The Pythagorean Theorem, a2+b2=c2, is a cornerstone of geometry. Proven over 2,000 years ago, it allows one to calculate any side of a right-angle triangle if the other two sides are known. In 2022, mathematical concepts in trigonometry were used as a new proof of the theorem. It was not a Nobel Prize winner in mathematics, a Fields Medal winner in mathematics or a professor at a prestigious university’s mathematics department that discovered this groundbreaking proof. The new proof was created by two teenagers in New Orleans, Louisiana that designed the solution to answer a bonus question in a high school math contest. The students attended St Mary’s Academy, one of the oldest Catholic schools for Black women in the United States. In 2022, Ne’Kiya Jackson and Calcea Johnson were the two brilliant minds that came up with the proof and were praised for it.
Now college students, Jackson and Johnson have published their proof,alongside nine other original proofs,in the October, 2024 issue of the scientific journal, American Mathematical Monthly. Mathematical experts find these proofs particularly impressive due to the complicated use of trigonometry, a branch of mathematics based on the Pythagorean Theorem, to prove the theorem again. In 2009, mathematician Jason Zimba also used trigonometry for a different proof, using a different approach.
The path to publishing Jackson and Johnson’s proof was strenuous. Despite the need for the proof to be trigonometric, only algebra and geometry have been used to prove the theorem over the last 2000 years. The two teenagers spent months, including their summer break, in 2022 making sure their work was correct. Once the proof was completed, the two young women presented their work at the American Mathematical Society in 2023. What impressed the mathematical community most was Jackson and Johnson’s ability to solve the proof using trigonometry by not assuming the Pythagorean Theorem to be true. Instead, they used the Law of Sines to prove the theorem. This approach that was utilized by Jackson and Johnson was necessary since the Pythagorean theorem is so fundamental to trigonometry, any proof that used a trigonometry-based solution would have to assume that the theorem is true, which would represent circular thinking.
Tom Murdoch, an honorary professor at the University of Bristol’s School of Mathematics in the United Kingdom, described the ingenious proof by stating, “Trigonometric functions are based on sine and cosine, which are expressed as ratios of certain lengths of a right-angled triangle. It’s quite easy to get into a circular argument, and what’s so appealing about this is they found a line of argument using sine and cosine that doesn’t assume Pythagoras is true.” With a fresh set of eyes uninhibited by “impossibility,” Jackson and Johnson created a revolutionary proof of the ancient theorem.
Source: Interesting Engineering
Over 300 proofs of the Pythagorean Theorem have been developed over the last 2000 years. Yet, Jackson and Johnson can say their unique proof, discovered independently, has broken new ground in the field of mathematics. Aside from the excitement of mathematicians about the new proof, Jackson and Johnson received praise from notable figures like Michelle Obama. Although the reaction to these young mathematicians’ proof has been universally positive, they acknowledge that they surprised most of the world. When Bill Whitaker asked during a 60 Minutes interview why their contribution shocked the public, Jackson responded, “Probably because we’re African American, one, and we’re also women and, oh, our age.” Jackson went on to say, “I’d like for it to be celebrated for what is; it’s a great mathematical achievement.” It’s clear that Jackson and Johnson’s achievement is a testament not to their race, gender or age, but to their ingenuity and hard work.
