Two New Orleans high school students think they’ve found a way to prove Pythagorean’s theorem in a way mathematicians previously thought impossible.
Calcea Johnson and Ne’Kiya Jackson said they proved the famous theorem using a special equation used in trigonometry.
The two students at St. Mary’s Academy – an all-girls school in New Orleans’ Plum Orchard – have now been encouraged by the American Mathematical Society to submit what they say is groundbreaking new evidence for peer review.
Earlier this month, on March 18, they showed their evidence at the Society’s biannual meeting in Georgia and were reportedly the only high schoolers in attendance.
Calcea Johnson (right) and Ne’Kiya Jackson (left) think they found a way to prove the Pythagorean Theorem in a way mathematicians thought impossible
The pair presented their proof this month at the biannual Southeastern conference of an American Mathematical Society
The pair appeared on local broadcaster WWL last week to talk about their presenting experience and proof.
“There’s nothing quite like it – to be able to do something that people think young people can’t do,” Johnson told WWL. “You don’t see that with kids like us. Normally you have to be of legal age to do that.”
“We have really great teachers,” Jackson added during the interview, which aired Thursday.
WWL reported that Jackson and Johnson will graduate this spring and that they plan to pursue careers in environmental engineering and biochemistry.
There are hundreds of different proofs for the Pythagorean theorem, which states that for a right triangle with sides a, b, and c, they are related by the expression a^2+b^2=c^2.
But Johnson and Jackson claim to have found entirely new evidence.
Since the Pythagorean theorem is one of the concepts that gave rise to trigonometry, it has been said that trigonometry cannot be used to prove the Pythagorean theorem.
That’s because it would entail circular reasoning – logic in which something is proved using an assumption based on what one is trying to prove.
The principles of trigonometry are laid out in a series of equations. One of these equations, the Pythagorean trigonometric identity, is expressed as (sinθ)^2 + (cosθ)^2 = 1, where θ is one of the angles in a right triangle plotted on a “unit circle.”
The students said in the abstract for their lecture that their proof is not based on this identity, but uses a different trigonometric equation, the Law of Sines, which is not based on the Pythagorean theorem.
“In the 2000 years since the discovery of trigonometry, it has always been assumed that any alleged proof of the Pythagorean theorem based on trigonometry must be circular,” reads the abstract of the lecture.
“Indeed, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis), the author states that “there are no trigonometric proofs because all the basic formulas of trigonometry are themselves based on the truth of the Pythagorean theorem .”‘
However, the two say that the rule of trigonometry they used in their proof is not based on the Pythagorean theorem.
“But that is not entirely correct: in our lecture we present a new proof of the Pythagorean theorem based on a fundamental result of trigonometry – the law of sines – and we show that the proof is independent of the Pythagorean trig identity \ sin is ^2x + \cos^2x = 1,” they added.
The book the pair referred to was published in 1927 and goes on to say that the two theories are truly intertwined: “Trigonometry is because the Pythagorean theorem is.”
What the two girls allegedly did was break down the principles of trigonometry into those that relied on Pythagoras and those that didn’t.
Catherine Roberts, executive director of the American Mathematical Society, told the Guardian she encourages St Mary’s students to have their work reviewed by a peer-reviewed journal.
“Members of our community can examine their results to determine if their proof is an accurate contribution to the mathematical literature,” Roberts told the publication.
She also said that members of the American Mathematical Society “celebrate these young mathematicians for sharing their work with the broader math community.”
“We encourage them to continue their mathematics studies,” she added.