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authorbryan newbold <bnewbold@snark.mit.edu>2008-06-10 10:48:24 -0400
committerbryan newbold <bnewbold@snark.mit.edu>2008-06-10 10:48:24 -0400
commit397cefef7929e6cd49959db1d891f5b0654ebd05 (patch)
treeec197475d62c9149288863f89c97e26131959815 /math
parent80b4628e5a94dfedd2c6ee91bdb608c531a78598 (diff)
downloadknowledge-397cefef7929e6cd49959db1d891f5b0654ebd05.tar.gz
knowledge-397cefef7929e6cd49959db1d891f5b0654ebd05.zip
added a bunch of math content based on alaska notes. added more ethernet content
Diffstat (limited to 'math')
-rw-r--r--math/algebra71
-rw-r--r--math/books to read33
-rw-r--r--math/good books5
-rw-r--r--math/integers5
-rw-r--r--math/logic18
-rw-r--r--math/numbers54
-rw-r--r--math/sets45
7 files changed, 226 insertions, 5 deletions
diff --git a/math/algebra b/math/algebra
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+====================
+Algebra
+====================
+
+.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+
+.. list-table:: Closure of binary operators on given sets of numbers
+ * Operation
+ - :latex:`$+$`
+ - :latex:`$\times$`
+ - :latex:`$-$`
+ - :latex:`$\divide$`
+ - :latex:`$^$`
+ - :latex:`$\sqrt{\text{ }}$`
+ * Positive Integers
+ - Y
+ - Y
+ - N
+ - N
+ - Y
+ - N
+ * Positive rationals
+ - Y
+ - Y
+ - N
+ - Y
+ - Y
+ - N
+ * Rationals (and zero)
+ - Y
+ - Y
+ - Y
+ - Y
+ - Y
+ - N
+ * Reals wrt positive integers
+ - Y
+ - Y
+ - Y
+ - Y
+ - Y
+ - Y
+ * Complex numbers
+ - Y
+ - Y
+ - Y
+ - Y
+ - Y
+ - Y
+
+Definitions
+=============
+
+*involution*
+ to raise a number to a given power
+
+*evolution*
+ to take a given root of a number
+
+*associative*
+ :latex:`$(a+b)+c=a+(b+c)$`
+
+*comutative*
+ :latex:`$a+b=b+c$`
+
+*distributive*
+ :latex:`$(a+b)c=ac+bc$`
+
+.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976
+
+.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.
diff --git a/math/books to read b/math/books to read
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+===============================================
+Math books that look interesting
+===============================================
+
+`On formally undecidable propositions of Principa Mathematica and related systems`:title:, by Kurt Godel.
+
+`Computability and Unsolvability`:title:, by Martin Davis.
+
+`Mathematical Foundations of Information Theory`:title:, by A.I. Khinchin.
+
+`Calculus of Variations with Applications to Physics and Engineering`:title:, by Robert Weinstock.
+
+`Relativity, Thermodynamics, and Cosmology`:title:, by Richard Tolman.
+
+`Mathematics Applied to Continuum Mechanics`:title:, by Lee Segel.
+
+`Optimization Theory and Applications`:title:, by Donald Pierre.
+
+`The Variational Principles of Mechanics`:title:, by Cornelius Lanczos.
+
+`Tensor Analysis for Physicists`:title:, by J.A. Schonten.
+
+`Investigations on the Theory of Brownian Movement`:title:, by Albert Einstein.
+
+`Great Experiments in Physics`:title:, ed. by ???.
+
+`Curvature and Homology`:title:, by Samuel Goldberd.
+
+`The Philosophy of Mathematics`:title:, by Stephan Korner.
+
+`The Various and Ingenious Machines of Agostino Ramelli`:title:, by A. Ramelli (!).
+
+`Experiments in Topology`:title:, by Stephan Barr.
diff --git a/math/good books b/math/good books
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+++ b/math/good books
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+==========================================
+Recommended Math Reading
+==========================================
+
+BLANK
diff --git a/math/integers b/math/integers
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+====================
+Integers
+====================
+
+definition via Peano's Postulates: meserev 1-4
diff --git a/math/logic b/math/logic
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+=======================
+Mathematical Logic
+=======================
+
+.. note::
+ Incomplete; in progress
+
+definition of induction: meserev 1-4
+
+Proofs
+===========
+
+Indirect Proof: "reductio ad absurdum"
+ Show a paradox or impossibility in all cases by assuming the proposition
+ is false; then the proposition is true.
+
+Proof by elimination
+ Propose a complete set of propositions and remove all but one.
diff --git a/math/numbers b/math/numbers
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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.
diff --git a/math/sets b/math/sets
index 6d75a55..42eb831 100644
--- a/math/sets
+++ b/math/sets
@@ -2,11 +2,46 @@
Sets
====================
-.. warning:: Under progress!
+.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
-.. note:: Most of the definitions and notation in the section are based on [rudin]_
+Basics
+=============
+If every element :latex:`$a \in A$` is also :latex:`$a \in B$`, then we call
+A a *subset* of B and write :latex:`$A \subset B$`. If there are elements of B
+which are not elements of A, then we call A a *proper subset* of B.
-If :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`; otherwise :latex:`$A \neq B$`.
+If :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`;
+otherwise :latex:`$A \neq B$`.
-.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin.
- McGraw-Hill, 1976
+The null or empty set, which has no elements, is a subset of all others.
+
+A relation on a space of sets S is something that can be definted as either
+true or false (holding or not holding) for any binary pair in S.
+
+Binary Operators
+==================
+Binary operators defined on a set apply to any two elements of that set; order
+may or may not be important. A set is *closed* with regards to a binary
+operator if it contains the result of the binary operator. A set is *uniquely
+defined* with regards to a binary operator if the result of the operator on two
+elements of the set is unique from the results from all other pairs of
+elements.
+
+Some equivalence relations are
+:latex:`$\identity$` (NOTE: = with three lines) (*identity*);
+:latex:`$\congruence$` (NOTE: = with tilde on top) (*congruence*; eg of
+geometric figures); and
+:latex:`$~$` (NOTE: tilde) (*similarity*; eg of geometric figures).
+
+Some properties of equivalence relations are
+
+*reflexive*
+ if :latex:`$a=a$` is true for all a
+*symetric*
+ if :latex:`$a=b$` implies :latex:`$b=a$`
+*transitive*
+ if :latex:`$a=b$` and :latex:`$b=c$` implies :latex:`$a=c$`
+
+.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976
+
+.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.