Real Analysis: Week 1 (and one really important line)

One week ago, I didn’t know what Real Analysis even was. I knew it must be important (it’s a requirement for the major, after all), and that it was probably a theory based course (since we’re “analyzing” how numbers work, rather than learning the methods that these workings allow). Going into the first day that way was exciting- after a summer spent mostly at home, it felt like I was throwing myself into some far off corner of the universe and waiting to find out what it looked like.

As it turns out, Real Analysis is a little bit more like every corner of the universe. Our professor, Molly, introduced the class as a course that would allow us to begin “proving calculus.” And calculus- that’s huge. Calculus may be only one branch of mathematics, but it’s the branch that most of math education in the US most naturally leads to. That makes sense, too, since calculus has so many applications in so many important fields. Proving the validity of calculus implicitly assures the validity of fields in economics, physics, chemistry, and pretty much any natural or social science which uses complicated statistics to understand a data set. Real Analysis, then, seemed like some pretty powerful stuff.

We started our first class with some of that powerful stuff, right away, talking about what the set of “real numbers” even is. It’s hard to wrap my head around it, to be honest, because here’s what the real numbers are- they’re the line of every possible number that exists in either the positive or negative direction of zero, with not a single gap in them. The numbers we deal with most frequently in our day to day life are “natural numbers,” the kinds of numbers you can count on your fingers. They’re every number you can get by starting with 1 and adding 1 to itself as many times as you could possibly want. It’s an infinite number of numbers, but compared to the real numbers, the natural numbers are missing a lot. To get to the real numbers from the natural numbers, you first step through the integers. The integers include the natural numbers, every negative version of a natural number, and zero. But we’re still not even close to the real numbers. Next we step through the rational numbers- these are every number you can get by taking a ratio of two integers. So numbers like 13/167, or 2/9, those are rational numbers. And as you can probably imagine, that’s also a really big set of numbers.

But it turns out that, for as many rational numbers as there are, there are far more numbers that can never be expressed as a ratio of two integers. So if we were to lay all of the rational numbers out in a line, there would be spaces, empty spots of numbers which we couldn’t express using that set. The real numbers, though, fix that problem. They include every rational and irrational number. Essentially, any number that you could find out in the world, only the set of real numbers is guaranteed to include it.

So the real numbers are pretty special. We learned that in the first fifteen minutes of class. And it turns out that because the real numbers are so special, mathematical systems that use the real numbers have some cool properties. We’ve been exploring those kinds of properties since them. For example, any interval of numbers (like the interval from 0 to 1, written as (0,1)) is guaranteed to have a single largest element, one unique number bounding the entire set below it. But if you don’t use the real numbers, you can’t guarantee such a thing. That’s just one example, but the essential thing I’ve learned so far is this: the simple fact that the real numbers exist allows for and dictates the behavior of entire fields like calculus . The rules of most math that I’ve ever seen in my life are themselves governed by a line.

For only the start of a class, that’s a pretty amazing thing to have already learned, and I can’t wait to see where in the universe we end up in the next few days.




Published by Anna

Hi! I'm a junior at Colorado College majoring in math and minoring in music. I'm originally from just outside of Philadelphia, but grew up spending time every year in Colorado, where my mom grew up. I transferred to CC after my first semester of college, and could not be more happy with where I ended up.