Louisiana students Ne鈥橩iya Jackson and Calcea Johnson wowed their teachers in 2022 when they discovered a new way to prove the 2000-year-old Pythagorean theorem in response to a bonus question in a high school math contest. But that was only the beginning.
A volunteer at their former school, New Orleans鈥 St. Mary鈥檚 Academy, encouraged them to submit their work on the famous mathematical theory to a professional conference, and in March 2023 they became the youngest people to present at the American Mathematical Society鈥檚 Southeastern Sectional conference in Atlanta. Their appearance elicited a wave of media coverage, including .鈥 The pair also received to the city of New Orleans and
Now Jackson and Johnson, who started college last year, have notched another achievement: authoring an academic paper detailing their original proof 鈥 plus nine more. Their work published Monday in the scientific journal .
鈥淚鈥檓 so surprised that we鈥檙e getting published in a paper at such a young age,鈥 Johnson, a sophomore studying environmental engineering at Louisiana State University, said in a shared by the journal publishers.
鈥淚 didn鈥檛 think it would go this far,鈥 said Jackson, who is pursuing a doctoral degree in pharmacy at Xavier University of Louisiana.
Pythagoras was an ancient Greek philosopher and mathematician who lived 2,500 years ago. It鈥檚 not clear whether he or his disciples came up with his eponymous theorem. But the principle is a mainstay of math class and enables the calculation of the length of any side of a right-angled triangle 鈥 as long as the lengths of the other two sides are known. It鈥檚 usually summarized as a^2 + b^2 = c^2, in which the square of the longest side (the hypotenuse) is equal to the squares of the two shorter sides added together.
Many mathematicians over the years have explained why the theorem works using algebra and geometry. In contrast, Jackson and Johnson were able to prove it using trigonometry 鈥 a subbranch of mathematics that focuses on the study of triangles.
Experts described Jackson and Johnson鈥檚 approach as particularly challenging because trigonometry as a field is essentially based on Pythagoras鈥 theorem; thus using trigonometry to prove the theorem typically requires what mathematicians call 鈥渃ircular reasoning.鈥 However, the new proofs aren鈥檛 circular, according to the study.
鈥淣one of the theorems we use in our proofs 鈥 have already assumed the Pythagorean theorem to be true,鈥 the authors wrote in the paper.
Tom Murdoch, an honorary professor at the University of Bristol鈥檚 School of Mathematics in the United Kingdom, described the study as 鈥渋mpressive.鈥
鈥淚 think what鈥檚 intriguing about it (this study) is that a lot of people thought it was impossible,鈥 he said.
鈥淭rigonometric functions are based on sine and cosine, which are expressed as ratios of certain lengths of a right-angled triangle. It鈥檚 quite easy to get into a circular argument, and what鈥檚 so appealing about this is they found a line of argument using sine and cosine that doesn鈥檛 assume Pythagoras is true,鈥 he added.
Jackson and Johnson鈥檚 study outlines five new ways of proving the theorem using trigonometry, and their method reveals five more proofs for 10 total. The pair presented only one of the proofs at the 2023 conference, meaning that nine are totally new.
鈥淚 think sometimes having very little knowledge of the problem means that you are not bound by what鈥檚 gone before. Looking at this with fresh eyes, which I think they鈥檝e done, that鈥檚 the really impressive thing,鈥 Murdoch said.
Preparing the academic paper while they started college was 鈥渢he most daunting task of all, since we had absolutely no experience writing for an academic journal,鈥 the pair noted in the study. Their road to publication included a standard hurdle known as peer review, or the stage during which eminent scientists in the field examine and comment on the work.
Johnson said she was happy to be able to show women and women of color in particular what can be achieved in a traditionally male-dominated field.
鈥淚 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, and to let other young women know that they are able to do whatever they want to do. So that makes me very proud to be able to be in that position,鈥 Johnson said
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