Other

Orgasmic Troll · 2003-12-12, 15:11

Quote:

Originally posted by philmoore here
Although the forum was started to give people a place to discuss GIMPS and primes-related topics, I think Mike (xyzzy) would probably agree that any topics are ok, we even have a Bush vs. Clinton thread in the soapbox! Why not start a thread in this math forum and call it "other" math and see what people post there?

Phil

I have a math related question that has nothing to do with primes .. Right now, I'm going to a junior college majoring in math. So far, I've finished 3 semesters of Calc and Linear Algebra, and I will be taking Diff. Eq's. I'm a member of the MAA and I own a few books on abstract algebra and elementary number theory (although I don't have a reason to push myself to study them, so I just graze through them), and I taught myself calc-based probability for an actuarial exam.

My question is, what can I expect from upper division math classes when I transfer? I'm definitely leaning strongly towards pure math, but the classes I'm taking now seem strongly biased towards engineering students. I don't have anything against Engineering, but I feel like I'm being taught how to use a hammer when I want to be making blueprints. So what does an upper division class look like? sound like? taste like? smell like?

Along the same vein, what is modern algebra used for? correct me if I'm wrong, but it just seems like definitions of mathematical structures (I understand that it allows you to abstract problems and see how they are related to other problems), but the only application I've seen is proving that the quintic equation and polynomials of higher degree don't have a general solution. One of my calc teachers specialized in Galois theory, and IIRC, he used it in some capacity as a consultant to businesses. How? What does he do?

BTW, I can see how my post might be misinterpreted as the "When am I *ever* going to use this?!" exasperation, but it's just that I am waiting with bated breath to get to the good stuff and I am eager and extremely curious :)

Thanks in advance!

Wacky · 2003-12-12, 19:06

Travis,

I'm sorry, but, in some sense, I'm not sure that there is an answer. Each department has its own strengths, depending on the faculty at the time.

Back eons ago, I took a large number of upper level math courses. We used to joke, only partially, that there were more math departments than there were colleges within the university. The engineers had theirs, the business school had another, education had a third, and the college of liberal arts had two. There were the "fifth floor" mathematicians and the "third floor" mathematicians. Those two groups were as different from each other as either of them was from the engineers.

An even within one group, the focus was constantly changing as the faculty changed.

I suggest that you ask your question of one of the advisors at the University(s) that you are considering. They can give you a much better idea of what you are likely to find at that institution.

In any case, I am sure that you will find it both interesting and challenging.

ewmayer · 2003-12-12, 19:35

Green's Theorem - it's all about Green's Theorem. (Though Mr. Stokes would beg to differ.

)

philmoore · 2003-12-12, 19:58

I was a math major as an undergraduate and loved it, but ended up feeling like I wanted to do something more "practical" after I graduated and became a computer programmer for a few years. My second programming job was at a university physics department, and as I began to study physics and discovered that quantum mechanics, in particular, made use of much of the analysis that had seemed so abstract and remote from application when I was an undergraduate, my interest in math was rekindled. I agree, that higher math books, in the interest of trying to teach the maximum amount of material that can be squeezed into an academic term, pay a price by presenting a refined picture of "pure math" consisting of abstract definitions and structure theorems. Absract algebra certainly has many applications in number theory and geometry. I was fortunate to take a course in projective geometry at the same time as my first semester in abstract algebra, and being able to see the geometry applications at the same that I was learning the language of algebra helped me in motivating the subject matter.

I'm now teaching math at a community college, and when I'm asked a question similar to yours, i.e., what is higher math all about, I usually mention that much of higher math divides into three general areas: algebra, geometry, and analysis. Descartes' coordinate systems enabled algebra to be used in solving geometrical problems, and laid the stage for the invention of calculus, the first course in analysis. There is so much interplay between the three areas, there are no clear borders between them, but you might think of the three major areas as being collections of different types of methods which are used to study mathematical problems. Topology was originally a branch of geometry, but its language is basic to the modern foundation of analysis, and a branch of topology called "algebraic topology" looks more closely related to abstract algebra than to geometry.

Differential equations is good stuff, and you will come out of that class with a solid feel for all that you have learned in your three semesters of calculus. Since laws of nature are often most easily expressed as differential equations, this area of analysis gives you an important tool for solving many "real world" problems, and it is interesting how much good math arose from consideration of such problems. Number theory is an area which seems more removed from the real world, but much math which has grown out of number theory has proven to have striking applications. And yet, I think that most people who get hooked on math find themselves attracted to the sheer beauty of mathematical reasoning and thought. One area which I found particularly appealing was complex analysis, where surprising properties follow from the simple fact that a function has a continuous derivative.

My advice would be to study what you find interesting, and maybe even try to use your summers to branch out a bit on your own, rather than just restrict yourself to what you learn in classes. If you want any recommendations about specific books or areas, post them here and see what responses you get.

Have fun!

Orgasmic Troll · 2003-12-16, 17:15

I have another "other" math question ..

I was thinking about the insolubility of the quintic, and I'll readily admit I don't know enough about the theory behind the theorem (it's group theory, right?) but I'm just trying to understand the result a little better.

The proof says that there is no general form for the quintic with a finite number of addition, multiplication and radicals. So does this mean that

x⁵+a₁x⁴+a₂x³+a₃x²+a₄x+a₅ cannot be factored into 5 terms of the form (x+b_i) or that the b_i's cannot be expressed with a finite number of addition, multiplication and radicals?

ewmayer · 2003-12-16, 17:45

Quote:

Originally posted by TravisT
I have another "other" math question ..

I was thinking about the insolubility of the quintic, and I'll readily admit I don't know enough about the theory behind the theorem (it's group theory, right?) but I'm just trying to understand the result a little better.

The proof says that there is no general form for the quintic with a finite number of addition, multiplication and radicals. So does this mean that

x⁵+a₁x⁴+a₂x³+a₃x²+a₄x+a₅ cannot be factored into 5 terms of the form (x+b_i) or that the b_i's cannot be expressed with a finite number of addition, multiplication and radicals?

I've just been reading Ian Stewart's "Galois Theory," but haven't gotten to part about qunitic insolubility in terms of radicals yet. From what I've read so far, when expressed in terms of group theory, the quintic simply has the "wrong" kind of group for an analytical solution (i.e. one expressible in terms of radicals) to be possible. I hope to have enough quiet time over the upcoming holidays to get my head around it more thoroughly - little formal training in group theory, so most of this is new to me.

Orgasmic Troll · 2003-12-16, 21:22

I forgot to add another option, is that the quintic can be factored into (x-b_i) and all the b_i's can be expressed with a finite number of operations, but it's impossible to figure out the b_i's given the a_i's

michael · 2003-12-19, 23:53

No, it means that there isn't a general formula to express the results of a quintic as in x_i=... for i=1 to 5

-michael

Orgasmic Troll · 2003-12-20, 06:04

Quote:

Originally posted by michael
No, it means that there isn't a general formula to express the results of a quintic as in x_i=... for i=1 to 5

-michael

I'm aware of that, I'm asking what is a consequence of that .. does that mean for the general quintic, x takes on values that are not expressable as a finite number of operations? or is it possible to say that quintics of the form (x-a)(x-b)(x-c)(x-d)(x-e) where a,b,c,d,e are all integers are insoluble as well?

michael · 2003-12-20, 09:04

The quintic can always be factored into c(x-a1)(x-a2)..(x-a5) where c=constant and the ai's are complex numbers.

The trouble ofcourse is to find those ai's and for quintic equations there is no finite way to express them using radicals.

If the ai's are all integers you have a special case, though if they're all distinct and different from 0 i don't think there is a solution either (i'm not sure of this!)

This was first proved by Niels Abel using Galois theorem.

-michael

2003-12-20, 09:04	#10
michael Dec 2003 Belgium 5·13 Posts	The quintic can always be factored into c(x-a1)(x-a2)..(x-a5) where c=constant and the ai's are complex numbers. The trouble ofcourse is to find those ai's and for quintic equations there is no finite way to express them using radicals. If the ai's are all integers you have a special case, though if they're all distinct and different from 0 i don't think there is a solution either (i'm not sure of this!) This was first proved by Niels Abel using Galois theorem. -michael Last fiddled with by michael on 2003-12-20 at 09:06

2003-12-12, 19:06	#2
Wacky Jun 2003 The Texas Hill Country 1089₁₀ Posts	Travis, I'm sorry, but, in some sense, I'm not sure that there is an answer. Each department has its own strengths, depending on the faculty at the time. Back eons ago, I took a large number of upper level math courses. We used to joke, only partially, that there were more math departments than there were colleges within the university. The engineers had theirs, the business school had another, education had a third, and the college of liberal arts had two. There were the "fifth floor" mathematicians and the "third floor" mathematicians. Those two groups were as different from each other as either of them was from the engineers. An even within one group, the focus was constantly changing as the faculty changed. I suggest that you ask your question of one of the advisors at the University(s) that you are considering. They can give you a much better idea of what you are likely to find at that institution. In any case, I am sure that you will find it both interesting and challenging.

2003-12-12, 19:35	#3
ewmayer ∂²ω=0 Sep 2002 República de California 10110101110111₂ Posts	Green's Theorem - it's all about Green's Theorem. (Though Mr. Stokes would beg to differ. )

2003-12-12, 19:58	#4
philmoore "Phil" Sep 2002 Tracktown, U.S.A. 3·373 Posts	I was a math major as an undergraduate and loved it, but ended up feeling like I wanted to do something more "practical" after I graduated and became a computer programmer for a few years. My second programming job was at a university physics department, and as I began to study physics and discovered that quantum mechanics, in particular, made use of much of the analysis that had seemed so abstract and remote from application when I was an undergraduate, my interest in math was rekindled. I agree, that higher math books, in the interest of trying to teach the maximum amount of material that can be squeezed into an academic term, pay a price by presenting a refined picture of "pure math" consisting of abstract definitions and structure theorems. Absract algebra certainly has many applications in number theory and geometry. I was fortunate to take a course in projective geometry at the same time as my first semester in abstract algebra, and being able to see the geometry applications at the same that I was learning the language of algebra helped me in motivating the subject matter. I'm now teaching math at a community college, and when I'm asked a question similar to yours, i.e., what is higher math all about, I usually mention that much of higher math divides into three general areas: algebra, geometry, and analysis. Descartes' coordinate systems enabled algebra to be used in solving geometrical problems, and laid the stage for the invention of calculus, the first course in analysis. There is so much interplay between the three areas, there are no clear borders between them, but you might think of the three major areas as being collections of different types of methods which are used to study mathematical problems. Topology was originally a branch of geometry, but its language is basic to the modern foundation of analysis, and a branch of topology called "algebraic topology" looks more closely related to abstract algebra than to geometry. Differential equations is good stuff, and you will come out of that class with a solid feel for all that you have learned in your three semesters of calculus. Since laws of nature are often most easily expressed as differential equations, this area of analysis gives you an important tool for solving many "real world" problems, and it is interesting how much good math arose from consideration of such problems. Number theory is an area which seems more removed from the real world, but much math which has grown out of number theory has proven to have striking applications. And yet, I think that most people who get hooked on math find themselves attracted to the sheer beauty of mathematical reasoning and thought. One area which I found particularly appealing was complex analysis, where surprising properties follow from the simple fact that a function has a continuous derivative. My advice would be to study what you find interesting, and maybe even try to use your summers to branch out a bit on your own, rather than just restrict yourself to what you learn in classes. If you want any recommendations about specific books or areas, post them here and see what responses you get. Have fun!

2003-12-16, 17:15	#5
Orgasmic Troll Cranksta Rap Ayatollah Jul 2003 641 Posts	I have another "other" math question .. I was thinking about the insolubility of the quintic, and I'll readily admit I don't know enough about the theory behind the theorem (it's group theory, right?) but I'm just trying to understand the result a little better. The proof says that there is no general form for the quintic with a finite number of addition, multiplication and radicals. So does this mean that x⁵+a₁x⁴+a₂x³+a₃x²+a₄x+a₅ cannot be factored into 5 terms of the form (x+b_i) or that the b_i's cannot be expressed with a finite number of addition, multiplication and radicals?

2003-12-16, 21:22	#7
Orgasmic Troll Cranksta Rap Ayatollah Jul 2003 641 Posts	I forgot to add another option, is that the quintic can be factored into (x-b_i) and all the b_i's can be expressed with a finite number of operations, but it's impossible to figure out the b_i's given the a_i's

2003-12-19, 23:53	#8
michael Dec 2003 Belgium 5·13 Posts	No, it means that there isn't a general formula to express the results of a quintic as in x_i=... for i=1 to 5 -michael