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“Mind-Blowing”: Young Mathematicians Publish 10 New Proofs for Pythagorean Theorem

Ne'Kiya Jackson and Calcea Johnson
Photo by Calcea Johnson

Two young mathematicians from Louisiana, Ne’Kiya Jackson and Calcea Johnson, have once again astonished the math community with their groundbreaking work on the Pythagorean theorem. After making headlines in 2022 with their initial, seemingly impossible proof using trigonometry, the duo has now presented nine additional proofs, further cementing their place in mathematical history, reported by Live Sciences.

As high school students, Jackson and Johnson tackled the Pythagorean theorem, which asserts that the square of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two sides. Traditionally, mathematicians believed that proving this theorem using trigonometry was impossible since trigonometric formulas rely on the theorem’s validity. However, the students proved otherwise when they answered a bonus question in a school math contest. Their work was initially presented at an American Mathematical Society meeting in 2023, but it had not yet undergone thorough scrutiny.

A breakthrough came when a new paper, published in the American Mathematical Monthly on October 28, confirmed their proof had successfully passed peer review. Not only did their original proof hold up, but Jackson and Johnson also unveiled nine more proofs using trigonometry. “To have a paper published at such a young age — it’s really mind-blowing,” Johnson, now studying environmental engineering at Louisiana State University, expressed in a statement. She added, “I am very proud that we are both able to be such a positive influence in showing that young women and women of color can do these things.”

In their approach, the young mathematicians circumvented a logical fallacy known as circular reasoning. While trigonometry often incorporates the Pythagorean theorem, Jackson and Johnson’s proof utilized the Law of Sines, avoiding the need to assume the theorem’s truth. Their findings, as published, include four entirely new trigonometric proofs, as well as an innovative method that produced five additional proofs, bringing their total to ten.

Remarkably, Jackson and Johnson are only the third and fourth individuals known to have proven the Pythagorean theorem using trigonometry without circular reasoning. The other two were professional mathematicians. “I didn’t think it would go this far,” Jackson, now a pharmacology student at Xavier University of Louisiana, admitted. “I was pretty surprised to be published.”

In their paper, the students highlight two distinct ways to present trigonometric functions such as sine and cosine. They argue that these methods are often mistakenly merged into one, creating confusion. By separating these methods, they were able to identify multiple new ways to prove the Pythagorean theorem. “Trying to make sense of trigonometry can be like trying to make sense of a picture where two different images have been printed on top of each other,” they wrote. This separation, they concluded, opens the door to discovering “a large collection of new proofs of the Pythagorean theorem.”

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