From 397cefef7929e6cd49959db1d891f5b0654ebd05 Mon Sep 17 00:00:00 2001 From: bryan newbold Date: Tue, 10 Jun 2008 10:48:24 -0400 Subject: added a bunch of math content based on alaska notes. added more ethernet content --- math/numbers | 54 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 54 insertions(+) create mode 100644 math/numbers (limited to 'math/numbers') diff --git a/math/numbers b/math/numbers new file mode 100644 index 0000000..541d174 --- /dev/null +++ b/math/numbers @@ -0,0 +1,54 @@ +======================== +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. -- cgit v1.2.3