Two high school students proved the Pythagorean theorem in a way that early 20th century mathematicians thought impossible.
Calcea Johnson and Ne’Kiya Jackson, both at St. Mary’s Academy in New Orleans, presented their work last month at a meeting of the American Mathematical Society. Johnson told her WWL-TV, her CBS affiliate in New Orleans, “Honestly, it’s an incomparable feeling. There’s nothing like it. It’s not like young people are thought to be able to do it.” I can do it,” he said.
If tested, Johnson and Jackson’s proof would contradict mathematician and educator Elisha Loomis in his 1927 book. Pythagorean proposition The trigonometry proof of the Pythagorean theorem is incorrect. Their work joins several other trigonometry proofs that have been added to the mathematical archives over the years. Each avoided “circular logic” to prove an important theorem. So what exactly is the trigonometry proof of the Pythagorean theorem, and why was his Loomis so closed to the idea?
The Pythagorean theorem provides a formula for calculating the long side of a right triangle by summing the squares of the other two sides.is often expressed as a2 + b2 = c2. In this expression, a, b and c Represents the lengths of the three sides of a right triangle (a triangle with a 90 degree angle between the two sides).quantity c The length of the longest side, called the hypotenuse. This theorem is named after the ancient Greek philosopher Pythagoras, but some historians believe it was known in Babylon about 1,000 years ago.
The theorem “brings algebra and geometry together,” says Stuart Anderson, professor emeritus of mathematics in the Texas A&M University School of Commerce. “The statement a2 + b2 = c2, which is an algebraic statement. But the figure it’s based on is geometric. ”
Trigonometry, on the other hand, focuses on angle-dependent functions. These functions, such as sine and cosine, are defined using right triangles. Imagine a right triangle with one side lying flat against the table and the other side perpendicular to the first side and straight up. The hypotenuse reaches diagonally between these two sides.
Then measure the angle between the hypotenuse and the table. Mathematicians define the sine of this angle as the height of the vertical side divided by the length of the hypotenuse. The cosine of this angle is the length of the horizontal side divided by the hypotenuse. Therefore, the Pythagorean theorem is equivalent to the equation sin.2 X +cos2 X = 1. “Many of the basic trigonometric theorems are just Pythagorean theorems,” explains Anderson, referring to equations that describe the relationships between various trigonometric functions.
Loomis believed that if these functions were used to prove the Pythagorean theorem, he would have assumed the theorem from the beginning.
But that’s not necessarily true. In a talk at the American Mathematical Society conference, Jackson and Johnson said that the trigonometric identity, called the law of sine, does not depend on the Pythagorean theorem and can be used to prove the theorem.
Anderson hopes that Jackson and Johnson’s proof will increase interest in mathematics among students. says.
Other trigonometric proofs of previous theorems include several described on mathematician Alexander Bogomolny’s website. One of these he was created by Jason Zimba, then a physicist and mathematician at his College of Bennington, geometry forum This proof used trigonometric identities that allow us to compute the cosine and sine of an angle. X – y Without using the Pythagorean theorem—if you know cosine and sine X and y By myself.
On October 26, 2009, Bogomolny added Zimba’s proof to his website, writing: error. “Over time, Bogomolny added more trigonometry proofs to the site. One such proof he could write in just four lines.
This story shows that even the simplest mathematics can surprise us. “I think mathematicians have learned not to make bold claims that something is impossible, because we do it over and over again over the years. I was embarrassed because I was so embarrassed,” says Anderson.
The American Mathematical Society encourages New Orleans students to submit proofs for publication in peer-reviewed scientific journals.
