What goes into a capable math mind?

[onnyctip]

Hi guys. I'm a high school drop-out. I went back and got a General Education Development degree. I'm also doing grade twelve courses because I want to goto university and study engineering. My math is very weak but I feel if I work hard I can catch up.

Right now I'm doing "Calculus and Vectors" and "Advanced Functions" at the high school level. So let's say a great mathematician's knowledge is like a pyrimad. At the base are basics and higher up are more advanced areas that are built up from the base...

I'm thinking it goes something like:

Numbers, simple operations and arithmetic

then calculus and using higher notations

then near the top are concepts like Fourier analysis and functions.

What should I know in terms of subjects?

Thanks.

 

[Dr. Seafood]

In my opinion, the "hierarchy of mathematics" is about which topics rely on the results of other topics. Thinking of it that way,

1. Set theory, logic, operations on sets (cardinality, denumerability, Cartesian product)
2. Definitions and properties of fundamental number systems (natural numbers, integers, rationals, reals, complex numbers); arithmetic; algebra
3. General functions on sets, and functions of real variables; elementary algebraic and transcendental functions (exponential, log, trigonometric)
4. Linear algebra (vector spaces, matrices, linear mappings)
5. Sequences of numbers and real analysis
6. Calculus of functions of one real variable (limits, continuity, derivative, integral, series); and later, complex variables

I think from here, one is ready to tackle a lot of different branches of mathematics: you can head down the real/complex analysis and topology route, abstract algebra, combinatorics/discrete math, geometry (differential, Euclidean and non-Euclidean), applied math (probability and statistics, optimization). The list goes on.

By the way, are you in Ontario, maybe?

 

[onnyctip]

In my opinion, the "hierarchy of mathematics" is about which topics rely on the results of other topics. Thinking of it that way,

1. Set theory, logic, operations on sets (cardinality, denumerability, Cartesian product)
2. Definitions and properties of fundamental number systems (natural numbers, integers, rationals, reals, complex numbers); arithmetic; algebra
3. General functions on sets, and functions of real variables; elementary algebraic and transcendental functions (exponential, log, trigonometric)
4. Linear algebra (vector spaces, matrices, linear mappings)
5. Sequences of numbers and real analysis
6. Calculus of functions of one real variable (limits, continuity, derivative, integral, series); and later, complex variables

I think from here, one is ready to tackle a lot of different branches of mathematics: you can head down the real/complex analysis and topology route, abstract algebra, combinatorics/discrete math, geometry (differential, Euclidean and non-Euclidean), applied math (probability and statistics, optimization). The list goes on.

By the way, are you in Ontario, maybe?

Yes, Ottawa. I'm guessing the course names give it away?

Thanks for your info btw!