A moment of mathematical delight

In my 2nd year of college, in the midst of taking Linear Algebra there was a homework problem that was just a delight. It opened my eyes to the possibilities and along with taking a course in Abstract Algebra, wooed me away from Systems Analysis and caused me to switch to a major in mathematics.

Up to this point mathematics had had the same flavor. Algebra, Trigonometry, Calculus, I really enjoyed it all. In fact I distinctly remember the Calculus classes feeling like I was being reminded of what I already knew. Computations, numerical manipulations, etc. Certainly there were lots of interesting ideas, but nothing like what was to come.

So then I took a combined Linear Algebra and Discrete Mathematics course. After learning all the mechanics of matrices and such, we hit this chapter that defined vector spaces. That was pretty neat, but the possibilities didn't hit me until I did homework problem number 14 at the end of the chapter.

This is from Howard Anton's Elementary Linear Algebra, 5th Edition. It's a simple enough problem. You had to demonstrate that the following defined a vector space:

The set consists of just the moon. The operations are moon + moon = moon and k(moon) = moon where k is a real number.

This was the first time I'd seen a problem like this. I dove into proving or disproving it was a vector space. As it turns out, it satisfies the definition of a vector space.

All of a sudden there were so many possibilities! This problem showed me what you could start playing with when you stepped away from the real world and stripped mathematics down to the ideas. I wanted to just hug this problem!

This problem had such an impact on me that I still remember it 20 years later, and was able to find the problem in the book in just a couple of seconds.

Much of the mathematics I took later built on this problem's promise and this ended up being the mathematics I loved. Abstract Algebra and Point Set Topology were the areas I savored, and I still like to think about the ideas we played around with in those classes all these years later.