Two teenagers have once again proved an ancient math rule

Two years ago, a pair of high school classmates each and every composed a mathematical marvel, a trigonometric proof of the Pythagorean theorem. Now, they’re unveiling 10 more.

For over 2,000 years, such proofs were regarded as not you desire to if truth be told. And yet, undeterred, Ne’Kiya Jackson and Calcea Johnson published their new proofs October 28 in American Mathematical Monthly.

“Some people have the impression which that you desire to be in academia for years and years earlier than you are ready to as it will be produce some new mathematics,” says mathematician Álvaro Lozano-Robledo of the University of Connecticut in Storrs. But, he says, Jackson and Johnson demonstrate that “you are ready to make a splash on the identical time as a high school student.”

Jackson is now a pharmacy student at Xavier University of Louisiana in New Orleans, while Johnson is studying environmental engineering at Louisiana State University in Baton Rouge.

Mathematical proofs are sequences of statements that demonstrate an assertion is correct or false. Pythagoras’ theorem — a² + b² = c², about the length of a right triangle’s hypotenuse to those of its other two sides — has been proven many times with algebra and geometry (SN: four/2/03).

But in 1927, mathematician Elisha Loomis asserted that the feat may per chance now not be done using rules from trigonometry, a subset of geometry that deals with the relationships between angles and side lengths of triangles. He believed that Pythagoras’ theorem is so fundamental to trigonometry, any trigonometry-based prove the theorem would should first assume it became true, thereby resorting to circular logic.

Jackson and Johnson conceived the first of their trigonometry-based proofs in 2022, while seniors at St. Mary’s Academy in New Orleans, a Catholic school attended mostly by young Black women folk. At that point, best two other trigonometric proofs of Pythagoras’ theorem existed, presented by mathematicians Jason Zimba and Nuno Luzia in 2009 and 2015, respectively. Working on the early proofs “sparked the creative process,” Jackson says, “and from there we developed additional proofs.”

After formally presenting their work at an American Mathematical Society meeting in March 2023, the duo set out to publish their findings in a peer-reviewed journal. “This proved to be essentially the most daunting task of all,” they said within the paper. Further to writing, the duo needed to develop new skills, all while entering college. “Learning a means to code in LaTeX [a typesetting software] is now not so straight forward while you’re also in quest of to write a 5-page essay with a gaggle, and submit an information analysis for a lab,” they wrote.

Nevertheless, they were motivated to accomplish what they started. “It became important to me to have our proofs published to solidify that our work is correct and respectable,” Johnson says.

Per Jackson and Johnson, trigonometric terms will likely be defined in two other ways, and this will likely per chance increasingly complicate efforts to prove Pythagoras’ theorem. By focusing on only one amongst those methods, they developed four proofs for right triangles with sides of kind of an excellent deal of of lengths and one for right triangles with two equal sides.

Among these, one proof stands out to Lozano-Robledo. In it, the scholars fill one larger triangle with an infinite sequence of smaller triangles and use calculus to search out the lengths of the larger triangle’s sides. “It looks as if nothing I’ve ever seen,” Lozano-Robledo says.

Jackson and Johnson also leave any other five proofs “for the interested reader to stumble on,” they wrote. The paper incorporates a lemma — a sort of stepping-stone to proving a theorem — that “provides a clear direction towards the additional proofs,” Johnson says.

Now that the proofs are published, “other people would per chance take the paper and generalize those proofs, or generalize their ideas, or use their ideas in other ways,” Lozano-Robledo says. “It just opens up an awful lot of mathematical conversations.”

Jackson hopes that the paper’s publication will inspire other students to “see that obstacles are element of the strategy. Stick to it, and yow will stumble on yourself achieving more than you thought you desire to if truth be told.”