# Introductory real analysis before or after introductory abstract algebra?

What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both?

I recognize that the overlap between the two courses is minimal, and therefore they are largely independent of one another. (A debatable minor exception could be discussing fields when introducing the real numbers axiomatically, when the notion of a field is discussed in more detail in an abstract algebra course.) That said, students might cope better with real analysis than abstract algebra because it initially covers familiar ground (calculus). On the other hand, they might find the challenges of epsilontics, including working with double quantifiers and inequalities particularly difficult in contrast to the early proofs in abstract algebra.

(By introductory real analysis, I mean primarily a course covering the theory of single variable calculus and possibly also introducing metric spaces; a typical textbook could be Abbott's Understanding Analysis; by introductory abstract algebra, I mean a first course on groups, rings and fields; a typical textbook might be Fraleigh's A First Course in Abstract Algebra or Gallian's Contemporary Abstract Algebra.)

• I had always considered these courses to be roughly equivalent upper-level courses that could be taken pretty much independently of each other --- either taken simultaneously or taken in the order that best fits one's schedule of classes. For example, if you want to take undergraduate quantum mechanics or numerical analysis or whatever, and the time it's offered conflicts with real analysis, then you take algebra, or if you're at a small college, it's common for these to be offered on a rotating basis every other year so you have no choice. However (+1), I'm curious as to what others will say. – Dave L Renfro Aug 3 '19 at 16:45
• If a student first has a course on "how to write proofs" then these two follow-up courses may be done in either order. But if one (and only one) of these course is also about how to write proofs, then that knowledge will be assumed by the other one. – Gerald Edgar Aug 3 '19 at 19:29

Despite the names of these fields, as a student I found real analysis more abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I can justify this, but let me give two examples:

• Lagrange's theorem in abstract algebra: The order of a subgroup $$H$$ of a finite group $$G$$ divides the order of $$G$$. Sure, this is abstract, but it is discrete and definite and understandable from a thorough grasp of cosets.
• Heine-Borel theorem in real analysis: Closed and bounded iff every open cover has a finite subcover. Requires understanding limit points, accumulation points, triangle inequality.

Certainly one can pluck out a theorem from abstract algebra that is decidedly more abstract than a particular theorem in real analysis, to make the opposite point. But to me abstract algebra as a whole was (and still is) more concrete than real analysis.

So I would argue: Abstract algebra before real analysis, just because proof sophistication would improve in abstract algebra to help with the more difficult (and abstract) proofs in real analysis.

• +1 I would have written nearly the exact same thing. – Daniel R. Collins Aug 4 '19 at 23:11
• +1 Although I suspect that many students struggle to develop "a thorough grasp of cosets" – J W Aug 5 '19 at 6:28
• @JW: See also, What makes cosets hard to understand?. – Joseph O'Rourke Aug 5 '19 at 12:39

Is 'at the same time' an option? I mean, by junior year, math majors should be taking at least two math classes per semester, right?

When I was an undergraduate at Penn State, these two courses were the only 300 level math courses, both designed to be taken first semester junior year. The introduction to abstract algebra used "Numbers, Groups, and Codes", and the introduction to real analysis used "Elementary Analysis: The Theory of Calculus".

Both courses were designed to teach how to write proofs, and I don't think having two courses focused on teaching this was a waste of time.

• I tend to side with you in the sense of it the answer being roughly "it doesn't matter much". In contrast, a second course in linear algebra would be a much bigger improvement and if the second course in linear was prerequisite to abstract, that could be very fun. – James S. Cook Aug 5 '19 at 5:15
• Thanks for your answer. Certainly, taking the two courses concurrently is an option, but I would like to focus on whether there are advantages (or disadvantages) of taking one particular course before the other. That said, I'm curious if in parallel could be argued or demonstrated to be a better option than in series. – J W Aug 5 '19 at 5:35
• @JamesS.Cook, it would be interesting if you would elaborate on your comment above about a second course in linear algebra prior to abstract algebra. Maybe it could even be framed as an answer or possibly a new question. – J W Aug 5 '19 at 6:25
• @J W, thanks for the question. I’m interested to find out everyone’s thoughts/experiences on this issue. Like @James S. Cook, my opinion is that they’re independent enough that the order doesn’t matter. It may be best to let a student take whichever they like first, based on their interests. Most importantly, I think having both teach proof writing would be beneficial. – Joe Aug 5 '19 at 10:44