83
edits
Difference between revisions of "Chapter 2: An ancient theorem and a modern question"
Jump to navigation
Jump to search
Chapter 2: An ancient theorem and a modern question (view source)
Revision as of 20:45, 16 May 2020
, 20:45, 16 May 2020→Community Explanations
| Line 37: | Line 37: | ||
Here are some [https://www.youtube.com/watch?v=COkhrDbNcuA animated proofs] as well as a [https://www.khanacademy.org/math/basic-geo/basic-geometry-pythagorean-theorem/geo-pythagorean-theorem/e/pythagorean_theorem_1 quiz] to test your understanding. | Here are some [https://www.youtube.com/watch?v=COkhrDbNcuA animated proofs] as well as a [https://www.khanacademy.org/math/basic-geo/basic-geometry-pythagorean-theorem/geo-pythagorean-theorem/e/pythagorean_theorem_1 quiz] to test your understanding. | ||
=== Euclidian Geometry === | |||
This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows. | |||
==== Euclidian Postualtes ==== | |||
# A straight line segment can be drawn joining any two points | |||
# Any straight line segment can be extended indefinitely in a straight line. | |||
# Given any straight line segment, a circle can be drawn having that segment as its radius | |||
# All right angles are congruent. | |||
# If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. | |||
A good video explaining these concepts can be found [https://www.youtube.com/watch?v=gLMIFRLw9LU here]. | |||
Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.) | |||
== Preliminaries == | == Preliminaries == | ||