Chapter 1: A New Perspective on the Pythagorean Theorem

Imagine the thrill of proving a core mathematical theorem once thought to be unprovable. This is precisely what high school students Calcea Johnson and Ne’Kiya Jackson achieved, overturning long-held beliefs about the Pythagorean theorem at St. Mary’s Academy in New Orleans. Their groundbreaking work was recently presented at a distinguished meeting of the American Mathematical Society, where they expressed their excitement about accomplishing what many believed was beyond the capabilities of young scholars.

The Pythagorean theorem, a formula for determining the longest side of a right triangle, is typically represented as a² + b² = c², with a, b, and c symbolizing the triangle's side lengths. In contrast, trigonometry focuses on angle-dependent functions, such as sine and cosine, which are derived from right triangles. This theorem serves as a bridge between algebra and geometry, as its algebraic expression a² + b² = c² originates from a geometric principle.

In the early 20th century, mathematician Elisha Loomis contended that any proof of the Pythagorean theorem using trigonometric functions would lead to circular reasoning, presupposing the theorem itself, which he deemed a mathematical fallacy. However, at the American Mathematical Society meeting, Johnson and Jackson introduced a fresh perspective by employing a trigonometric identity known as the law of sines, asserting that it did not rely on the Pythagorean theorem, thus making it a valid means of proving it.

This isn’t the first occurrence of trigonometric proofs for the Pythagorean theorem. Mathematician Alexander Bogomolny’s website showcases numerous such proofs, including one from physicist Jason Zimba, published in Forum Geometricorum in 2009. Zimba's work involved using a trigonometric identity to calculate cosine and sine of an angle without invoking the Pythagorean theorem. Bogomolny, who initially doubted the possibility of a trigonometric proof, acknowledged his oversight and included Zimba’s proof on his site along with additional subsequent proofs.

The journey of Johnson and Jackson’s extraordinary accomplishment highlights that even in mathematics, a field where certainty and proof reign supreme, unexpected developments can arise. Stuart Anderson, an emeritus professor of mathematics at Texas A&M University-Commerce, aspires that their proof will spark a renewed interest in mathematics among students, encouraging further inquiry and innovation in the discipline. Supported by the American Mathematical Society, Johnson and Jackson are preparing to submit their proof for publication in a peer-reviewed journal, marking a pivotal moment in the exploration of the Pythagorean theorem and its link to trigonometry.

As emerging mathematicians continue to challenge established norms and expand the horizons of mathematical understanding, the narrative of Johnson and Jackson's pioneering achievement exemplifies the impact of curiosity, determination, and innovation in the realm of mathematics.

New Orleans teens make mathematical discovery unproven for 2,000 years - This video explores how two high school students challenged long-standing mathematical beliefs by proving the Pythagorean theorem in a novel way.

Section 1.1: Shattering Preconceptions

The remarkable success of these students serves as a powerful reminder that the world of mathematics is dynamic and full of surprises.

Subsection 1.1.1: Visual Representation of Their Journey

Chapter 2: The Legacy of Their Discovery

What did we all miss in this 2000 year old problem? - This video discusses the implications of the students' proof and what it means for the future of mathematical education and exploration.