(C) Daily Kos This story was originally published by Daily Kos and is unaltered. . . . . . . . . . . Math open thread: your favorite proofs of the Pythagorean theorem [1] ['This Content Is Not Subject To Review Daily Kos Staff Prior To Publication.', 'Backgroundurl Avatar_Large', 'Nickname', 'Joined', 'Created_At', 'Story Count', 'N_Stories', 'Comment Count', 'N_Comments', 'Popular Tags'] Date: 2023-04-25 Ne’Kiya Jackson (left), and Calcea Rujean Johnson stop for a photo at the welcome sign for this year's conference of the American Mathematical Society. Two black girls recently came up with a new proof of the Pythagorean theorem using trigonometry. Ne'Kiya Jackson and Calcea Johnson should be very proud of themselves. Their achievement should not be diminished by news outlets misunderstanding what the the two young mathematicians have accomplished. Some headline writers make it sound Jackson and Johnson proved something that had never been proved before, but in fact, several other mathematicians have proved it before in lots of different ways. A portrait of President James A. Garfield (R, 1881) overlaid with a trapezoid split into three triangles. In 1876, the future president published a new proof of the old Pythagorean theorem. Other headlines make it sound like Jackson and Johnson came up with the first ever trigonometric proof, which they haven’t, and they haven’t claimed they did. I also heard in the news on TV that they solved it without “circular geometry,” which was very confusing, because the theorem is about triangles, namely triangles with one right angle, not circles. The Pythagorean theorem states that, given a right triangle (a triangle with one interior angle being exactly 90 degrees), the square of the hypotenuse (the longest side of the triangle) is equal to the the sum of the squares of the other two sides. Consider for example a triangle with the hypotenuse measuring 5 units and the other two sides measuring 3 and 4 units respectively. Then we see that 32 + 42 = 9 + 16 = 25 = 52. However, trigonometry does have a lot to do with circles. When I think of sines, cosines and tangents, I think of circles, but these functions are trigonometric functions and they’re defined in terms of right triangles. In fact, the Pythagorean theorem is fundamental to trigonometry. So, to use trigonometry to prove the Pythagorean theorem would seem to be an example of what we colloquially call “circular reasoning.” It would be like trying to prove that zero exists by first assuming that it does exist. This might explain why some of the headline writers are getting confused: Jackson and Johnson have proven the Pythagorean theorem using trigonometry, but they have avoided the logical fallacy of circular reasoning. Jackson and Johnson were not the first to do this but it does look like only one other has done this before. There is a lot of value in proving things that have already been proven but in a different way. A new proof of an already proven theorem might be easier to understand than the original proof. For example, the Pythagorean theorem inspired Pierre de Fermat to make a conjecture centuries ago that became famous as his so-called “last theorem” and wasn’t proven until 1994 by Andrew Wiles. The Wiles proof of the Fermat conjecture requires many relatively new and advanced mathematical concepts that Fermat couldn’t have known about. I can barely comprehend that proof, so I take an expert’s word that the proof is valid. I would appreciate an elementary proof like the one Fermat claimed to have found. Given that x2 + y2 = z2 has many solutions in integers, like the aforementioned 32 + 42 = 52, Fermat wondered if equations with higher integer exponents, like x3 + y3 = z3 or x4 + y4 = z4, might also have solutions in integers. After pondering the question, Fermat’s intuition must have been that xn + yn = zn has no solutions in integers if n > 2. Whether or not he turned that intuition into a mathematically valid proof is a matter of speculation, but historical evidence strongly suggests he never did. However, Fermat did prove a theorem related to the Pythagorean theorem: x4 + y4 = z2 has no solutions in positive integers. In other words, if the side lengths of a right triangle are integers, the area of the triangle is not the square of a rational number. Demonstrations of theorems are also helpful. I might not understand every proof of the Pythagorean theorem, but I can understand water flowing from one tank to another. x YouTube Video The open thread question: What is your favorite proof or demonstration of the Pythagorean theorem? [END] --- [1] Url: https://www.dailykos.com/stories/2023/4/25/2165245/-Math-open-thread-your-favorite-proofs-of-the-Pythagorean-theorem Published and (C) by Daily Kos Content appears here under this condition or license: Site content may be used for any purpose without permission unless otherwise specified. via Magical.Fish Gopher News Feeds: gopher://magical.fish/1/feeds/news/dailykos/