Where to start studying math?
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Where to start studying math?
Ok, so, I just registered, and so I suppose firstly I should introduce myself. I'm a 23 year old student, I have endured 3 years of computer science (mostly learning to program with idiotic teachers... sigh) and 2 years in biology (mostly a lot of stuff about a lot of stuff). Yes, it's not your typical career change, but it's what I did, nonetheless.
Anyway, I'm here in the math board because it is a subject that has always interested me. When I was a CS student I took various math courses, such as diferential and integral calculus, an introductory course to linear algebra, a totally disastrous course of discrete mathematics, and a course on diferential equations. All of those, I feel, where taught as procedures that should be done in a rather mechanical fashion, and as such I never put too much thought into what I was actually doing. Still, I'm motivated enough to relearn these subjects and learn new things to get a real grasp of mathematics, if only for pure personal enjoyment. The question is, where should I start studying in this much more serious way?
I want to hear suggestions on where to start, what subjects and it'd be great if some of you could mention any books I could use.
Thanks a lot in advance!
Anyway, I'm here in the math board because it is a subject that has always interested me. When I was a CS student I took various math courses, such as diferential and integral calculus, an introductory course to linear algebra, a totally disastrous course of discrete mathematics, and a course on diferential equations. All of those, I feel, where taught as procedures that should be done in a rather mechanical fashion, and as such I never put too much thought into what I was actually doing. Still, I'm motivated enough to relearn these subjects and learn new things to get a real grasp of mathematics, if only for pure personal enjoyment. The question is, where should I start studying in this much more serious way?
I want to hear suggestions on where to start, what subjects and it'd be great if some of you could mention any books I could use.
Thanks a lot in advance!
Re: Where to start studying math?
There's a good way of building up an intuition for formal mathematics that I've been teaching myself with, and it's paralleled the undergraduate curriculum at my school fairly well too. This is mostly a list of topics but it could be constructed best as a heap.
Edit: Thought I'd put in the heap for viewing.
 12=Logic
 11=Naive Set Theory
 10=Abstract Algebra
 9=General Topology
 8=Intermediate Set Theory and Orderings
 7=Metric Spaces
 6=Analysis (real and complex, some hypercomplex)
 5=Ordinal Theory
 4=HigherDimensional Analysis
 3=Geometry
 2=Number Theory
 1=Everything Else
Code: Select all
1
10  9  7  5 <
12  11 < 3
8  6  4 2
Edit: Thought I'd put in the heap for viewing.
Re: Where to start studying math?
Thanks for the reply. And, I view that diagram not as a heap, but as a list which nodes can be binary in nature, which would mean it is a rather weird list which can contain binary trees or other lists. I guess the data structure course did serve its purpose haha =d

 Posts: 27
 Joined: Sat Aug 01, 2009 12:14 am UTC
Re: Where to start studying math?
At my undergrad we had a course that taught "math as proofs instead of calculation" to transition math majors from calculus to abstract algebra and analysis. It was a blast, and sounds like it might be about where you're at.
I wouldn't necessarily recommend the textbook from my course for independent study, though.... perhaps someone else knows of a good resource for this?
I wouldn't necessarily recommend the textbook from my course for independent study, though.... perhaps someone else knows of a good resource for this?
Re: Where to start studying math?
I would say, start with algorithms and proofs of them, if you haven't taken that course in CS already. It's probably the closest thing to what you know, and it is really just math. You might then consider graph theory after that, to make more sense of what you did in CS, and graph theory in itself is an interesting topic anyways.
Re: Where to start studying math?
I have a book on algorithms and data structures in C++, would that work for learning algorithms? It also has some theory on trees and graphs... =D
Re: Where to start studying math?
bobbeathome wrote:At my undergrad we had a course that taught "math as proofs instead of calculation" to transition math majors from calculus to abstract algebra and analysis. It was a blast, and sounds like it might be about where you're at.
I wouldn't necessarily recommend the textbook from my course for independent study, though.... perhaps someone else knows of a good resource for this?
I recommend this route as well. For abstract algebra, a lot of people will recommend Dummit and Foote's text (simply called Abstract Algebra) it's used at my school for the graduate abstract class but it's certainly an accessible text and from what I've seen, presupposes little to no previous knowledge of abstract algebra. Also it goes over more material (and is cheaper) than a lot of people's recommendation of Gallian's Contemporary Abstract Algebra. Analysis, on the other hand, is a bit more difficult to selfstudy. It's harder to know if/when you go wrong in analysis than abstract algebra.
 imatrendytotebag
 Posts: 152
 Joined: Thu Nov 29, 2007 1:16 am UTC
Re: Where to start studying math?
I would recommend:
1 Learn some basic number theory (modular arithmetic, Fermat's little theorem, unique factorization) and Naive set theory. The former helps familiarize you with rigorous proofs, and the latter gets you accustomed to the language (and, in some sense, philosophy) of higher formal mathematics.
2 Learn how to do 1dimensional calculus formally (ie, epsilondelta proofs of limits, formal definition of derivative, proof of the Mean Value theorem, formal definition of Riemann Integral, proof of the fundamental theorem of calculus)
3 Linear Algebra, here I recommend Axler's book: "Linear Algebra done Right". Just go through the book and do a lot of the exercises.
By now you should be fairly comfortable with writing formal proofs and with working with abstract ideas and symbols. Next:
4,5 Real Analysis, Abstract Algebra. Do these in either order. I would recommend Dummit and Foote for Algebra.
Others PointSet Topology, Logic, Representation Theory, More Advanced Number Theory, Differential Equations, Measure Theory (or Probability Theory or rigorous Statistics): If, after Real Analysis and Abstract Algebra you're hungry for more, pick and choose from these topics based on what interests you. I recommend Munkres for PointSet Topology.
This should satiate your appetite for mathematics for a good while! Good luck!
1 Learn some basic number theory (modular arithmetic, Fermat's little theorem, unique factorization) and Naive set theory. The former helps familiarize you with rigorous proofs, and the latter gets you accustomed to the language (and, in some sense, philosophy) of higher formal mathematics.
2 Learn how to do 1dimensional calculus formally (ie, epsilondelta proofs of limits, formal definition of derivative, proof of the Mean Value theorem, formal definition of Riemann Integral, proof of the fundamental theorem of calculus)
3 Linear Algebra, here I recommend Axler's book: "Linear Algebra done Right". Just go through the book and do a lot of the exercises.
By now you should be fairly comfortable with writing formal proofs and with working with abstract ideas and symbols. Next:
4,5 Real Analysis, Abstract Algebra. Do these in either order. I would recommend Dummit and Foote for Algebra.
Others PointSet Topology, Logic, Representation Theory, More Advanced Number Theory, Differential Equations, Measure Theory (or Probability Theory or rigorous Statistics): If, after Real Analysis and Abstract Algebra you're hungry for more, pick and choose from these topics based on what interests you. I recommend Munkres for PointSet Topology.
This should satiate your appetite for mathematics for a good while! Good luck!
Hey baby, I'm proving love at nth sight by induction and you're my base case.
Re: Where to start studying math?
Since elementary number theory (note that 'elementary' does not mean it's always easy) is very accessible, especially with what you've already covered, and since you've got a computer science background, you might find the applications of number theory to cryptography quite interesting.
Either way, I would definitely recommend starting with some number theory, rather than leaving it for last.
Either way, I would definitely recommend starting with some number theory, rather than leaving it for last.
I came here to read a cool post, a witty dialogue, a fresh joke, but stumbled upon a "bump"...
Way to go, jerk... ~CordlessPen
Way to go, jerk... ~CordlessPen

 Posts: 7
 Joined: Fri Jan 07, 2011 5:31 am UTC
Re: Where to start studying math?
My first question for the original poster is  what do you want to do?
These are obviously not mutually exclusive.
If you want to be exposed to math proper, try a proof based course. You could do worse than an abstract algebra course. Frankly, everything from linear algebra to relational algebra will make more sense after you take it. However, most schools have a intro to logic, number theory, or noneuclidean geometry course that gently introduces mathematical rigour to the student. You may find that more suitable.
If you're looking to read a book for fun or want to see how math could apply to CS and biology, you might find a book on combinatorics. I suggest combinatorics because more than likely (on the undergraduate level) it will include an introduction to graph theory. All the books I've seen specifically on graph theory are either overly simple or designed for specialists (i.e. graduate discrete math seminars, not even CS students).
In particular, I liked this book on Combinatorics and Graph Theory.
http://www.amazon.com/CombinatoricsGraphTheoryUndergraduateMathematics/dp/1441927239
If the original poster wants to elaborate on his question, we might be able to provide more helpful advice.
 Are you considering completing an undergraduate degree in mathematics?
Do you just want to take a class/read a book for fun?
Do you want to explore how math could apply to CS and biology?
These are obviously not mutually exclusive.
If you want to be exposed to math proper, try a proof based course. You could do worse than an abstract algebra course. Frankly, everything from linear algebra to relational algebra will make more sense after you take it. However, most schools have a intro to logic, number theory, or noneuclidean geometry course that gently introduces mathematical rigour to the student. You may find that more suitable.
If you're looking to read a book for fun or want to see how math could apply to CS and biology, you might find a book on combinatorics. I suggest combinatorics because more than likely (on the undergraduate level) it will include an introduction to graph theory. All the books I've seen specifically on graph theory are either overly simple or designed for specialists (i.e. graduate discrete math seminars, not even CS students).
In particular, I liked this book on Combinatorics and Graph Theory.
http://www.amazon.com/CombinatoricsGraphTheoryUndergraduateMathematics/dp/1441927239
If the original poster wants to elaborate on his question, we might be able to provide more helpful advice.
Re: Where to start studying math?
Well, I'd have to study most of what I already covered again, because I don't remember most of it. Although I'm sure if I start doing exercises again it would come back bit by bit. I'd like to apply what I learn, so finding applications for cs and biology (computational biology interests me =D) and studying them while learning more how math works would be great.

 Posts: 7
 Joined: Fri Jan 07, 2011 5:31 am UTC
Re: Where to start studying math?
Rosen's book is probably the 'best' discrete math book out there. For a discrete math course (i.e. a glorified leveling course/sequence of oddundergraduate math for CS majors) it's good book. However, I'd recommend you take a look at the aforementioned combinatorics book if you want to learn graph theory/combinatorics the 'right' way. You'd be surprised how much you'll remember anyway.
Rosen's Discrete Math book:
http://www.mhhe.com/math/advmath/rosen/
You can find some videos for the first half of Rosen's book here:
http://www.aduni.org/courses/discrete/index.php?view=cw
These lectures cover logic, naive set theory (very basic), summations (for algorithm analysis), recursive relations, and basic combinatorics.
Also, you may find the following video/audio lectures helpful. I have his algorithms book which I used for self study. His discrete math lectures focus on basic number theory, recursive relations, and graph theory.
Steven Skiena's Discrete Math lectures (the other half of discrete math)
http://www.cs.sunysb.edu/~algorith/mathvideo/
Once again, here is a good undergraduate book on Combinatorics (and Graph Theory) suitable for self study:
http://www.amazon.com/CombinatoricsGraphTheoryUndergraduateMathematics/dp/1441927239/ref=sr_1_2?ie=UTF8&qid=1294790709&sr=82
Additionally, you may find these books helpful.
David MacKay's Information Theory book:
http://www.inference.phy.cam.ac.uk/mackay/itila/
The New Turing Omnibus (A wideranging survey of CS topics)
http://www.amazon.com/NewTuringOmnibusSixtySixExcursions/dp/0805071660/ref=sr_1_2?ie=UTF8&s=books&qid=1294791943&sr=82
Rosen's Discrete Math book:
http://www.mhhe.com/math/advmath/rosen/
You can find some videos for the first half of Rosen's book here:
http://www.aduni.org/courses/discrete/index.php?view=cw
These lectures cover logic, naive set theory (very basic), summations (for algorithm analysis), recursive relations, and basic combinatorics.
Also, you may find the following video/audio lectures helpful. I have his algorithms book which I used for self study. His discrete math lectures focus on basic number theory, recursive relations, and graph theory.
Steven Skiena's Discrete Math lectures (the other half of discrete math)
http://www.cs.sunysb.edu/~algorith/mathvideo/
Once again, here is a good undergraduate book on Combinatorics (and Graph Theory) suitable for self study:
http://www.amazon.com/CombinatoricsGraphTheoryUndergraduateMathematics/dp/1441927239/ref=sr_1_2?ie=UTF8&qid=1294790709&sr=82
Additionally, you may find these books helpful.
David MacKay's Information Theory book:
http://www.inference.phy.cam.ac.uk/mackay/itila/
The New Turing Omnibus (A wideranging survey of CS topics)
http://www.amazon.com/NewTuringOmnibusSixtySixExcursions/dp/0805071660/ref=sr_1_2?ie=UTF8&s=books&qid=1294791943&sr=82
Re: Where to start studying math?
how about looking into applied math (computational analysis, operations research)?
O.R. might be a great way to use some of the skills you've already gained in CS. In short, you think/design using math but you express your models/ideas computationally. logistics, resource management (planning/scheduling/recovery), supply chain ... it's not just MBA jibberish. it's how airlines are run, it's what determines who shows up (and when) while you're at the hospital, in short, it's at the intersection between math and everyday life.
O.R. might be a great way to use some of the skills you've already gained in CS. In short, you think/design using math but you express your models/ideas computationally. logistics, resource management (planning/scheduling/recovery), supply chain ... it's not just MBA jibberish. it's how airlines are run, it's what determines who shows up (and when) while you're at the hospital, in short, it's at the intersection between math and everyday life.
Last edited by gmalivuk on Thu Jan 13, 2011 6:03 am UTC, edited 1 time in total.
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Re: Where to start studying math?
thejesus wrote:how about looking into applied math (computational analysis, operations research)?
O.R. might be a great way to use some of the skills you've already gained in CS. In short, you think/design using math but you express your models/ideas computationally. logistics, resource management (planning/scheduling/recovery), supply chain ... it's not just MBA jibberish. it's how airlines are run, it's what determines who shows up (and when) while you're at the hospital, in short, it's at the intersection between math and everyday life.
I'm interested in this, actually, have you got any books or articles or websites I can start up with? Resource management and logistics, recommendations for that would be great.
Thanks!
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