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 recommended section if you are interested but are not strictly necessary for understanding this chapter.

Pythagorean Theorem [math]\displaystyle{ a^2 + b^2 = c^2 }[/math]

For any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Here are some animated proofs as well as a quiz to test your understanding.

Euclidian Geometry

This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows.

Euclidian Postualtes

1. A straight line segment can be drawn joining any two points
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having that segment as its radius
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

A good video explaining these concepts can be found here.

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

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 and negative powers
  • Understanding Exponents (Why does [math]\displaystyle{ 0^0 }[/math]=1)?
  • What Do Fractional Exponents Mean?
  • Netaive Exponents and the Decimal System
• 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