calling Math effiecienados

Pre-College Topics High School Life
1

<p>Can some one/people explain the following thories/ideas to me:
*
-Number Theory
-Complex Rank
-Category Theory
-Any concepts required to understand these*</p>

<p>Thanks I really appreciate it :)</p>

<p>Notes: -This is for the benefit of my own knowledge
-I am in Geometry in school although have taken math courses(out of school) up to where I understand integration and differetiation and how it can be used to find volumes and area</p>

2

<p>Entire books are written to explain those subjects.</p>

3

<p>Hm… Number Theory in one sentence?</p>

<p>You’ve got to be kidding me…</p>

4

<p>Number theory is an entire branch of mathematics with numerous fields and sub-categories. It is essentially the study of real integers, and there are numerous books and college-level courses you can take in it. Given it’s very broad nature, and the fact that it is essentially a more in-depth extension of an algebra course, it’s not something that is taught in all high school math curricula. There’s no way anyone anywhere could teach you all of number theory. If you happen to take a discrete math course, it will introduce you to some topics, such as prime numbers, polynomials, modular arithmetic, operations in base-n, logic, and computer number systems, and game theory. </p>

<p>Complex rank and category theory are further extensions of discrete mathematics, and would also take up a lot of time to learn.</p>

<p>If you want, you could find information on these on your own.</p>

5

<p>actually, you see, the problem is that it is very difficult for a high school freshie to understand the mathematical language they use, so i was hoping someone could sum it up in a short essay to me. Or if possible, does anyone know good books that explain these things?</p>

6

<p>First, it might be better if you ask the question on [Art</a> of Problem Solving](<a href=“http://www.artofproblemsolving.com%5DArt”>http://www.artofproblemsolving.com)</p>

<p>I’m into subjects that come up on math competitions, so although I can tell you about number theory, I don’t even know other two subject you are talking about(I know complex numbers, sets, and single variable calculus well - if they are related to what you are talking about).</p>

<p>Number theory is basically a study of integers. Topics include:

  1. Divisibility
  2. Primes
  3. GCD and LCM
  4. Modular Arithmetics(and theorems like Fermat’s little theorem, Wilson’s theorem)
  5. Floor Functions
  6. Special Numbers such as Fermat Numbers, Mersenne Numbers, and perfect numbers.
  7. Divisors( including sums of divisors and number of divisors)
  8. Diophantine Equations - equation where the solutions are integers. For example, find integer solutions for 2x+3y = 7.
    and I can go on almost forever, but I’ll stop.</p>
7

<p>I forgot to include 2 major topics.
Prime Factorization and Base Number.</p>

<p>For books,
104 Number Theory Problems by Titu Andreescu - This book is for competition preparation. It does not go deep into number theory, but the problems in this book are generally hard.
Introduction to Number Theory by Mathew Crawford - I don’t know how good this book is, but it’s probably good as an introduction to the topic since it is “Introduction” and is made by AoPS.
An Introduction to the Theory of Numbers by Ivan Niven - This book is less oriented toward competitions. I heard that it’s a good book and got an online copy, but I haven’t read the book yet. </p>

<p>I don’t know what you’re looking for and how good you’re at math, so you’ll have to choose.
Also, you can find the first one and third one online.</p>

8

<p>You spelled aficionados incorrectly.</p>

9

<p>^Maybe effcienados were really for what the OP was looking. If he finds a person who is efficient at math, s/he will explain these complex ideas in a sentence or thereabouts.</p>

<p>Great neologism OP!</p>

10

<p>I cannot explain those to you in spite of being good at mathematics. You would have to look them up in a textbook.</p>