A brief ebook about the mathematician Pythagoras and the theorem that shares his name.
Pythagoras and the Pythagorean Theorem
By Dylan Olson
Pythagoras of Samos lived over two and a half thousand years ago in ancient Greece, and is one of the earliest known mathematicians. For how important he is to the field of mathematics, not much is actually known about the man, most of what we know of him comes from writing from others as he left no mathematical writings himself. In around 530 BCE he founded a school in Croton called the Pythagorean School, which led to the creation of a group of mathematicians and philosophers called the Pythagoreans, who tried to find ways to represent everything in the universe in terms of mathematics. They were actually quite mystical in nature, with a variety of strange, religious and ritualistic aspects to their practices.
Pythagoras is known for contributing several different concepts to mathematics. His Pythagoreans used the ‘tetractys’ as a mystical symbol of worship. The tetractys is a triangular shape that consists of four rows that add to the number 10, which was considered by the Pythagoreans to be the perfect number. There is some apocryphal evidence to suggest that Pythagoras was also responsible for discovering properties of string length and how different musical notes are created by differences in string length and vibration.
Regardless of their methodology, however, Pythagoras and the Pythagoreans were responsible for introducing more rigor to the field of mathematics than what had existed beforehand. Prior to Pythagoras and the other Greek mathematicians that came after him mathematics was generally only looked at through the lens of how it could be applied (primarily through taxation), whereas Pythagoras studied math for the sake of advancing mathematics itself, building principles of mathematics using rules called ‘axioms’ and the process of logical thinking.
To demonstrate the use of this formula, we’ll look at the following problem:
Perhaps the thing Pythagoras is most well remembered for, however, is the theorem that bears his name: the Pythagorean theorem. Briefly summarized, it states that: ‘in a right triangle, the sum of the squares of the two legs are equivalent to the square of the hypotenuse.’ In mathematics, this is often symbolically represented by the formula to the right.
We will use the formula above, substituting 4 and 3 for a and b respectively. We then get that a2 + b2 = 42 +32 = 16 + 9 = 25 = c2. At this point we have our c2 value, but to find c we take the square root of 25 and see that we get a value of 5 for c. Aside from finding the answer to the problem, there is another interesting thing to note about this result. The fact that the sum of the two squares gives us another perfect square tells us that the numbers 3, 4, and 5 are what we call a Pythagorean Triple.
The Pythagorean Theorem was one of the most fundamental results in what would become Euclidean Geometry. There is evidence that the relationship characterized by the theorem was actually known by other societies before the time of Pythagoras. The Babylonians seem to have had some knowledge of the formula, and the cultures of Mesopotamia, India, and China all discovered the theorem independent of one another, but Pythagoras is credited with providing the first recorded ‘proof’ of the theorem, a proof being a way to say that the result is always true. It has since been proven in a numerous amount of ways, a book called The Pythagorean Proposition documents 367 different proofs of the theorem, more than any other result in mathematics.
Chinese geometric proof of Pythagorean Theorem, 1st Century BCE.
Obviously, the sheer number of proofs of the Pythagorean Theorem, coupled with the fact that new proofs are still being developed (a new proof was proposed as recently as 2012), is solid evidence of how important Pythagoras and his theorem was to the study and development of mathematics.
Pretty impressive for a guy who might not have even existed!
Swayne, Steve. The Parthenon Athens. August 26, 1978. Flickr. Parthenon - Wikipedia. Photograph. March 16, 2017. https://upload.wikimedia.org/wikipedia/commons/d/da/The_Parthenon_in_Athens.jpg
Galilea. "Bust of Pythagoras." May 7, 2005. The Capitoline Museums. Pythagoras - Wikipedia. Photograph. March 16, 2017. https://upload.wikimedia.org/wikipedia/commons/1/1a/Kapitolinischer_Pythagoras_adjusted.jpg
Mastin, Luke. The Pythagorean Tetractys. 2010. Unknown. Pythagoras - Greek Mathematics - The Story of Mathematics. Digital. March 16, 2017. http://www.storyofmathematics.com/images2/pythagoras_tetractys.gif
Unknown. The Formula/Theorem. Unknown. Unknown. The Pythagorean Theorem: a necessity for Math. Digital. March 16, 2017. https://slate.adobe.com/a/WAVdV/images/20A6996F-CCF0-4423-8E4E-132BC5343D22.jpg
Olson, Dylan. "A Problem using the Pythagorean Theorem." 2017. PNG file type.
Unknown. Geometric proof of the Pythagorean theorem from the Zhoubi Suanjing. April 11, 2005. Unknown. Zhoubi Suanjing - Wikipedia. Ink. March 16, 2017. https://upload.wikimedia.org/wikipedia/commons/c/c3/Chinese_pythagoras.jpg
Prassad, C. Pythagoras. January, 2009. Unknown. The Attic of Gallimaufry. Pencil. March 16, 2017. http://2.bp.blogspot.com/_jeSuLa5ZscU/TDenS0PBd-I/AAAAAAAAlp8/A089efUzSu4/s1600/pythagoras.JPG