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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)$.

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Dror Bar-Natan 2003-09-29