Main branches of math definitions

[ErichFranz]

https://www.physicsforums.com/showthread.php?t=466170

I'm currently trying to get better at math. I've decided to do research on every facet of mathematics, starting at the source: what is a number? And going from there, getting progressively more advanced.

Instead of just trying to memorize formulas or learn how to do problems from a math book, like in school, I want to focus on the WHY. Why do the formulas work? Why does this theorem work? etc.

I think if I understand the underlying logic, I will start to understand all aspects of math better.

I've broken math into the following main branches:

Algebra
Geometry
Statistics
Calculus
Number theory
Combinatorics
Topology

How would you define each of those branches?

 

[TylerH]

Combinatorics is part of the huge catch all topic you left out, discrete math.

You can get overviews of those topics from Wikipedia. The first paragraph of the articles on these topics do a good job at giving broad overview of their respective topics.

 

[micromass]

Logic and set theory is another huge class you've left out...
Calculus is actually a part of a bigger field: analysis.

And I consider number theory to be a part of algebra...

 

[TylerH]

Do you mean college algebra? IMHO, number theory isn't a subcategory of college algebra. If you mean elementary algebra, then I'd say that it's the other way around, since number theory is the study of the relation of numbers, and elementary algebra could loosely be classified as the study of exploiting those relations.

 

[micromass]

I knew people were going to complain when I said that number theory is a part of algebra I always considered number theory as algebra. But you're probably right, I don't know much about number theory, so I'm not really qualified to say that it is or is not algebra...

But I'd like to hear from you why you consider (college) algebra and number theory to be different?? It's always fun to hear somebody's opinion about that...

 

[TylerH]

Well... seeing as the only topic of algebra that I know is the simple properties of groups... I can say less about college algebra than you can about number theory.

My understanding of algebra, is that it is the study of operators.

 

[micromass]

Well, in a sense I agree with you... Number theory is not really the study of operators, so maybe it can't be considered algebra.

It's just that algebra and number theory overlap very much. And they use thesame techniques and methods. So I'll always consider number theory closer to algebra, then something like topology... But maybe it isn't really algebra...

 

[czelaya]

Algebra: The study of structures (lines, circles,... not limited but deals heavily with the concept of the function). A very important sub field in physics is Group Theory: basically, how structures have innate symmetry. A concrete grasp of algebra, especially the function, is crucial in developing more advanced mathematics (calculus).

Geometry: The study of shapes and how the stuctures of those shapes are interrelated. It includes not only understanding basic shapes but also deals with trigonometry that is a sub-field in its own right.

Statistics: The study of defining probability and interpreting data.

Calculus: Calculus is huge stepping stone in studying physics and chemistry. It has this notion of being "impossible" because it takes a radical departure in notation. However, it's not difficult once you understand the central basics. Actually, it's quite easy.

-Calculus is the study of change and evaluating volumes based on functions. The central themes in calculus are the derivative and integral. The change (rate) is the derivative and determining volumes is integration.

-Differential equations is the study of equations with derivatives. Vastly important in all fields in physics, engineering, chemistry, economics, and anything evolving change. However, I'm not sure whether it is a sub-field of calculus. Overall, differential equations are one of the most important fields in mathematics in understanding physical systems. Differential equations are broken down into their own respective fields: partial differential equations, complex differential equations, schoastic differential equations, and the list goes on.

Number theory: ?(haven't taken)

Combinatorics: ?(haven't taken)

Topology: The study of shapes and how there properties change when they undergo some spatial change.

There are sub-fields of the above that are combinations of the above that are important in their own right.

For instance:

Differential geometry: Geometry + calculus. The importance of differential geometry is crucial in understanding so many concepts in physics (Hamiltonians dynamics, thermodynamics, and so forth).

Differential topology: Topology + calculus.

Then you have mathematical fields combined with physics.

Statistical mechanics: statistics + physics (thermodynamics + quantum mechanics)

I'm not a mathematician but overall that's my interperation.

 

[ErichFranz]

hows this for a list?

Algebra
Discrete math
Logic
Set theory
Geometry
Statistics
Calculus
Topology

 

[micromass]

Change calculus by analysis and your list sounds good...