What's the difference? "Types" of Algebra, etc

[DS2C]

In my goal to self study up to Calculus, I've utilized the very useful Insights page on what books are best for a real intuitive understanding in basic high school mathematics such as Algebra, Geometry, Trigonometry. I purchased all the recommended books and I was wondering what some of the differences are between some of the "subjects".

To cut to the chase, what's the difference between:
1. Algebra and Abstract Algebra
2. Synthetic Geometry and Analytic Geometry

I searched around but really all I could find was problems and books for them. I'm really just looking for a quick idea of what their differences are in a quick simple manner ie "Synthetic Geometry is about abc and Analytic Geometry is about xyz."

Thank you for any help.

 

[fresh_42]

In my goal to self study up to Calculus, I've utilized the very useful Insights page on what books are best for a real intuitive understanding in basic high school mathematics such as Algebra, Geometry, Trigonometry. I purchased all the recommended books and I was wondering what some of the differences are between some of the "subjects".

To cut to the chase, what's the difference between:
1. Algebra and Abstract Algebra
2. Synthetic Geometry and Analytic Geometry

I searched around but really all I could find was problems and books for them. I'm really just looking for a quick idea of what their differences are in a quick simple manner ie "Synthetic Geometry is about abc and Analytic Geometry is about xyz."

Thank you for any help.

Not sure that I know the difference between algebra and abstract algebra. Perhaps the former is how to deal with equations of any kind, as if someone says "do the algebra". To me there is no difference.

For short: synthetic geometry is without coordinates, i.e. only with axioms like "two points define a straight", and analytic geometry is with coordinates.

You'll find a bit more text here:
https://en.wikipedia.org/wiki/Synthetic_geometry
https://en.wikipedia.org/wiki/Analytic_geometry
https://en.wikipedia.org/wiki/Abstract_algebra

 

[DS2C]

Thank you for your response that was quick. After reading a bit of those articles I am still entirely confused as the terms it uses to describe the terms still make no sense to me! Kind if ridiculous hah. I guess I just need to start reading the books from square one and fill in the blanks.

 

[fresh_42]

(Abstract) Algebra is about theories of groups and fields and some ring theory, too. It rather splits in linear algebra (vector spaces), commutative algebra (commutative rings) and algebra (the rest) in general.

Synthetic geometry is by compass and ruler, analytic means computational. But the classification isn't a strict one, and to be honest, not important either. Important is whether your needs will be covered or not.

 

[DS2C]

Thank you. The Geometry makes more sense now but I now think that no matter how the Algebra is described I will be clueless until I just read the books. I don't know what a single word you said regarding Algebra meant lol.

 

[jbriggs444]

Thank you. The Geometry makes more sense now but I now think that no matter how the Algebra is described I will be clueless until I just read the books. I don't know what a single word you said regarding Algebra meant lol.

At a simple level, abstract algebra is where you stop talking about numbers and start talking about abstract objects and operations.

For instance, you write down $a+b=b+a$, without requiring that a and b denote numbers. Maybe they are "rock", "paper" and "scissors" and addition is given by the table:

 +         rock     paper    scissors
           --------------------------
rock      |rock     paper    scissors
paper     |paper    scissors rock
scissors  |scissors rock     paper

The set {rock, paper, scissors} together with the "+" operation defined above form a "commutative group".

https://en.wikipedia.org/wiki/Group_(mathematics)#Definition

 

[TeethWhitener]

In the US among the general population, "algebra" commonly refers to a middle-to-high school level math course that deals with solving polynomial equations, simplifying expressions, etc. Sometimes there's a series (Algebra I, Algebra II) where algebra I refers to simple aspects of solving equations with unknowns and algebra II provides an introduction to things like vectors and matrices.

On the other hand, "abstract algebra" is typically just called algebra by research mathematicians. It refers to examination of mathematical structures (groups, rings, fields, etc.) and the relations between them. It's called abstract algebra to distinguish it from the more common non-mathematicians' notion of algebra.

Edit: Just to be clear, in the US, you don't really see any abstract algebra being taught in high school (higher-level high school math generally focuses on calculus instead). It's more commonly first encountered by math majors in early- to mid-level undergraduate studies.

 

[DS2C]

At a simple level, abstract algebra is where you stop talking about numbers and start talking about abstract objects and operations.

For instance, you write down $a+b=b+a$, without requiring that a and b denote numbers. Maybe they are "rock", "paper" and "scissors" and addition is given by the table:

 +         rock     paper    scissors
           --------------------------
rock      |rock     paper    scissors
paper     |paper    scissors rock
scissors  |scissors rock     paper

The set {rock, paper, scissors} together with the "+" operation defined above form a "commutative group".

https://en.wikipedia.org/wiki/Group_(mathematics)#Definition

In the US among the general population, "algebra" commonly refers to a middle-to-high school level math course that deals with solving polynomial equations, simplifying expressions, etc. Sometimes there's a series (Algebra I, Algebra II) where algebra I refers to simple aspects of solving equations with unknowns and algebra II provides an introduction to things like vectors and matrices.

On the other hand, "abstract algebra" is typically just called algebra by research mathematicians. It refers to examination of mathematical structures (groups, rings, fields, etc.) and the relations between them. It's called abstract algebra to distinguish it from the more common non-mathematicians' notion of algebra.

Edit: Just to be clear, in the US, you don't really see any abstract algebra being taught in high school (higher-level high school math generally focuses on calculus instead). It's more commonly first encountered by math majors in early- to mid-level undergraduate studies.

Thanks guys that was actually very helpful. Definitely clears thing up a bit.