Editing Chapter 2: An ancient theorem and a modern question

Exponents can be thought of as repeated multiplication, meaning:

Now, you may notice that this doesn't help if we are interested in numbers like <math>2^{\frac{1}{2}}</math> or <math>2^{-1}</math>. These cases are covered in the recommended section if you are interested but are not strictly necessary for understanding this chapter.

=== Pythagorean Theorem <math>a^2 + b^2 = c^2</math> ===

# A straight line segment can be drawn joining any two points.

# Given any straight line segment, a circle can be drawn having that segment as its radius.

Euclid's fifth postulate cannot be proven as a theorem (by assuming only the first four), 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> ===

 

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:

So just remember, <math>\pi = 180^\circ</math>. Further explanations are given in the [[Preliminaries| preliminaries]] section.

 

A [https://en.wikipedia.org/wiki/Geodesic geodesic] is a curve representing the shortest path between two points in a space. It is a generalization of the notion of a "straight line". In a "flat" space, the straight line is indeed the shortest distance between two points, but in a curved space, this no longer holds true; the shortest distance between two points inherits some of the curvature from the space in which it exists. Geodesics are well explained in the videos pertaining the hyperbolic geometry in the essential section.

A type of geometry which can emerge when the fifth postulate is no longer taken to be true. Objects like triangles obey different rules in this type of geometry. For instance, [https://en.wikipedia.org/wiki/Hyperbolic_triangle hyperbolic triangles] have angles which sum to '''less''' than <math>\pi</math> radians. In fact, we have we have a triangle with an area represented by <math>\triangle</math> and three angles represented by <math>\alpha, \beta, \gamma</math> then by the ''Johann Heinrich Lambert formula'':

where <math>C</math> is just some constant determined by the ''units'' by which we measure a give length or area. The ''units'' we use can always be chosen such that <math>C=1</math>.

* An addictive puzzle game where you do Euclidean constructions

* Follow a steppable motion graphic that proves the Pythagorean theorem

** [https://timalex.github.io/royal-road/squareangle/ Squaring a triangle] by Community Contributor @TimAlex

** [https://betterexplained.com/articles/understanding-exponents-why-does-00-1/ Understanding Exponents (Why does <math>0^0=1)</math>?]

** [https://www.youtube.com/watch?v=jG7vhMMXagQ Pi Is (still) Wrong.]