added stereographic projection

1 files changed, 9 insertions, 5 deletions

diff --git a/ClassJune28.page b/ClassJune28.page
index bb2a229..f2b1c23 100644
--- a/ClassJune28.page
+++ b/ClassJune28.page
@@ -6,7 +6,7 @@
 
 A Fourier series is a function of the form 
 $$\sum_{n=-\infty}^\infty a_n e^{inx}$$ or $$\sum_{n=-\infty}^\infty b_n \cos(n x) + c_n \sin (n x),$$
-depending on one's taste for the imaginary. The rumor on the street is that any periodic function (well, any nice one) can be expressed as a Fourier series: you hand me a function $f:[0,2\pi]\rightarrow \mathbb{C}$ and I hand you a list of real numbers $b_0,c_0,b_1,c_1,b_2,c_2,b_3,c_3,\dots$ such that
+depending on one's taste for the imaginary. The rumor on the street is that any periodic function (well, any nice one) can be expressed as a Fourier series: you hand me a function $f:[0,2\pi]\rightarrow \mathbb{C}$ and I hand you a list of real numbers $\{b_0,c_0,b_1,c_1,b_2,c_2,b_3,c_3,\dots\}$ such that
 
 $$f(x) = \sum_{n=-\infty}^\infty b_n \cos(n x) + c_n \sin (n x).$$
 
@@ -14,13 +14,17 @@ If this exchange is always possible, there must be at least as many different li
 
 Infinity, of course. So to perform a meaningful check that the set of nice periodic functions is the same size as the set of lists of real numbers, we need a more refined notion of a size of a set than just giving its number of elements, or saying that it is infinite. We say that two sets have the same cardinality (our new word for size) if we can pair off their elements, giving a perfect matching between the sets.
 
-For example, the open interval $(0,1)$ has the same cardinality as $\mathbb R$. To see this, consider 
+For example, the open interval $(-\pi,\pi)$ has the same cardinality as $\mathbb R$. To see this, take the interval and bend it into a semicircle, then balance it on the real line. To find the point on the real line corresponding to a given number in the interval/on the semicircle, $\theta \in (-\pi,\pi)$, draw the straight line emanating from the centre of the circle at an angle $\theta$ with the vertical; it hits $\theta$ on the semicircle and its partner $x$ on the real line. Explicitly, the map takes $\theta \mapsto \tan \theta$.
 
-## To show that $(0,1) \sim \mathbb R$
+![alt text](/stereographic circle.jpg)
 
+There are, however, fewer natural numbers in $\mathbb N = \{1,2,3,\dots\}$ than there are real numbers in $\mathbb R$. Suppose to the contrary that it is possible to list the real numbers, or even just those in the interval $(0,1)$. Write that list down in binary, forming a big block extending infinitely downward (each row is a number), and to the right (the binary digits of each number). Read off the digits on the diagonal to get a number $x=0.010100110101110101011\dots$. Now toggle every digit of that number to get a new number $y$, say $0.101011001010001010100\dots$. This number $y$ is not anywhere on the list--it disagrees with the $n^{th}$ number on the list in its $n^{th}$ digit. Hence the list was incomplete.
 
-## Cantor's proof for $\mathbb R > \mathbb N$
-Suppose that we had a bijection between $\mathbb N$ and $\mathbb R$. Since $\mathbb R \sim \mathbb N$ In other words, we have a list of all the real numbers
+A set is called countable if it has the same cardinality as the natural numbers, if its elements can be placed in an infinite list. The real numbers, we've just seen, are not countable.
+
+Cardinality holds some surprises, though. For example, there are as many pairs of natural numbers as there are natural numbers; the set $\mathbb{N}\times\mathbb{N}$ is countable. They can be enumerated by following the zig-zag path through the cartesian grid of pairs below.
+
+![alt text](/orderedpairs.0.png)
 
 1-->0.
 ## Proof that no. of available functions is greater than number of functions required to define the periodic function