Pythagorean Theorem: Difference between revisions

Latest revision as of 16:41, 19 February 2023

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

[math]\displaystyle{ a^{2}+b^{2}=c^{2} }[/math]

where [math]\displaystyle{ c }[/math] represents the length of the hypotenuse and [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the ancient Greek thinker Pythagoras.

Resources:[edit]

Pythagorean theorem

Discussion:[edit]

Anon: I spent a great deal of my career as a student at St. John's College SF thinking about this theorem in particular. To those out there who see this, I offer another portal. If you feel as though you lack the tools to begin your journey, if what you seek is somewhere to begin, I recommend that you attempt to digest the entire first book of Euclid's Elements. I suggest you take it seriously and discuss it with friends, but more importantly, I suggest you display the proofs on a chalkboard, without the book or notes, to anyone you can. When you can teach it to someone else you will have made it further than nearly anyone else, and you might be close to really understanding it. Euclid built one of the first "universal" geometrical models, and he did it with just a few definitions, some common notions, a straight-edge and a compass. It also took us about 1700 years to realize that we could build a complete model with even fewer givens, specifically postulate 5, allowing us our first glimpse of hyperbolic and elliptical space. enjoy!

https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html

@@ Line 2: / Line 2: @@
 [[File:Visual_proof_pythagoras.png|center|class=shadow|300px]]
 </div>
-From Wikipedia, the free encyclopedia
 In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":
-:$$ {a^{2}+b^{2}=c^{2},} {a^{2}+b^{2}=c^{2}}$$
+:<math>a^{2}+b^{2}=c^{2}</math>
-where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the ancient Greek thinker Pythagoras.
-The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.
+where <math>c</math> represents the length of the hypotenuse and <math>a</math> and <math>b</math> the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the ancient Greek thinker Pythagoras.
 == Resources:==
 *[https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]
 == Discussion:==
+Anon: I spent a great deal of my career as a student at St. John's College SF thinking about this theorem in particular. To those out there who see this, I offer another portal. If you feel as though you lack the tools to begin your journey, if what you seek is somewhere to begin, I recommend that you attempt to digest the entire first book of Euclid's Elements. I suggest you take it seriously and discuss it with friends, but more importantly, I suggest you display the proofs on a chalkboard, without the book or notes, to anyone you can. When you can teach it to someone else you will have made it further than nearly anyone else, and you might be close to really understanding it. Euclid built one of the first "universal" geometrical models, and he did it with just a few definitions, some common notions, a straight-edge and a compass. It also took us about 1700 years to realize that we could build a complete model with even fewer givens, specifically postulate 5, allowing us our first glimpse of hyperbolic and elliptical space. enjoy!
+https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html
+[[Category:Pages for Merging]]