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>. These cases are covered in the recommended section if you are interested but are not strictly necessary for understanding this chapter.


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

Essential

Recommended

Further Exploration