Mathematical Concepts

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I collect here notes on a number of mathematical concepts that I believe an undergraduate mathematics major should understand.  In some notes I also describe modern applications.
 LaTex (my webpage on LaTex with many links!)
Geometric sums and series (pdf)
In a geometric sequence there is a ratio r such that each term is a multiple of the previous term by that ratio.  Infinite geometric series give us ways to write repeating decimals, compute the log function and the inverse tangent function, among other things....
 Notes on complex numbers (pdf)
Complex numbers explain much of the phenomena underlying mathematical computations at the undergraduate level.  The complex numbers are the “right way” to view the number system.
 Euclidean Algorithm (pdf)
This algorithm, over 2000 years old, is useful in number theory and has applications to modern security and is used in modern internet credit card transactions.
 Chinese Remainder Theorem (pdf)
This computational theorem relies on the Euclidean algorithm.  It also has modern applications to digital communications and digital commerce.  These brief notes on the CRT assume knowledge of the Euclidean Algorithm.
 Notes on functions & permutations (pdf)
Brief notes on functions and permutations, from an abstract point of view, in preparation for Galois’ definition of a group.
 Solving cubic equations (pdf)
We all have learned the quadratic formula. But what about solving cubic equations or even polynomial equations of higher degree?  Here is an introduction to the cubic equation, along with a few examples.
 Graphs (under construction)
Finite graphs provide an easy introduction to combinatorics and have many applications in our digital age.  Here are some basic notes on graph theory.
 Using LaTex to write mathematical papers (link)

(Last updated on March 2, 2012)