Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (17) |
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This document in PDF: Visualization.pdf

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:

- A number , the order relation and the absolute value of a
difference .
- Intervals such as
,
,
,
and
.
- A point in the plane. (Notice the sad clash of notation).
- The graphs of the functions , and
.
- The Euclidean distance function
.
- The parabola and the graphs of for several 's.
- The graphs of
,
,
and
.
- The graphs of
,
,
and
.
- The graphs of
,
and
.
- 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