<p>As we know, there are three major branches of mathematics: algebra, geometry, and analysis. The question then is - is understanding one of those branches necessary in understanding another one of those branches? Is it conducive? Is it possible that someone can do analysis on the research level without knowing any abstract algebra at all? Or differential geometry?</p>
<p>I know a little bit about this…</p>
<p>I have a friend who’s finishing his freshman year in HS. Their school had used an integrated math program for years, as they felt having algebra and geometry together helped to increase the knowledge of both. I think this ended if they got to Calculus, maybe even pre-Calc. He just informed me that this is now going to be a thing of the past, with their school going to the same post-elementary progression that my school had: Pre-Algebra, Algebra I, Geometry, Algebra II, Pre-Calculus, and Calculus. (Calc was optional at my school, you took it if you took Algebra I in eighth grade instead of ninth, and even then you weren’t obligated to take it).</p>
<p>By looking at this, it seems as if Geometry is deemed necessary to take Algebra II. However, many found Algebra II tougher after taking Geometry because they had a year off from the method of thinking you need for Algebra and we also found that not a whole lot of Geometry applied to Algebra II. It was more relevant with Precalculus (which was also called Algebra III/Trig). The Algebra aspect was easy for most in that class as we were fresh off Algebra II, but the trig stuff was hard for people who weren’t taking/already took Physics and hadn’t seen it since Geometry.</p>
<p>There were also math electives at my high school in both problem solving and probstat. I think Algebra II was a requirement for both. The challenge I had when taking probstat is that I was in pre-Calc at the same time, in fact, I had those classes back to back and I was a second semester senior (a problem of itself). Going from the mode of thinking needed for probstat to the different world of precalc often was challenging, since I had exactly four minutes in between to change gears.</p>
<p>Then there are the college math courses. I’m assuming you are listing Calculus under the branch of analysis here. Calculus, to me, isn’t even close to algebra or geometry. Sure, there were traces of both in there, but the concepts of stuff like limits and derivitives were extremely tough for me because I was trying to interpret them in algebraic sense. I wound up dropping Calculus, barely passing with a 62%, and then took the required College Algebra course instead and got a 97%.</p>
<p>To me, the answer is yes and no. I think that if I took a more advanced Algebra course, I’d probably get an A. I got an A in Pre-Algebra, Algebra I, Algebra II, and the first half (the Algebra III) part of pre-Calc.</p>
<p>Geometry was even more up my alley. I got an A in that class, did very well on the proofs, and seemed to do well with terms.</p>
<p>With the Analysis branch of math, I don’t fare well. I got a B in the second half of pre-Calc, which was the Calc concept stuff (though I would have had a low C without trig, I always did well on trig).</p>
<p>In all three types of classes I saw some sort of crossover. Not so much using analysis in Algebra or Geometry, but more of using the others with each other. Probstat was almost a fourth type of math.</p>
<p>The fact that I was a consistent A student at all levels of Algebra and in Geometry yet did poorly in Calculus makes me think that they’re not related. Bottom line is that you need to take them in order to get the concepts that may crop up, but just because you get one branch doesn’t mean you’ll get another.</p>
<p>An end note: I have a friend who hates geometry and proofs. He usually doesn’t get A’s in those classes (and doesn’t use proofs in calc unless forced to). But he usually aces Algebra and Calc-type courses. The one exception was linear algebra, although he said it was more like geometry than algebra. I have another friend who loves all math, proofs included, and complained that she got “only a 95%” on a proof test. She went in early every morning for extra help in Geometry so she could match the 98-99 she normally got in Algebra courses. Time will tell if she can put up those numbers in a Calc course, she takes her first one next year.</p>
<p>err AFPrep I’m pretty sure Inquiline is talking about a different kind of algebra and geometry than you… like abstract algebra and topology.</p>
<p>To answer the original question, I think knowledge in one would be conducive to learning another, just because you would have done more mathematical thinking. i think someone could do real work in one of them without much knowledge of another, but it’s just more likely that someone doing real work would know about them all.</p>
<p>Just do number theory so you have to learn everything :-)</p>
<p>I think representation theory has the more to do with all branches of mathematics than number theory does. You can pretty far into number theory without having too much of a background in algebra or analysis (and no differential geometry whatsoever).</p>
<p>As far as the original question, knowledge of multiple areas isn’t needed for some questions, but a lot of research problems spread across areas, and in general it’s best to be familiar with them all. If anything, you learn different methods for attacking various kinds of problems.</p>
<p>“You can [go] pretty far into number theory without having too much of a background in algebra or analysis”
this isn’t true. unless by ‘pretty far’ you mean just elementary nt, which isn’t pretty far.</p>
<p>
</p>
<h1>haha - that’s a generic response that can be used for any field of knowledge (be it math, physics, chem, psych,etc). ;)</h1>
<p>Thanks for the long post AFPrep850 - though it wasn’t what I was looking for, it was still very interesting. I’m talking about abstract algebra, real/complex analysis, and differential geometry/topology.</p>
<p>Speaking of math…how difficult is Intro to Stats at college for a non-math person? I know that’s a subjective question. Is it the type of class that’s doable in an online format?</p>