Impossible? Not for Them!
Calcea Johnson and Ne’Kiya Jackson, two high school students from New Orleans, astonished mathematicians by proving the Pythagorean theorem using trigonometry. Many mathematicians thought a trigonometric proof of this foundational theorem of geometry was not possible without using circular reasoning. Jackson and Johnson, however, created one.
The Pythagorean theorem, a cornerstone in our understanding of spatial relationships, relates the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Although many people have proven this theorem many times in many ways, no one had ever proved it using trigonometry without using circular reasoning.
In fact, Elisha Loomis, a noted math professor, wrote a book called “The Pythagorean Proposition” containing many proofs of the Pythagorean theorem. Loomis noted that there are no trigonometric proofs of the theorem that do not at some point rely on the theorem itself, thus asserting the belief that a non-circular trigonometric proof of the Pythagorean theorem would be impossible. The mathematical community generally accepted Loomis’ pronouncement.
Johnson and Jackson knew of Loomis’s book and his claim. Intrigued, they took up the challenge of devising a trigonometric proof for a math contest they entered. After months of hard work and study, they created an ingenious new proof. Not only did they win the contest, but conference organizers of the American Mathematical Society’s Spring meeting invited them to present their proof in Atlanta, Georgia, on March 18, 2023. Their presentation attracted the attention of seasoned mathematicians worldwide and is currently undergoing the rigorous process of peer review.
Excited by the news of this novel proof, I discussed Jackson and Johnson’s proof with a friend who is a math student. “It’s really cool,” Laura remarked. “I had to study it a couple of times just to make sure I had it correctly in my head before talking about it.” Johnson and Jackson proved the Pythagorean theorem using the Law of Sines, a principle in trigonometry that relates the lengths of sides of a triangle to the sines of its angles and avoids circular reasoning. “They were able to use the Law of Sines to prove the Pythagorean theorem, which was crazy cool. No one ever tried to do that.”
They also introduced concepts of infinite series to their proof, suggesting a deep understanding of mathematical sequences and series. “They took a right triangle and mirrored it. So now you have a double triangle, right? And then you make another smaller triangle next to that, and then another one here, and it keeps going down in a shape they call a ‘waffle cone.’ And you can make an infinite series to describe the side length of this infinitely large right triangle. Using calculus, you can solve for this infinite series. It ultimately reduces back to the side lengths being A2 + B2 = C2. This combination of trigonometry and calculus principles enabled them to present a proof that is both unique and rooted in fundamental mathematical concepts.”
Their proof is a testament to their ingenuity and will enrich the theorem’s extensive collection of proofs. Will their proof hold up to intense scrutiny by mathematicians all over the world? I wonder aloud. “Yeah, I think their proof will be validated.”
Calcea Johnson and Ne’Kiya Jackson are great examples, showing us all what passion and hard work can achieve. With a growth mindset, people can develop their talent in math. Age and gender are no limits. As Laura says, “There will be plenty of other people who see what they’ve done and think, ‘I never thought I could do something like this, but I can see myself in you.”
Next post on Friday, March 1st.