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This article is intended to provide a brief introduction to what [[college]] [[mathematics]] is all about.  It is aimed at [[high school]] students who have enjoyed mathematics in high school, and are starting to think about what to study in college.  This article is a work in progress -- please expand and leave suggestions at the talk page.  The theorems and ideas mentioned here aren't all things that you'll necessarily encounter in classes; indeed, most of them you probably won't.  Instead, they are presented to indicate the sorts of thinking and methods that go into mathematics, and the sorts of problems that mathematicians work on today.

This article is intended to provide a brief introduction to what [[college]] [[mathematics]] is all about.  It is aimed at [[high school]] students who have enjoyed mathematics in high school, and are starting to think about what to study in college.  This article is a work in progress—please expand and leave suggestions at the talk page.  The theorems and ideas mentioned here aren't all things that you'll necessarily encounter in classes; indeed, most of them you probably won't.  Instead, they are presented to indicate the sorts of thinking and methods that go into mathematics, and the sorts of problems that mathematicians work on today.

Mathematics is a strong and marketable major for students, relatively (but not completely) free of [[liberal bias]].

Mathematics is a strong and marketable major for students, relatively (but not completely) free of [[liberal bias]].

== What College Mathematics is Not ==

== What College Mathematics is Not ==

Before we look at the sorts of problems that actually ''are'' dealt with by mathematics in college, let's look at some that are not.  Rest assured, many of the most painful aspects of math classes you've taken before aren't so bad in college!

Before we look at the sorts of problems that actually ''are'' dealt with by mathematics in college, let's look at some that are not.  Rest assured, many of the most painful aspects of math classes you've taken before aren't so bad in college!

# Simply learning more advanced versions of mathematical ideas from high school.  Tired of learning tricks for integration?  Majoring in math won't just teach you new methods to integrate functions -- if you've taken an introductory calculus class, chances are you already know the important ones!  What it will do is force you to think carefully about basic questions about what an integral is.  Given some subset of space, what does it really mean to find its volume?

# Simply learning more advanced versions of mathematical ideas from high school.  Tired of learning tricks for integration?  Majoring in math won't just teach you new methods to integrate functions—if you've taken an introductory calculus class, chances are you already know the important ones!  What it will do is force you to think carefully about basic questions about what an integral is.  Given some subset of space, what does it really mean to find its volume?

# Doing lots of computations.  Chances are you'll have to do a few, but most mathematics in college is based around "proofs" -- very careful and rigorous arguments which show that mathematical statements are true.  Many proofs won't involve any computations at all!

# Doing lots of computations.  Chances are you'll have to do a few, but most mathematics in college is based around "proofs"—very careful and rigorous arguments which show that mathematical statements are true.  Many proofs won't involve any computations at all!

# Learning a bag of tricks to do Olympiad-type problems.  If you've ever taken the AMC, Math League, or other similar competition tests, and not done well, don't worry!  Much of these tests check nothing more than whether you're familiar with some collection of techniques for solving special problems.  One math professor at Harvard keeps in his desk a copy of a math competition on which he scored a perfect 0/120.  Mathematical theory goes far beyond these problems.

# Learning a bag of tricks to do Olympiad-type problems.  If you've ever taken the AMC, Math League, or other similar competition tests, and not done well, don't worry!  Much of these tests check nothing more than whether you're familiar with some collection of techniques for solving special problems.  One math professor at Harvard keeps in his desk a copy of a math competition on which he scored a perfect 0/120.  Mathematical theory goes far beyond these problems.

Given that a math major won't just be getting more practice at the sorts of math encountered in high school, what is it about?  Here's a list of some of the major fields that are the foundation of a mathematics education.   

Given that a math major won't just be getting more practice at the sorts of math encountered in high school, what is it about?  Here's a list of some of the major fields that are the foundation of a mathematics education.   

<i>For each field, I plan to give a very general description and to describe a few problems/theorems that are nice results and capture some of the essence of a subject.  I'll aim for a few paragraphs about each one.  Then I will give a problem or two that is a much more open-ended question that has motivated the development of large amounts of theory.  For now I have no written about any of these -- please leave feedback and suggestions on the talk page and I will try to arrive at the best set of motivating problems!</i>

''For each field, I plan to give a very general description and to describe a few problems/theorems that are nice results and capture some of the essence of a subject.  I'll aim for a few paragraphs about each one.  Then I will give a problem or two that is a much more open-ended question that has motivated the development of large amounts of theory.  For now I have no written about any of these—please leave feedback and suggestions on the talk page and I will try to arrive at the best set of motivating problems!''

=== Algebra ===

=== Algebra ===

==== Mathematical Billiards ====

==== Mathematical Billiards ====

Imagine that you're playing a game of pool on a standard rectangular table, but with a twist: there's no friction, so after you hit the ball, it continues along its trajectory forever.  Obviously if you hit the ball so that its path is perpendicular to one of the walls of the table, it will bounce straight back, then off the opposite wall, straight back again, etc.: the ball will follow a repeating pattern of bouncing between the two walls forever.  Similarly, if you aim just right, you can hit the ball so that it moves in a diamond-shaped path forever, bouncing off the middle of the sides of the walls repeatedly.  A trajectory of the cue ball is called "periodic" if it's like one of these, in that the ball eventually ends up back where it started, and moving in the same direction.  A trajectory is said to have "period p" if it hits the wall exactly <math>p</math> times in the course of this repeating cycle.  The two examples given above have periods of two and four, respectively.  One question we might ask is whether there is an orbit of period <math>p</math> for every possible choice of p. Can you come up with a path on the standard billiards table that has period 3?  If not, can you prove that one doesn't exist?

Imagine that you're playing a game of pool on a standard rectangular table, but with a twist: there's no friction, so after you hit the ball, it continues along its trajectory forever.  Obviously if you hit the ball so that its path is perpendicular to one of the walls of the table, it will bounce straight back, then off the opposite wall, straight back again, etc.: the ball will follow a repeating pattern of bouncing between the two walls forever.  Similarly, if you aim just right, you can hit the ball so that it moves in a diamond-shaped path forever, bouncing off the middle of the sides of the walls repeatedly.  A trajectory of the cue ball is called "periodic" if it's like one of these, in that the ball eventually ends up back where it started, and moving in the same direction.  A trajectory is said to have "period p" if it hits the wall exactly <math>p</math> times in the course of this repeating cycle.  The two examples given above have periods of two and four, respectively.  One question we might ask is whether there is an orbit of period <math>p</math> for every possible choice of p.&nbsp;Can you come up with a path on the standard billiards table that has period 3?  If not, can you prove that one doesn't exist?

As is often the case is mathematics, we attempt to generalize these results on periodic orbits to other situations.  What if the billiards table is not rectangular, but triangular?  It's true, but extremely difficult to prove, that given any triangular billiards table, there exists a periodic orbit of '''some''' period.  The existence of periodic orbits is an extremely subtle question.  This can be generalized even further, to ask whether all polygons admit periodic orbits.  What about polyhedra?  Can you imagine some periodic billiards orbits on a cube-shaped "table" (no gravity! think billiards in space).

As is often the case is mathematics, we attempt to generalize these results on periodic orbits to other situations.  What if the billiards table is not rectangular, but triangular?  It's true, but extremely difficult to prove, that given any triangular billiards table, there exists a periodic orbit of '''some''' period.  The existence of periodic orbits is an extremely subtle question.  This can be generalized even further, to ask whether all polygons admit periodic orbits.  What about polyhedra?  Can you imagine some periodic billiards orbits on a cube-shaped "table" (no gravity! think billiards in space).

The orbits that aren't periodic turn out to be just as interesting as those that are.  Imagine that you place the cue ball at random on your billiards table, and smack it in some random direction.  Chances are it won't end up in a periodic orbit -- putting the ball in a periodic orbit requires a very careful shot.  So the ball will bounce around forever without retracing its path.  But will it ever come back to the point where it started, perhaps moving in a different direction?  The answer to this question is "no", but the Poincare Recurrence Theorem gives a somewhat satisfactory replacement: given any distance (e.g., an eighth of an inch), and any angle (e.g. two degrees), after some amount of time, the ball will be within that distance of its starting point, and directed within that angle of its initial path!  This is a very suprising fact.

The orbits that aren't periodic turn out to be just as interesting as those that are.  Imagine that you place the cue ball at random on your billiards table, and smack it in some random direction.  Chances are it won't end up in a periodic orbit—putting the ball in a periodic orbit requires a very careful shot.  So the ball will bounce around forever without retracing its path.  But will it ever come back to the point where it started, perhaps moving in a different direction?  The answer to this question is "no", but the Poincare Recurrence Theorem gives a somewhat satisfactory replacement: given any distance (e.g., an eighth of an inch), and any angle (e.g. two degrees), after some amount of time, the ball will be within that distance of its starting point, and directed within that angle of its initial path!  This is a very suprising fact.

While this problem seems extremely elementary, sophisticated mathematical techiques from a range of fields of geometry and analysis have been brought to bear on it.  Even so, many easily-stated questions remain open.

While this problem seems extremely elementary, sophisticated mathematical techiques from a range of fields of geometry and analysis have been brought to bear on it.  Even so, many easily-stated questions remain open.

*K-12 teaching

*K-12 teaching

Some mathematicians enter entirely unrelated fields.  Math majors have the highest average scores of students in any field on the LSAT and GMAT, the tests required for entrance into law school and business school <ref>http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1430654</ref>: the skills acquired in studying mathematics in invaluable in the logic section of the LSAT and for subsequent careers in law.

Some mathematicians enter entirely unrelated fields.  Math majors have the highest average scores of students in any field on the LSAT and GMAT, the tests required for entrance into law school and business school:<ref>http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1430654</ref> the skills acquired in studying mathematics in invaluable in the logic section of the LSAT and for subsequent careers in law.

== Further Reading ==

== Further reading ==

Numerous expository books cover the topics mentioned on this page, along with many others, in a very accessible way.  Below are a few recommended books whose content is comparable to that of this article.

Numerous expository books cover the topics mentioned on this page, along with many others, in a very accessible way.  Below are a few recommended books whose content is comparable to that of this article.

# ''Mathematics: The New Golden Age'', by Keith Devlin.  Accessible discussions of many important ideas.

# ''Mathematics: The New Golden Age'', by Keith Devlin.  Accessible discussions of many important ideas.