Pure vs. Applied

“So, what’s the difference between pure math and applied math?”


“Tyler and I are in the same program, but he’s pure and I’m applied.”

I’ve said this exact sentence (or something to its effect) countless times to others, but usually it raises confusion from non-math folks. (Tyler, by the way, is one of my close friends in the department, and we’ll be rooming together next year.)

“So, what’s the difference between pure math and applied math?”

That’s the question I’m going to answer for all of you non-math folks out there. Some of it will probably end up sounding like gibberish, but I’ll try to limit that to a minimum.

The basic distinction

At the end of the day, pure math could be described as “math for math’s sake”. It doesn’t mean it’s not applicable to real-world scenarios or irrelevant; it just means that the work being done is more abstract in nature and is meant to advance the study of mathematics in and of itself. On the other hand, applied math has the application of mathematics to real-world problems or scenarios as its end goal. This doesn’t mean that applied math can’t be abstract or theoretical nor contribute to the study of mathematics in and of itself; it just means that applied mathematicians are motivated primarily by real-world applications.

Every applied mathematician needs to learn some pure math before they can start doing the work of applied mathematics. The converse, however, is not always true; the pure mathematician could have his or her nose in books on pure mathematics their whole education and still contribute to the field. However, they could suffer from one or two disadvantages. The first is that they could lack motivation for doing their research and might not see how it actually matters. Perhaps an algebraist1 would get more excited about his research if he learned that his studies would be benefiting knot theory, a branch of mathematics that (in its applied side) is currently working on ways to identify protein chains in the human body. The second is that they might lack intuition as to how to go about solving the types of problems that are placed in front of them. (Think about how much easier it was in high school to work math problems once you had seen an example or two!) Far more often than you might think, methods or techniques from one area of mathematics often get borrowed in another area, and the pure mathematician who has seen “living, breathing” problems worked out in the everyday might be better equipped to tackle his or her own problems.

In short, applied math needs pure math, but to say that pure math needs applied math would be an overstatement, but perhaps not a grave one.

Practical Implications

The way it works at the University of Iowa, every math student is either enrolled in the Department of Mathematics or the division of Applied Mathematical and Computational Sciences (AMCS). The former is pure and the latter is obviously applied. However, the first-year pure and applied students share some of the same courses. At least, they usually share Real and Complex Analysis – two classes that attempt to generalize and make rigorous some of the results seen in undergraduate Calculus. They will also share Differential Equations (which builds on undergraduate Calculus) and Topology (an abstract study of spaces and their properties).

Now, the pure math students (those who want to do research and pursue a PhD) must select an adviser from within the Department of Mathematics, while AMCS students may pick an adviser from either inside or outside the department. And following the basic distinction given between pure and applied math above, the pure students will be studying “math for math’s sake”, while the applied math students may choose to tackle problems that are perhaps more tractable or “real-world”.

Myself: Mathematical Mutt

As for me, while I am in fact an AMCS student (and hence applied), I have almost always straddled the line between pure and applied math. For example, my undergraduate degree2 was in applied, but I had also fulfilled the requirements to graduate with a degree in pure math (including two semesters of abstract algebra).  Similarly, this past year I took Numerical Analysis with AMCS students, Topology with the pure students, and Analysis with both. One area of research I am looking into doing (but not committed to yet) is called Topological Data Analysis (TDA), which both pure and applied students are working on.3


Sometimes, pure math people can exhibit a (perhaps feigned) sense of smug superiority for inhabiting the rarefied domain of abstract truth, not recognizing that applied mathematics is making great contributions to the field and is more necessary today than perhaps at any time in history. On the other hand, aspiring applied mathematicians can fall into the trap of viewing pure mathematics as a necessary evil which can be discarded in their future, not realizing that pure mathematics gives them both a foundation for study and methods of thought for solving problems.

I’m thankful that I often get to do both.


Still confused? Feel free to leave your questions in the comments section below!

  1. An algebraist is someone who studies abstract algebra, one of the core divisions of pure mathematics. 
  2.  University of St. Thomas ’16 – Go Tommies! 
  3. Ironically, the professor I am currently reading with on TDA has told me that her pure students are doing the most applied work while her applied students are doing the most pure work. 

Concerning Qualifying Exams

Let’s just get this all out on the table. Yes, I’m staying in Iowa City for the summer and doing math the whole time, but to what purpose? Two words: Qualifying Exams. It seems that few people outside of academia understand what I mean by “Qualifying Exams” (or “quals” for short), so hopefully this entry will be a good reference point for what I’m to undergo and how it will be affecting my summer in Iowa City.

The first year: Pre-qualifying

When you enter graduate school in Math (at Iowa at least), your first year is kind of like a super-college rigmarole in which you’re taking classes that should be mostly review (but usually they aren’t) so that you can be brought up-to-speed on the mathematical foundations you’ll need to continue on in your program, start researching, get some results, write them down, and get a PhD in anywhere from 4 to 6 years. The best adjective I’ve used to describe this year is “whirlwind-ish” – at least that’s how it was for most of us. There’s five two-semester courses you could take in your first year:

  • Abstract Algebra,
  • Topology,
  • Analysis,
  • Differential Equations, and
  • Numerical Analysis.

Every graduate student in math, whether they’re pure or applied (I’ll make a post about pure math vs. applied later and then hyperlink it here) has to take three of these in their first year. Taking four at once is possible but not recommended. Five would be suicide. If you’re one of those “poor unfortunate souls” in the pure program, you usually take the first three in your first year and then take Differential Equations your second year. If you’re one of those impertinent folks the AMCS track, you usually take the bottom three your first year and then Topology your second year. If, however, you’re in the AMCS track (AMCS itself merits its own blog post) and your name is either Alex Bates or Rajinda Wickrama, you ended up taking Topology, Analysis, and Numerical Analysis your first year and then must take Differential Equations your second year.

This first year is, on the whole, not the most pleasant experience in the world. One of my colleagues wrote an excellent article on her experiences in her first semester of graduate school in math, much of which I would echo. I would not, however, say that things improved much for me during second semester, other than a greater emotional, mental, and spiritual stability gained mostly through having a daily routine and being a part of my church community. On the whole, things seemed to be a little tougher academically even though we had mostly transitioned to the pace of graduate life. And being a TA wasn’t getting any easier. (Being a TA also merits a post.)


But now, mercifully, I am writing from the other side. I finished my finals and am done with that first year. But now a darker and more ominous challenge lay on the horizon: the aforementioned Qualifying Exams. So what are quals, exactly? And, as the name suggests, what do they qualify you for?

In short, quals are exams that you as a graduate student must pass by the end of your second year to continue on in the program, pick an adviser, start researching, etc. They are offered every Fall and Spring, and usually students take the qualifying exams at the beginning of their second year (for me, August 2017) and in the three subjects that they studied during their first year of graduate school (for me, Topology, Analysis, and Numerical Analysis). There are three “grades” given for performance on the qualifying exams: PhD pass, Master’s pass, or fail. You need to get a “PhD pass” on the three that you take to qualify yourself to continue on in the PhD program. Since I’m in the PhD program, that’s my goal- pass all three at the PhD level. However, there’s a convenient exception if you take them all in August after your first year: it suffices then to PhD pass only two of them and Master’s pass the third.

Moving forward

So, this is essentially what I will be doing all summer: studying for the qualifying exams. At the time of this post’s writing, I have 79 days to go before my first qualifying exam, but it’s not as if I don’t know the material yet – I studied it all this past year. Nor will I be like one running aimlessly or like a boxer beating the air (1 Cor 9:26), but my TA’s from my courses this past year will be holding preparation classes for the qualifying exams during the summer. And the folks in AMCS have generously awarded me with a summer fellowship so I can singlemindedly pursue my studies without having to worry [too much] about money.

So that’s why I’m in Iowa City this summer. And that’s what I’ll be spending my time working on most hours of most days from now until August. But since Iowa City is (apparently) a wonderful place in the summertime, when I’m not indoors studying for 50+ hours a week, you’ll probably be able to catch me outside at the various farmers’ markets, movies on the Pentacrest, or random events I’ll do with folks from my church.

So if you’re someone who knows me personally, please cut me some grace if I ever seem frazzled this summer or “out of it”. If you’re a person of faith, I’d ask you to pray for me that I would do my work with God-glorifying diligence, competence, and creativity. If you’re someone in my program, feel total freedom to and by all means push me this summer to be a better mathematician; I will do my best to reciprocate such efforts. And if you’re not a mathematician, enjoy your summer of not having to do math!