Chapter 2: An ancient theorem and a modern question: Difference between revisions
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=== Pythagorean Theorem <math> a^2 + b^2 = c^2 </math>=== | === Pythagorean Theorem <math> 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.'' | |||
== Preliminaries == | == Preliminaries == | ||
Revision as of 20:33, 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 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.
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