Difference between revisions of "Chapter 2: An ancient theorem and a modern question"

Chapter 2: An ancient theorem and a modern question (view source)

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 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.)
+=== Radians and <math> \pi </math> ===
+<math> \pi </math> is introduced in the books as the sum of all angles of a triangles, which is <math> 180^\circ</math>. This might be confusing to those who know that <math> \pi = 3.14 \cdots </math>.
+The explanation for this is simple. <math> \pi </math> is simply used as a shorthand for <math> \pi R </math> where <math> R </math> stands for radian. An arc of a circle with the same length as the radius of that circle subtends an '''angle of 1 radian''' (roughly 57.29). Adding three radians together brings you almost '''180 degrees''' around. <math> \pi </math> radians brings you ''exactly'' 180 degrees around. The circumference subtends an angle of <math> 2\pi </math>. To summarize:
+<math> 1 radian\pi </math>
 == Preliminaries ==