Chapter 2: An ancient theorem and a modern question

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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 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
  • Rethinking Arithmetic: A Visual Guide
• Know what squares (powers of two) and square roots are
  • Squares and Square Roots
• Know what logarithms are
  • Logarithms, Explained
• 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
  • Triangle of Power
• And quick a introduction to radians
  • What are radians? Simply explained

Essential

• An additcting puzzle game where you do Euclidian constructions
  • Euclidia
• An animated version of a proof of the Pythagorean Theorem
  • Pythagorean Theorem Proof by Community Contributor @TimAlex
• Hyperbolic geometry
  • Playing Sports in Hyperbolic Space
  • Ditching the Fifth Axiom

Recommended

• Understanding fractional powers
  • What Do Fractional Exponents Mean?
• A more in-depth description of the logarithms and exponents with applications
  • Logarithm Fundamentals
• For those who want an additional explanation of radians
  • https://www.youtube.com/watch?v=tSsihw-xPHc
• For those who want an additional explanation of radians and are mad about it
  • Pi Is (still Wrong).
• A spot of linear algebra
  • The Essence 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
  • Euclid’s Elements
• A more in depth introduction to linear algebra
  • Linear Algebra Done Right by Sheldon Axler