A question from an engineering undergraduate My question primarily concerns the necessary transition from an undergraduate program in electrical engineering to graduate program in applied mathematics or pure mathematics.    

I'm an electrical engineering student. During the first year in my university life, I found myself really fascinated with mathematics, and this summer after my first year of school, I self-studied Velleman's "How to Prove it", and analysis from Spivak's book.
As someone who had never been engaged in the circle of serious mathematics, I am lost as to the purpose of my studying: is it too late/highly improbable for me now to actually pursue a future in applied mathematics or pure mathematics while remaining in engineering as an undergraduate? Although I do have good reasoning skills, and finished Spivak's book in two months, I'm know I have much too long a way to go. Hence my question: should I try to take some mathematics courses outside my program such that I could partially fill the gap of my knowledge and basic abilities of mathematics? If so, is there any general area of math courses I should take? And should I actually complete a math minor or major degree (in my school specialist is ranked higher than major)?
You're probably in your early 20s, so: relax, you're okay. There's plenty of time to learn math. You have not missed some sort of mandatory train for becoming a mathematician or even just doing math at a reasonably advanced level. College is a great time to start doing mathematics. 

If you finished Spivak in just two months (depending on the detail you went into), then you're probably already a great deal more capable than most of the students actually doing the math major. So don't think that you have too long of a way to go; you're obviously able and interested, and between the two of those, you can probably complete some sort of math curriculum pretty quickly. I know that I was in that situation as well, at one point. I ended up knocking out the entire math major in a single year by taking all core sequences concurrently. Caution: your mileage may vary.

If you can (i.e. if it fits into your program in terms of scheduling), I'd definitely recommend doing a math major. If you're looking into going into engineering, you must know linear algebra, so take that. Multivariate calculus is also recommended. Beyond that, real and complex analysis are also staples of applied mathematics, along with courses in differential equations.
Aren't engineering students supposed to know algebra and calculus? While mathematics is certainly a better major than, say, the study of 15th century Armenian literature (which is fairly useless as far as today's jobs are concerned), you would be better off studying something with the greatest likelihood of landing you a good job after you graduate. In that regard, electrical engineering is better than pure mathematics.

If I were you, I'd concentrate on the math courses required for the electrical engineering degree, while cultivating the study of things like algebraic number theory as a hobby. Once you have a good degree under your belt and a steady paycheck coming your way, then you can start to explore more philosophical fields of study.

That's the advice I'd give my son.