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. 

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