Chapter 2: An ancient theorem and a modern question: Difference between revisions
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<math> 2^a \cdot 2^b = 2^{a+b} </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| | 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 but if you are interested but are not strictly necessary for understanding this chapter. | ||
== Preliminaries == | == Preliminaries == | ||
Revision as of 20:29, 16 May 2020
Description goes here.
Community Explanations
Translation
In Euclidean geometry, a translatio is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.
Exponents
Exponents can be though of as repeated multiplication, meaning:
[math]\displaystyle{ 2^3 = 2 \cdot 2 \cdot 2 }[/math]
and:
[math]\displaystyle{ 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 }[/math]
Multiplying these together we also see that:
[math]\displaystyle{ 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:
[math]\displaystyle{ 2^3 \cdot 2^5 = 2^{3+5} }[/math]
Or more generally:
[math]\displaystyle{ 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]\displaystyle{ 2^{\frac{1}{2}} }[/math] or [math]\displaystyle{ 2^{-1} }[/math]. These cases are covered in the recommended section but if you are interested but are not strictly necessary for understanding this chapter.
Preliminaries
- Know how to visually represent addition, subtraction, multiplication, and powers
- Know what squares (powers of two) and square roots are
- Know what logarithms are
- Know what an equation and the solution of an equation is (note that an equation can have more than one solution!)
- Now tie it all together
- And quick a introduction to radians
Essential
- An additcting puzzle game where you do Euclidian constructions
- An animated version of a proof of the Pythagorean Theorem
- Pythagorean Theorem Proof by Community Contributor @TimAlex
- Hyperbolic geometry
Recommended
- Understanding fractional and negative powers
- A more in-depth description of the logarithms and exponents with applications
- For those who want an additional explanation of radians
- For those who want an additional explanation of radians and are mad about it
- A spot of linear algebra
Further Exploration
- To understand what geometry really is
- The Four Pillars of Geometry by John Stillwell
- A guide through Euclid's Elements
- A more in depth introduction to linear algebra
- Linear Algebra Done Right by Sheldon Axler