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> 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>

Now, you may notice that this doesn't help if we are interested in numbers like <math> 2^{\frac{1}{2}}<\math> or <math>2^{-1}<\math>. This is covered in the [[Recommended]| the recommended section] but is not strictly necessary for this chapter.

Preliminaries

Essential

Recommended

Further Exploration