Just because something hasn’t been done doesn’t mean it can’t be done...
And that is exactly what two young students from Louisiana have proved.
Calcea Johnson and Ne’Kiya Jackson, both students at St. Mary’s Dominican High School in New Orleans, Louisiana, say they have solved a math problem that has baffled mathematicians for over 2,000 years.
The pair believes they have found a new proof for the Pythagorean theorem that uses trigonometry without resorting to circular logic. The duo presented their findings at the American Mathematical Society’s Annual Southeastern Conference in Georgia on March 18th, 2023.
Johnson and Jackson first became interested in Pythagoras’ theorem after entering a math contest meant to encourage students’ further interests in the field, according to St. Mary’s. Their study for the contest led them to believe the theory’s original proof was inaccurate.
The Pythagorean theorem is one of the most famous formulas in mathematics and has been taught in schools for centuries. The principle — which dates back over two centuries — is named after the Greek philosopher Pythagoras and explains the relationship between the three sides of a right-angled triangle.
It is expressed with the formula a² + b² = c².
The Pythagorean theorem is a central formula in trigonometry that has applications in fields as diverse as engineering, physics, and architecture.
The principle and its significance to trigonometry have previously led mathematicians to argue that any alleged proof that uses trigonometry is guilty of circular reasoning. It has been largely accepted that there is no way to define the theorem using trigonometry — which is what Jackson and Johnson claim they have now done.
In an abstract posted on the American Mathematical Society's website, the two said that they used the "Law of Sines," which is used to find angles of a general triangle.
"In the 2,000 years since trigonometry was discovered, it's always been assumed that any alleged proof of Pythagoras's Theorem based on trigonometry must be circular," the pair wrote. "In fact, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis), the author emphatically 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 that isn't quite true: in our lecture, 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 \sin^2x + \cos^2x = 1."
Although a number of mathematicians have argued that the Pythagorean theorem can be verified without circular logic by instead using the notion of similar triangles, they have been praising the gifted students for their groundbreaking work.
“It appears these two students at least managed to 1) recognize Loomis’ statement was too sloppy/general to be true, and 2) produce a PT proof for which either the method or its described relation to trig is not replicated in Loomis’ book,” Cambridge mathematician Sarah Rasmussen wrote on Twitter. “Both of these are excellent accomplishments, and the students should be very proud of the work they did.”
Ne’Kiya and Calcea, both 17, were the only high school students at the conference, and their achievements were commended by the Executive Director of the American Mathematical Society Catherine Roberts.
Roberts added that the 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 further encouraged the pair to submit their findings to a scientific journal. By doing this, their findings will undergo an arduous academic peer-review process and be confirmed by various experts in the field.
The girls’ achievement is momentous, not just for St. Mary’s Academy, but for young Black female students everywhere.
In an interview with New Orleans’ WWL-TV, Johnson explained the remarkable feeling of presenting their work alongside world-renowned researchers. “There’s nothing like it," she said. "Being able to do something that people don’t think that young people can do. You don’t see kids like us doing this. It’s usually adults that do this.”
Johnson explained that she and Jackson were inspired by the slogan of their school, "no excellence without hard labor."
The girls attributed their success to the dedication and hard work of the teachers at St. Mary’s Academy, with Jackson adding, “we have really great teachers.”
The dynamic duo intends to continue with mathematics, with Johnson aiming for a career in environmental engineering and Jackson in biochemistry.
