Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I:

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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 $a$, the order relation $a<b$ and the absolute value of a difference $\vert a-b\vert$.

2. Intervals such as $(a,b):=\{x: a<x<b\}$, $[a,b):=\{x: a\leq x<b\}$, $[a,b]:=\{x: a\leq x\leq b\}$, $(a,\infty):=\{x:x>a\}$ and $(-\infty,a]:=\{x:x\leq a\}$.

3. A point $(a,b)$ in the plane. (Notice the sad clash of notation).

4. The graphs of the functions $f_1(x)=c$, $f_2(x)=cx$ and $f_3(x)=cx+d$.

5. The Euclidean distance function $d((a,b), (c,d)):=\sqrt{(a-c)^2+(b-d)^2}$.

6. The parabola $y=x^2$ and the graphs of $f(x)=x^n$ for several $n$'s.

7. The graphs of $f_1(x)=\frac{1}{x}$, $f_2(x)=\frac{1}{x^2}$, $f_3(x)=\frac{1}{1+x^2}$ and $f_4(x)=\frac{x}{1+x^2}$.

8. The graphs of $f_1(x)=\sin x$, $f_2(x)=\sin\frac{1}{x}$, $f_3(x)=x\sin\frac{1}{x}$ and $f_4(x)=x^2\sin\frac{1}{x}$.

9. The graphs of $f_1(x)=\begin{cases}x^2 & x<1 2 & x\geq 1\end{cases}$, $f_2(x)=\begin{cases}x^2 & x\leq 1 2 & x>1\end{cases}$ and $f_3(x)=\begin{cases}1 & x\in{\mathbb{Q}} 0 & x\not\in{\mathbb{Q}}\end{cases}$.

10. The circle $(x-a)^2+(y-b)^2=r^2$, the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

Just for fun. For $x\in[0,1]$, the number $f(x)$ is defined to be the result of the following process: Write $x$ in binary, replace every $1$ in the resulting expansion by a $2$, and interpret the result as a number written in base 3. For example, $x=\frac13=0.01010101_2\ldots\to 0.02020202_3\ldots=\frac14=f(x)$.

• Draw the graph of $f$.
• Draw the range of $f$ as a subset of $[0,1]$. (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 $\frac{\log 2}{\log 3}$.)

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