# Visualization

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Our task for this week is to master the axiomatically meaningless task of visualization of numbers and functions. We will learn how to interpret graphically all of the following:

1. A number , the order relation and the absolute value of a difference .

2. Intervals such as , , , and .

3. A point in the plane. (Notice the sad clash of notation).

4. The graphs of the functions , and .

5. The Euclidean distance function .

6. The parabola and the graphs of for several 's.

7. The graphs of , , and .

8. The graphs of , , and .

9. The graphs of , and .

10. The circle , the ellipse and the hyperbola .

Just for fun. For , the number is defined to be the result of the following process: Write in binary, replace every in the resulting expansion by a , and interpret the result as a number written in base 3. For example, .

• Draw the graph of .
• Draw the range of as a subset of . (The answer, called the Cantor set'' plays a major role in much of analysis and in particular in the theory of fractals. In some sense its dimension is the irrational number .)

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Dror Bar-Natan 2003-09-29