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

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

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Exponents can be thought of as repeated multiplication, meaning:
Exponents can be thought of as repeated multiplication, meaning:


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


and:  
and:  


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


Multiplying these together we also see that:
Multiplying these together we also see that:


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


This is known as the additive property of exponentiation. It can be written as:
This is known as the additive property of exponentiation. It can be written as:


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


Or more generally:
Or more generally:


$$ 2^a \cdot 2^b = 2^{a+b} $$
<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 \( 2^{\frac{1}{2}}\) or \(2^{-1}\). These cases are covered in the recommended section if you are interested but are not strictly necessary for understanding this chapter.
Now, you may notice that this doesn't help if we are interested in numbers like \( 2^{\frac{1}{2}}\) or \(2^{-1}\). These cases are covered in the recommended section if you are interested but are not strictly necessary for understanding this chapter.
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The explanation for this is simple. \( \pi \) is simply used as a shorthand for \( \pi R \) where \( R \) 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. \( \pi \) radians brings you ''exactly'' 180 degrees around. The circumference subtends an angle of \( 2\pi \). To summarize:
The explanation for this is simple. \( \pi \) is simply used as a shorthand for \( \pi R \) where \( R \) 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. \( \pi \) radians brings you ''exactly'' 180 degrees around. The circumference subtends an angle of \( 2\pi \). To summarize:
$$ 1 Radian = 1R = 57.29^\circ $$:
<math> 1 Radian = 1R = 57.29^\circ </math>:
$$ \pi \cdot 57.29 = \pi r = 180^\circ $$
<math> \pi \cdot 57.29 = \pi r = 180^\circ </math>


So just remember, \( \pi = 180^\circ \). Further explanations are given in the [[Preliminaries| preliminaries]] section.
So just remember, \( \pi = 180^\circ \). Further explanations are given in the [[Preliminaries| preliminaries]] section.
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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 \( \pi \) radians. In fact, we have we have a triangle with an area represented by \( \triangle \) and three angles represented by \( \alpha, \beta, \gamma \) then by the ''Johann Heinrich Lambert formula'':
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 \( \pi \) radians. In fact, we have we have a triangle with an area represented by \( \triangle \) and three angles represented by \( \alpha, \beta, \gamma \) then by the ''Johann Heinrich Lambert formula'':


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


where \( C \) 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 \( C=1\).
where \( C \) 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 \( C=1\).
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