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

<math> 2^3 = 2 \cdot 2 \cdot 2 </math>

<math> 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 </math>

<math> 2^3 \cdot 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2  = 2^8</math>

<math> 2^3 \cdot 2^5 = 2^{3+5} </math>

<math> 2^a \cdot 2^b = 2^{a+b} </math>

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| 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>===

=== 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 = 1R = 57.29^\circ </math>:

<math> \pi \cdot 57.29 = \pi r = 180^\circ </math>

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

<math> \pi - (\alpha + \beta + \gamma) = C \triangle </math>

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>.

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