In their scholarly article, Calcea Johnson and Ne’Kiya Jackson introduce five innovative ways to validate Pythagoras’ Theorem using trigonometry, along with a new technique that uncovers at least five additional proofs.
In 2022, two high school students from the U.S., Calcea Johnson and Ne’Kiya Jackson, surprised their teachers by finding a novel method to establish Pythagoras’ theorem through trigonometry during a local high school competition. Their impressive achievement earned them the keys to the city of New Orleans and personal acknowledgment from Michelle Obama.
Now, they have become published authors of a new peer-reviewed article detailing their findings, which appears in the journal American Mathematical Monthly.
Pythagoras’ well-known theorem, existing for over 2,000 years, can be succinctly expressed as a2 + b2 = c2. This theorem states that in a right-angled triangle, if you know the lengths of two sides, you can calculate the length of the third side, known as the hypotenuse. Essentially, the square of the hypotenuse is equivalent to the sum of the squares of the other two sides.
Throughout history, numerous mathematicians have used algebra and geometry to demonstrate this theorem. However, proving it through trigonometry was believed to be impossible because trigonometric definitions rely on the assumption that Pythagoras’ theorem is valid, presenting a case of circular reasoning.
Despite this challenge, Johnson and Jackson independently found a solution to this mathematical dilemma and proved Pythagoras’ theory without cyclical logic, a milestone achieved only twice before by professional mathematicians.
Following their individual accomplishments, Johnson and Jackson teamed up to present their findings at a regional American Mathematical Society meeting in Atlanta in March 2023. After receiving positive feedback, they decided to submit their work for final peer review and publication. Their research features five distinct methods for validating the theorem through trigonometry, along with an approach that reveals an additional five proofs, totaling ten proofs in all. Of these, only one had previously been shared at the conference, meaning nine are entirely original.
“I was pretty surprised to be published,” remarks Ne’Kiya Jackson. “I never thought it would escalate to this level.”
“Having a paper published at such a young age is truly astonishing,” agrees Calcea Johnson.
“This is such an exciting moment for me, especially because when I was younger, STEM [science, technology, engineering, and math] wasn’t seen as something ‘cool.’ Knowing that so many people now have an interest in STEM and mathematics gives me hope and excitement for the future of these fields,” she adds.
In their paper, the authors suggest that one reason high school students find trigonometry confusing and stressful is due to the existence of two distinct versions of the subject, both using similar terminology. This can make understanding trigonometry feel like interpreting an image with two overlapping pictures.
Johnson and Jackson propose that by distinguishing between these two versions and concentrating on just one, it is possible to uncover a wealth of new proofs for the Pythagorean Theorem.
Currently, Jackson is enrolled at Xavier University of Louisiana, where she is pursuing a doctoral degree in pharmacy, while Johnson is studying environmental engineering at Louisiana State University’s Roger Hadfield Ogden Honors College.
“I take great pride in being a positive role model, showing that young women, particularly women of color, can achieve these accomplishments, and encouraging other young women to pursue their goals,” Johnson shares.
Della Dumbaugh, editor-in-chief of American Mathematical Monthly, commented on Johnson and Jackson’s success, saying, “The Monthly is privileged to publish the work of these two young scholars.
“Their findings emphasize the potential of fresh perspectives from students in the field. Additionally, they underscore the significant contributions of educators and schools in nurturing the next generation of mathematicians.
“Moreover, this research embodies the spirit of Benjamin Finkel, who established the Monthly in 1894 to make mathematics accessible to both teachers and students.”