nothing we’ve ever seen before,” Lozano Robredo said.
Jackson and Johnson also left five more pieces of evidence “for the interested reader to discover,” they wrote. The paper includes a lemma that serves as a sort of stepping stone to proving the theorem, and “provides a clear direction for additional proofs,” Johnson says.
Now that the proof has been published, “other people may take the paper and generalize the proof, generalize their own ideas, or use their ideas in other ways.” says Lozano Robredo. “It just sparks a lot of mathematical conversation.”
Jackson hopes the publication of this paper will encourage other students to “understand that disability is part of the process.” If you keep at it, you may be able to accomplish more than you ever thought possible. ”
