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

Preliminaries

Essential

Recommended

Further Exploration