increasing complexity

There is a trend about my math studies that goes from grade 1 to now and perhaps beyond. Other than the fact that things get more difficult over time (duh), there is a clear trend towards complexity, and by the time you reach calculus, you start to realize that there is a limit to the computational power of us humans, not to mention the fact that there are so many things that simply cannot be done in closed form.

In the most elementary of levels, you start the journey of mathematics with the "four arithmetics", i.e. addition, subtraction, multiplication, and division. Things were very simple at the time: the only thing we ever had to worry about was not dividing by zero. Everything else is straightforward. All the linear equations are solvable (with a few trivial exceptions).

But as I learned more and more advanced stuff in math, I soon realized that everything becomes a lot more complicated. For example, quadratic equations are not always solvable in terms of real numbers. The only way to properly solve them is to invoke a new dimension of imaginary numbers, which complicate things a lot. A well-known property of complex numbers is that complex exponentials are often multivalued (such as "i" to the power of "i").

Nonetheless, numeric algebra is not as difficult as what I was to learn next. Calculus. It is one of the most powerful tools in math, yet it is also so complicated. Differentiation: straightforward to do, but for any expression that is more than a handful, it will certain produce a rather overwhelmingly long and often ugly expression. But the real trouble here is integration: not everything can be integrated to produce a closed-form expression. There are lot's of integrable functions, but we just don't have the symbols to describe them.

As if that wasn't enough, then came differential equations. It was pretty much like entering a thick quagmire. There are very very few differential equations that can be solved in closed-form; nearly everything else must be done in numerical solutions, and numerical solutions aren't always correct. This is in direct contrast to numeric equations, which do often have closed-form solutions and are often easy to find (this is typically true of low-order polynomial equations). Indeed, numeric equations often have a very simple set of rules that allow me to solve them, while differential equations often require the use of ad hoc 'tricks' or certain very limited methods (that only work in specific scenarios, such as separating the variables).

This is where I am now: slowly treading through the differential equations. Most of the time, the actual problem sets given are not too difficult to solve, since they were designed that way. But what about in real life? Most real phenomena are not that easy to solve.

Looking far ahead, I believe that I'll be studying other more exotic things like linear algebra and differential geometry, since they'll be absolutely essential for the physics that I intend to learn. They are very daunting, and they still appear to be so. Hopefully, by then I would be prepared to learn them.

And the conclusion? Nature is very complicated, but I must admit that this is exactly what makes it so interesting to study. If it weren't so complicated, it would be just "simple harmonic motion" everywhere, which - frankly - isn't all that fun.