Where Should I Start with Theoretical Math?

[Jkohn]

Hello all, so to keep things short and simple, let's assume I just discovered my passion for math, just for math-not because of the modern applications.

I want to learn math where what you do is proven. I really don't know where to start.

Please recommend!

ty!

-John

 

[homeomorphic]

Seems like you must have started already, considering what you are saying. Which makes it hard to say where to start. It really depends a lot on what your goal is. There may be some general comments that can be made, but everyone is different.

I've already mentioned Visual Complex Analysis several times, and I haven't posted that many times, so I feel silly bringing it up again, but it can't really be recommended too highly. The only prerequisite is high school math and calculus. Actually, not everything in the book is proven rigorously, but there aren't many claims in the book for which no argument is presented. It's just that the arguments are only intuitive ones. However, it is a good introduction to the kind of thinking (as well as to Complex Analysis) that you need to do in order to come up with and conceptually understand the proofs, especially geometric ones. Another book in the same spirit is Geometry and the Imagination by Hilbert and Cohn-Vossen. Books like those give you can idea of what math is really about, and I think that ought to be a prerequisite to reading more technical Definition, Theorem, Proof-type books. If you can't see the concepts behind at least some of the proofs, there's not much point.

After a couple of books of that nature, you could try tackling real analysis, which is a pretty basic subject that any mathematician needs to know, for which I think I would recommend A Radical Approach to Real Analysis because it makes some attempt to show where the subject might come from and what it might be good for (and the need for rigor in proofs), although I think the book may have its drawbacks. Maybe supplement it with a more standard text.

 

[micromass]

What kind of math did you already do?? Did you study calculus already??

I would start with the nice book "a book on abstract algebra" by Pinter. It is a very soft (but rigorous) introduction to abstract algebra and proofs. The book gives a nice idea of what math really is about.

 

[Jkohn]

Im 19, in high school I learned to memorize math, never to logically understand it. In geometry, the notes and tests were joke, I only learned by reading the textbook--I loved geometry.

Im currently in community college taking "college algebra", which is practically memorize this and youll get an A on the test. For example, on the exam I got a question wrong because when I graphed the function I didnt use the correct notation to point out that the line starts after 1.

Ive been reading history of math, and I really connect with the Greeks because of the fact that they didnt really care about the application, but mostly about proving through reason.

From what I understand the principles of mathematics are based off axioms which build onto theorems. I want to learn the axioms and then begin to understand the nature of numbers. From the first axiom...theorem..etc!

Next semester I am taking a class called Pre-calc/trig. So half the semester we do pre-calc and the other trig. I am really excited for trig, I LOVE shapes, that's all I think about.

As an example, take Pi. In school we are told Area of circle=Pir^2..why the **** do we use pi?

I hope I've been able to explain better what it is that I need.

ty!

-john

As an exampe, Id like to learn how the Greeks prooved Pi. Not just Pi defined as

 

[Broccoli21]

Im 19, in high school I learned to memorize math, never to logically understand it.

So you want to learn how to do proofs and perform rigorous math?

I was in a similar situation as you a while ago, and I found this to be great:

How to Prove It: A Structured Approach (Velleman)
https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

 

[homeomorphic]

Another good book is Lines and Curves: A practical geometry handbook. Again, not rigorous, but I think it's good not to get too obsessed with proof. Nothing wrong with proof, but you should understand there is more to math than definitions and axioms and formal proofs.

Often, the kind of math that is done by practicing mathematicians isn't actually 100% rigorous all the time, in the sense of each statement following strictly from only the definitions, axioms and previously proven statements in a step by step manner. The thing is, sometimes you want to prove things that are so complicated and deep that if you insist on writing down every last detail, it will just take too long to do it. So, there are certain standards of proof in different fields for when things just get too complicated to write out completely. But of course, every mathematician needs to be able to write out complete logical proofs, at least in principle. When you start doing theoretical undergraduate math classes, where you are supposed to write pretty complete proofs, sometimes, you skip a step here or there, if it's easy enough (you don't want to get too careless, though). Or, you can say x is small, instead of x is less than epsilon. It's not like you have to write it as if you were talking to a computer, although that's kind of what you try to approximate.

If you are really interested in axioms and proofs, I guess maybe there's no harm in starting on that, too. Maybe try a book on naive set theory. Halmos wrote one and he's supposed to be a good writer, but I haven't read any of his books. It will probably discuss building up the natural numbers, starting only from the empty set. I don't know that many books for that stuff.

 

[Nano-Passion]

Hey buddy, the math can get confusing at times so don't be discouraged. You need to absolutely get the basics of algebra down. If your into history you should also know that algebra was an incredible feat of the human mind back then. Things such as negative numbers and regular equations (x-3=8) are really taken for granted. It took centuries to come to the status of where we are today. Just the negative number itself was shunned out of mathematics at one point.

You might think that algebra is all memorization, and it is to a point-- but only if you don't know the history and theory behind it. Try to do pick up your algebra book and do the hardest problems there. When your algebra gets very solid calculus I will be a breeze (and very fun)!

I was at the same scenario as you are. I only started liking mathematics about last summer. I wish I would have went back and went above and beyond in my pre-calculus class, payed attention to the proofs, and solved the harder problems. By the way, I'm also into the theoretical side of math (along with physics).

Im 19, in high school I learned to memorize math, never to logically understand it. In geometry, the notes and tests were joke, I only learned by reading the textbook--I loved geometry.

Im currently in community college taking "college algebra", which is practically memorize this and youll get an A on the test. For example, on the exam I got a question wrong because when I graphed the function I didnt use the correct notation to point out that the line starts after 1.

Ive been reading history of math, and I really connect with the Greeks because of the fact that they didnt really care about the application, but mostly about proving through reason.

From what I understand the principles of mathematics are based off axioms which build onto theorems. I want to learn the axioms and then begin to understand the nature of numbers. From the first axiom...theorem..etc!

Next semester I am taking a class called Pre-calc/trig. So half the semester we do pre-calc and the other trig. I am really excited for trig, I LOVE shapes, that's all I think about.

As an example, take Pi. In school we are told Area of circle=Pir^2..why the **** do we use pi?

I hope I've been able to explain better what it is that I need.

ty!

-john

As an exampe, Id like to learn how the Greeks prooved Pi. Not just Pi defined as

Since your into the history of math (as I am), I'll tell you this. Its is told that philosophers/mathematicians first encountered pi when when they wanted to find the circumference of a circle. They might have layed out a piece of string in which they knew the length of, arranged it in the shape of a circle, and then figured out the radius. I'm sure they were astonished when they saw that one number ALWAYS kept coming up.. pi!

 

[Dembadon]

...

Im currently in community college taking "college algebra", which is practically memorize this and youll get an A on the test. For example, on the exam I got a question wrong because when I graphed the function I didnt use the correct notation to point out that the line starts after 1.

...

I don't like memorization either, but precision is extremely important in mathematics. Strict and non-strict inequalities are very different, and it will become increasingly important as you progress to write exactly what you mean when doing mathematics. It is a beautiful subject that rewards diligence and curiosity. I look forward to hearing of your progress and wish you the best of luck on your educational journey.

 

[lostcauses10x]

"I didnt use the correct notation"

It happens. Yet learning the notation alows you to read math as a language unto its self.
Play with it, play with numbers and of course read what others have already done. The history of math is a great subject unto itself.
theoretical Hmm
It was Einstein I believe that said "forget your preconceived ideas."
Such is never easy, and harder to do in math than one would think.

 

[Deveno]

Im 19, in high school I learned to memorize math, never to logically understand it. In geometry, the notes and tests were joke, I only learned by reading the textbook--I loved geometry.

Im currently in community college taking "college algebra", which is practically memorize this and youll get an A on the test. For example, on the exam I got a question wrong because when I graphed the function I didnt use the correct notation to point out that the line starts after 1.

Ive been reading history of math, and I really connect with the Greeks because of the fact that they didnt really care about the application, but mostly about proving through reason.

From what I understand the principles of mathematics are based off axioms which build onto theorems. I want to learn the axioms and then begin to understand the nature of numbers. From the first axiom...theorem..etc!

Next semester I am taking a class called Pre-calc/trig. So half the semester we do pre-calc and the other trig. I am really excited for trig, I LOVE shapes, that's all I think about.

As an example, take Pi. In school we are told Area of circle=Pir^2..why the **** do we use pi?

I hope I've been able to explain better what it is that I need.

ty!

-john

As an exampe, Id like to learn how the Greeks prooved Pi. Not just Pi defined as

as a matter of fact, the Greeks didn't "prove pi". what they DID do, is reason that the ratio of a circle's circumference to its diameter is always constant, no matter how large, or how small the circle. they knew some rough ideas of the size of pi, but they had calculational limitations (pi is difficult to calculate without some fairly sophisticated mathematics at your disposal...the accuracy of approximations didn't really get going well until the 17th century).

archimedes' "method of exhaustion" (which foreshadows the development of calculus by over a thousand years) used hexagons, then 12-gons, then 24-gons, then 48-gons and then 96-gons to approximate a circle's circumference by using perimeters. it is an impressive display of calculation, given the limited resources he had.

to answer why we use pi, it is a way to relate "circular" to "straight". and this comes up a lot more often then it seems it might. if you imagine standing on a large flat plane (plain? ok, silly pun), the points all at a constant distance from you, form a circle (which is why you can make a compass with a piece of string, a pencil, and a thumbtack). but if you want to describe how to reach these points, by going left/right/up/down, you wind up describing things in terms of triangles. which means trigonometry (which literally means "measurement of triangles"). and pi crops up all over the place with triangles (if you think about it, an angle is comparable to: how much of a circle you turn through).

by the way, i second micromass's recommendation of pinter's book on algebra.

 

[Jkohn]

Hey guys! Thanks for the reply's and for being so encouraging! Next semester I am taking Pre-Calculus/Trigonometry. In the Pre-calc description it says "The student will use mathematical induction to prove statements regarding theproperties of natural numbers."..so i am hoping that I will be learning more on how to proof things etcc..

What I was wondering was..while majoring in math is it worth seeking any type of internships?? Id like to do something productive, I was thinking intern at a local bank. But intern with a math related field would be awesome..
thoughts??
-cheers

 

[Danton]

If you want to get a general idea of what upper division mathematics is then I would recommend "A Concise Introduction to Pure Mathematics." It gives a nice introduction to basic set theory, proofs and induction, though the sections can be a bit short.

I would also recommends Pinter's book as-well, though if you've never done a proper mathematical proof it might be a bit daunting.

A good place to get experience proving theorems would be number theory since a lot of the proofs are straightforward and easy to understand. I would also recommend Combinatoric's, simply because it's a fun subject to study.

As far as internships for math majors I honestly can't think of any. I know there are summer REU's sponsored by the NSF but I think most of them require you to have taken upper division math courses.

 

[sandy.bridge]

I personally wish I had taken Linear Algebra before I had taken any calculus courses. I will have completed four Calculus courses before even touching Linear Algebra, and now that I look back, it would have been exceptionally beneficial to do i the other way around (although, for some reason it is not recommended that way at my university).