From 397cefef7929e6cd49959db1d891f5b0654ebd05 Mon Sep 17 00:00:00 2001
From: bryan newbold <bnewbold@snark.mit.edu>
Date: Tue, 10 Jun 2008 10:48:24 -0400
Subject: added a bunch of math content based on alaska notes. added more
 ethernet content

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 math/numbers | 54 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
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+========================
+Numbers
+========================
+
+.. note::
+    incomplete
+
+.. note:: 
+    Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+
+.. contents::
+
+*incommensurable*
+    objects are incommensurable when their ratio isn't rational
+
+Real Numbers
+==================
+The *real numbers* are defined via Dedakind cuts in [rudin]_, or [meserve]_ 
+(1-12).
+
+Complex Numbers
+==================
+The *complex numbers* are constructed as an ordered pair of real numbers.
+
+Algebraic and Transendental Numbers
+===============================================
+*Algebraic numbers* are solutions of polynomials, such as x in
+:latex:`$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$`, where all a are
+real numbers. *Transcendental numbers* are not solutions to any such 
+polynomials.
+
+All real numbers are either algebraic or transcendental. 
+
+Some algebraic numbers aren't real (such as :latex:`$i = \sqrt{-1}$`). They
+can be rational or irrational. All transcendental numbers are irrational;
+some are not real.
+
+Exersize: is the square root of 5 algebraic or transcendental?
+
+e
+========
+:latex:`$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$`
+
+Infinities
+==================
+*aleph-zero* (:latex:`$\aleph_0$`) is the countably infinite set.
+
+Positive integers, integers, and rational numbers are all countably infinite.
+
+It is unproven that the real numbers are *aleph-one* (:latex:`$\aleph_1$`).
+
+.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976
+
+.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
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