Chapter 2: An ancient theorem and a modern question: Difference between revisions

From The Portal Wiki
Line 35: Line 35:


''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.''
''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.''
[[File:Pythagorean.svg|thumb|The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).]]


== Preliminaries ==
== Preliminaries ==

Revision as of 20:38, 16 May 2020

Description goes here.

<< Previous Chapter Table Of Contents next chapter >>

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

Essential

Recommended

Further Exploration