Algebra Road map?

Rhino

I've been taking college courses and also a lot of self-learning. I really don't get a feel of how it all fits together or what an overview of all things "Algebra" would look like. (well, somethings, but feels like I don't have a clear picture yet).

Is there any type of huge big book or even just a index/chapter overview of everything from pre-algebra to the highest levels [abstract algebra?]?

What comes after College Algebra II and Abstract Algebra?

Moses555

As far as I know, after "College" algebra, typical courses would include triginometry or calculus. I don't think I would consider Abstract/Modern algebra as a follow follow up to those, but I get the feeling we have different ideas of what is abstract algebra.

As far as developing an overall picture of these topics, I think it is important to consider what the goals of these courses are. Algebra is all about tools to find the "value" of unknowns in equations. College algebra just includes more tools to the same end, and relates some of these concepts to advanced applications.

Rhino

Moses555 wrote: »

As far as I know, after "College" algebra, typical courses would include triginometry or calculus. I don't think I would consider Abstract/Modern algebra as a follow follow up to those, but I get the feeling we have different ideas of what is abstract algebra.

As far as developing an overall picture of these topics, I think it is important to consider what the goals of these courses are. Algebra is all about tools to find the "value" of unknowns in equations. College algebra just includes more tools to the same end, and relates some of these concepts to advanced applications.

So if I take up to say College Algebra II and Abstract Algebra (highest levels of ungrad Algebra they have) do you think that would cover say ~90% of currently known Algebra? The only higher course I've seen at my school was "Advanced Algebra" [for masters], but my school isn't heavy on the math department.

In regards to road map, I'll looking for more of an Index. Were do say Quardic formulas and radical equations fit into the "big picture"? [just using those as an example]

Moses555

Rhino wrote: »

So if I take up to say College Algebra II and Abstract Algebra (highest levels of ungrad Algebra they have) do you think that would cover say ~90% of currently known Algebra? The only higher course I've seen at my school was "Advanced Algebra" [for masters], but my school isn't heavy on the math department.

In regards to road map, I'll looking for more of an Index. Were do say Quardic formulas and radical equations fit into the "big picture"? [just using those as an example]

I'll try to explain my perspective on this.

I graduated college with a degree in math, and I was required to take 1 course in abstract algebra. It was a junior/senior level, math majors/minors only course dealing with the thoery of groups and focused on proof writing. While there is some historical relationship with "high school" algebra, they really are different subjects altogether. So for most people, abstract algebra isn't something they can use (or enjoy).

As for the quadratic formula, it is a tool, for solving equations with x^2 (x squared), and is covered in early algebra courses. Radical equations are really a special case of equations with powers, so I guess this would require logarithms when very complicated, so it sounds like a college algebra topic. The takeaway here is that logarithms and the quadratic equation are tools to "find x," for certain styles of equations. I could write an equation with x that required using these, or any other skill from algebra, that might just require applying them in a particular order or way.

Moses555

Now, if you're thinking of taking abstract algebra, I say more power to you. It was my favorite course in college, I learned a ton, and it really advanced my ability to think abstractly.

GoodOmens

Abstract algebra (I took it as "Modern Algebra," and I'm sure it goes by countless other names) really bears little in common with basic algebra. Instead of manipulating numbers and variables, you are manipulating more complex mathematical objects which don't immediately relate to most people's conception of "math." We're talking things like fields, rings, vector spaces.

It's really when math leaves the orbit of normal use, and enters another dimension.

I took it as a physics major because I had to take a certain number of math courses as part of the program. Personally I found it unpleasant and tedious, but I've always been more interested in math as a set of tools to solve real problems, as opposed to thinking of it as an internally consistent set of principles that don't necessarily have to have anything to do with the outside world.