In High School and Want to Do Advanced Mathematics?

This insight is written for high school students who don’t feel very challenged by their high school courses. Or maybe those high school students who want to have a taste of what university math is like. The good news is that you don’t have to wait until university to see exciting mathematics, you can learn it right now! Of course, plenty of math will require quite heavy prerequisites like three calculus courses, so it’s not like you can do everything. However, there are plenty of options still available to you.

Do not neglect your high school courses though. I get it, factorizing 50 polynomials for homework is pretty boring. But those skills are very important. You need to master them or you will get in trouble. Do not think that being able to do university mathematics somehow enables you to skip high school math. It doesn’t.

You should probably also seriously look into mathematics competitions. They can be great fun if you’re into this kind of stuff. I personally never was, and I failed every single competition badly. So yes, it is possible to be successful in mathematics and not like/not do well in competitions. Also, competitions are not university mathematics. They’re something completely separate.

So what are some of the things you could do? Let me give some examples. There are plenty more possibilities though.

1) Abstract algebra

The good news is that this doesn’t require any calculus or trigonometry at all. Maybe some of the examples or exercises use these notions, but not in a very profound way (i.e. you can easily skip them without any harm). A great book is Pinter’s “A Book of Abstract Algebra”“. http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178 A high schooler should most definitely be able to handle this book (but it won’t be easy). Armstrong’s “Groups and Symmetry” offers a more geometric approach http://www.amazon.com/Groups-Symmetry-Undergraduate-Texts-Mathematics/dp/0387966757

So what should you know:
– You should be acquainted with solving polynomial equations. The quadratic formula is a must.
– You should be acquainted with some set theory. For example the language of sets, functions, unions, intersections, and cartesian products. Nothing too advanced.
– You should be comfortable with some really basic number theory: prime numbers, and fundamental theorem of arithmetic.
– You should be comfortable with some proofs methods like mathematical induction and contradiction.
– You should be a bit comfortable with ruler and compass constructions (but this is not a hard requirement)
– In geometry, you should be comfortable with vectors and equations of lines and planes.

What will you learn?
– Group theory, which is used to describe symmetries. All kinds of symmetries. Symmetries of a geometric object like cubes, to symmetries of a physical theory.
– Ring theory, which can be used in code theory and algebraic geometry. In-ring theory, you will go very deeply into the fundamental theorem of arithmetic.
– Field theory, which eventually leads to a deep study of polynomial equations. In particular, you will show that most polynomial equations do not have a solution we can (easily) find exactly.

2) Linear Algebra

Linear algebra is geometry in arbitrary dimensions. You will see what points, lines, and planes are in 1000-dimensional space. Furthermore, many of the methods of linear algebra are very important in all kinds of science and engineering. So if you’re looking for a more applicable subject, this is it. I really like MacDonald’s “Linear and Geometric Algebra” since it gives a very cool introduction to the (very cool and useful, but often ignored) theory of geometric algebra. http://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932

What should you know?
– You should be very comfortable with vectors, vector sums, and inner product (aka scalar or dot product) as seen in geometry.
– You should be very comfortable with matrices, like solving linear equations using Gauss’s method, multiplying matrices, and inverting matrices.
– Some notions of proofs and set theory would be nice. Nothing too advanced.

What will you learn?
– The abstract notion of a vector space is amazingly useful. It shows up everywhere in physics and engineering.
– The notion of geometric algebra is a very useful tool to know, but it is often ignored in undergrad (not because it is too difficult).
– The theory of linear maps aids in geometry, and also in many linear problems. For example, matrix diagonalization is used everywhere.

3) Euclidean Geometry

Why not learn from the masters? Why not read the most important math book EVER written? I’m talking of course about Euclid’s Geometry. This book has been used for centuries to teach students and it still is not outdated. I highly recommend the following site http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html. If you prefer a book form, then try Casey and Callahan’s “Euclids Elements Redux” http://www.amazon.com/Euclids-Elements-Redux-Daniel-Callahan/dp/1312110783 which is also available for free online. This book also contains cool exercises. In fact, I recommend using both resources. If you want a more advanced commentary, then try the first chapter of Hartshorne’s Geometry: Euclid and beyond (beyond the first chapter will likely be too advanced for you unless you study some abstract algebra). http://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/0387986502

What do you need to know?
– Not anything specific. Just be motivated and willing to work hard.

What will you learn?
I highly recommend going through the first four volumes of Euclid. The other volumes are much more outdated and (in my very humble opinion) did not stand the test of time all that well. You will do geometry starting from scratch and you will go up to the construction of the regular pentagon, which is an amazingly beautiful construction in mathematics.

4) Affine, projective, spherical, and hyperbolic geometry

I learned this in high school. It was the deciding factor for me: it was so beautiful that I decided to go into math. I highly recommend Brannan, Esplen, Gray’s “Geometry”. The problem is that it requires somewhat more prerequisites than to be expected from the typical high school curriculum, but nothing you can’t learn in a few days.

What do you need to know?
– Obviously, a good knowledge of high school (Euclidean) geometry is needed. This includes vector computations and equations of planes and lines.
– A solid knowledge of matrix computation knowledge is necessary. This includes multiplying matrices, inverting matrices, and diagonalization. Also solving linear equations. I guess only a minority of people have seen matrix diagonalization, but it is not so difficult to learn the basics if you are willing to ignore the proofs for now. Khan Academy videos should get you acquainted with enough of the basics.
– Algebra knowledge including solving equations and polynomial equations.
– Some very basic group theory. The first few chapters of Pinter’s group theory or Armstrong’s “Groups and Symmetry” are nice http://www.amazon.com/Groups-Symmetry-Undergraduate-Texts-Mathematics/dp/0387966757 and http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178 Notions of groups and subgroups are necessary.

What will you learn?
– Conic sections and their basic applications. Including very fun stuff like why the properties of parabolas are so useful in practice.
– Projective geometry is amazingly beautiful and is actually applied in image processing (although sadly many image processing courses never mention projective geometry).
– Spherical geometry. I hope I don’t need to tell you about the applications of spherical geometry.
– Hyperbolic geometry is a fun and weird geometry. It was a mathematical curiosity for a long time until such weird geometries gained a lot more importance with relativity theory.

5) Other

Of course, there are a lot of topics I didn’t mention here and a lot of books I didn’t mention. An honorable mention is the truly beautiful and excellent books by John Stillwell, including “The Four Pillars of Geometry” and “Roads to Infinity”. Check them out, there surely is a subject you like. Stillwell is a master in exposition as well!