Chapter 2: An ancient theorem and a modern question
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> 2^3 = 2 \cdot 2 \cdot 2 </math>
and:
<math> 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 </math>
Multiplying these together we also see that:
<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:
<math> 2^3 \cdot 2^5 = 2^{3+5} <\math>
Or more generally:
<math> 2^a \cdot 2^b = 2^{a+b} <\math>
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 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