Portal:Chapter 2: An ancient theorem and a modern question

Description goes here.

<< Previous Chapter Table Of Contents  next chapter >>

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:

$$ 2^3 = 2 \cdot 2 \cdot 2 $$

and:

$$ 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 $$

Multiplying these together we also see that:

$$ 2^3 \cdot 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^8$$

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

$$ 2^3 \cdot 2^5 = 2^{3+5} $$

Or more generally:

$$ 2^a \cdot 2^b = 2^{a+b} $$

Now, you may notice that this doesn't help if we are interested in numbers like 2^{\frac{1}{2}}<\math> or 2^{-1}<\math>. This is covered in the [[Recommended]| the recommended section] but is not strictly necessary for 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