What to study to prepare me for Spivak?

[G037H3]

I'm currently studying Euclid's Elements and Elements of Algebra by Euler. I'd like to know what others recommend I study before Spivak/Apostol. I've been researching a lot, and have had major difficulty in discerning proper texts.

 

[G037H3]

anyone? x.x

 

[qspeechc]

Here are a few, if you're still around...

If you are shaky on proofs, or don't know anything about how to prove stuff, I recommend:
Principles of Mathematics by Allendoerfer and Oakley
How to Prove It by Velleman
You are already studying Euclid's Elements so if you can understand that it means you understand proofs at some level. A modern Euclidean geometry book for high schoolers is Geometry by Jacobs (but the 2nd or 1st edition), or Geometry by Moise. For a more sophisticated look at Euclidean geometry there is Moise's Elementary Geometry From an Advanced Standpoint.
If instead you're interested in algebra there is A Survey of Modern Algebra by Birkhoff and MacLane, which is fairly elementary, depending on how well you handle Euclid :lol:

 

[G037H3]

Here are a few, if you're still around...

If you are shaky on proofs, or don't know anything about how to prove stuff, I recommend:
Principles of Mathematics by Allendoerfer and Oakley
How to Prove It by Velleman
You are already studying Euclid's Elements so if you can understand that it means you understand proofs at some level. A modern Euclidean geometry book for high schoolers is Geometry by Jacobs (but the 2nd or 1st edition), or Geometry by Moise. For a more sophisticated look at Euclidean geometry there is Moise's Elementary Geometry From an Advanced Standpoint.
If instead you're interested in algebra there is A Survey of Modern Algebra by Birkhoff and MacLane, which is fairly elementary, depending on how well you handle Euclid :lol:

Oh I'm still around. I've already seen others recommend Principles of Mathematics, and I hope that I can get a copy and work through it, if I even know enough mathematics convention to do so. How to Prove It may be beyond my current education. After Euclid's Elements I intend on reading La Geometrie and Geometry Revisited. I love the style of Elements of Algebra (Euler) though, and really hope to find other books of similar lucidity. I looked up "A Survey of Modern Algebra", and you feel that it would be a good pre-Spivak book?

 

[qspeechc]

Principles of Mathematics assumes you know Algebra I and geometry (which you know) and nothing else, so don't worry about knowing enough mathematics. How to Prove It is pretty much the same, it only assumes typical pre-calc math, and even that is not essential, from what I recall. I would say neither book is beyond your abilities.
I do not know anything about the books La Geometrie, Geometry Revisited, and Euler's Elements of Algebra.
I suppose you want a book to read before Spivak because you are worried that you don't know how to do proofs or you don't have enough practice in proofs, is that right? If you are worried you don't know how to do proofs then I recommend How to Prove It or Principles of Mathematics. After that you should be prepared for Spivak, but if you want to get some practice in proving things before you get on to Spivak (which is to say you feel you do not have enough experience in proving things) I would recommend:
Elementary Geometry from an Advanced Standpoint by Moise. You already know geometry and this book is more rigorous than a high school geometry book and uses proof a lot more. Perhaps the familiarity of geometry will help you to get better at proving things.
A Survey of Modern Algebra is not as difficult as an undergraduate algebra book, but covers modern algebra. It always relates the new material to stuff you are familiar with from high school mathematics so will make a good transition to doing proofs.

However, with all that said, if you only read How to Prove It you will be ready to start reading Spivak.

 

[deluks917]

As a dissenting opinion I don't think you need anything to prepare you to read Spivak. I'd just save time and start reading his book. The problems in there are almost all proofs (or at least not computations) and they form a very good "introduction to proofs."

 

[G037H3]

Well, I had to leave high school (public education/my childhood was a nightmare), and I am now studying mathematics because I intend on self-actualizing instead of despairing over the past and things outside of my control. I used to read quite a bit (until about age 14; I'm 19 now, and feel very old) and I have a large amount of humanities knowledge (comparatively; its nowhere near where I should be ofc ;l), but I want mathematics to be one of my majors in university (I haven't applied yet :/). I recently moved to WA, and intend on applying to UW. >_<

I read a review on Principles of Mathematics and it seems ideal for filling in the huge gaps in my knowledge. I'm still not sure about How to Prove It. Geometry Revisited is simply something that I saw on Amazon that seemed to be a good book. La Geometrie is the work by Rene Descartes in which he formulates the idea of the Cartesian coordinate plane. Euler's Elements of Algebra is a book that develops algebra from simple arithmetic and ratios, through the solving of quadratics, and also discusses imaginary numbers and complex numbers. The writing style is very lucid and relaxed.

I had not intended on studying mathematics specifically until rather recently, when I determined that I must develop myself to the best of my abilities and avoid neglecting any essential intellectual field (mathematics, literature, history, philosophy, physics, music, visual arts, languages/linguistics). As my mathematics knowledge is severely lacking, I've decided that in order to reach my potential that I must develop a proper program of study with no real gaps. Thus I am studying Euclid. I want hard books...the problem is that I'd be unable to understand difficult material without the necessary prerequisites. =/ At this point I'm essentially a blank slate; I simply wish to follow the best possible path that can be discerned. :shy:

 

[qspeechc]

Well that explanation makes your situation much clearer to me. So having read Elements and Elements of Algebra, you know some algebra and geometry; if you want to plug the gaps in your knowledge of pre-calculus mathematics (algebra, trigonometry, coordinate geometry) before moving on to calculus then I strongly recommend Principles of Mathematics. It will cover the math you need before doing calculus. It starts will a chapter on logic and set theory, so it starts by formally introducing you to proofs. Then it goes on to number fields, groups and such things. Then algebra, trigonometry, coordinate geometry, limits, derivatives etc., probability and boolean algebra. So you will cover all the math you need to know before doing calculus, but everything is done rigorously, which is what you want. You can skip the chapters on probability and boolean algebra if you want to.
After Principles of Mathematics you will be ready to read Spivak, no problem.

 

[G037H3]

I stated that I'm 'currently studying' them, not that I have mastered them yet. And yes, Principles of Mathematics seems best (I read elsewhere on this forum that the 1963 version is better; unfortunately, I couldn't find a version on Amazon that even had a picture, and it would also be nice to preview a bit of the book). I intend on studying the results of Archimedes, as well as some advanced Euclidean geometry and intermediate algebra, either before, or concurrently, with a study of Calculus. I also intend on studying both Spivak and Apostol (Spivak first because it seems to me that if I do what I would enjoy more first, then I would enjoy Apostol more if I understand it), though the order of the two isn't critical to me.

I was relatively certain that disclosing a bit about my personal situation would clarify things, as this forum is much more mature than others I have been a part of in the past. The proper material is also a major concern because of some of my idiosyncrasies and personality features...MBTI: INFJ, if anyone has knowledge of that. Highly intuitive, easily depressed, overly perfectionistic, etc. In other words, I am a global thinker, and hate details/nitpicking and material that doesn't explain things in terms of conceptual relationships. =)

 

[qspeechc]

I'm not sure what you mean by 'intermediate algebra', but you need pre-calculus algebra (high school algebra) before studying calculus, this is very important. I would suggest you finish Elements and Elements of Algebra then to read Principles of Mathematics (btw, this is the book by Allendoerfer and Oakley; I checked on amazon.com and there is a similarly titled book by Russell, which is not the book I am recommending). Then you can start with Spivak, and you can read Archimedes and advanced geometry at the same time, if you want.
I have the second edition of Principles of Mathematics and from what I read in the preface it seems to be an improvement over the first edition: better organization, more problems etc. I cannot off-the-top-of-my-head remember what year the 2nd edition is from, I shall have to check when I get home.
I do not see the point of reading Spivak and Apostol. They cover pretty much the same material. If you want to read more about calculus after Spivak check out the suggestions in this thread:
https://www.physicsforums.com/showthread.php?t=424563
But don't get ahead of yourself :p. Study all those books, and when you're done with Spivak see how you feel and where you want to go from there. Those books should keep you busy for at least a year.

 

[G037H3]

By intermediate algebra, I suppose I mean Diophantine equations and other more abstract algebraic ideas. Modular arithmetic is yet another requisite subject, if it is not covered in Principles of Mathematics. I know of the book by Russell (I've done quite a bit of reading recently about the history of mathematics, in order to decide in what order the subjects should be studied). As for reading both Apostol and Spivak, it was pointed out by mathwonk that Apostol introduces the concepts in a more historically accurate way. The general vibe that I get from reading what others say about Spivak is that Spivak is a good book because the problems are illuminating.

To be honest, it would likely take a year if I didn't take it all that seriously. I don't think that a sustained study of Elements of Algebra, Elements, La Geometrie, Principles of Mathematics, Archimedes, Geometry Revisited, and Spivak's Calculus should take me more than roughly 9 months. I feel that with proper focus that this is enough time for me to get a proper grounding in mathematics.

 

[mathwonk]

you might just start reading any of the recommended choices and come here with questions that arise.

 

[G037H3]

I'm pretty sure that Elements of Algebra -> Elements -> Principles of Mathematics (with La Geometrie as a supplement before the coordinate geometry section) is the correct initial path. I'm almost completely sure that I'll have plenty of questions, so, considering that it would be rude to start a thread every time I have a question, I suppose that the best option would be to have one thread that I post in when I have questions? "help G037H3 not be a n00b" >_>

 

[mathwonk]

the problem with prerequisites is they take too long to go through and thus only delay your beginning with what you really want to do. so if you want to read spivak, start there. you'll find out soon enough if you can read it. but don't give up too soon. it is hopeless to prepare so throughly that reading spivak (and doing the problems) will become easy.

 

[G037H3]

the problem with prerequisites is they take too long to go through and thus only delay your beginning with what you really want to do. so if you want to read spivak, start there. you'll find out soon enough if you can read it. but don't give up too soon. it is hopeless to prepare so throughly that reading spivak (and doing the problems) will become easy.

That isn't my only goal though. I have very little mathematics knowledge, and studying some other material first will help rectify that problem.

 

[deluks917]

I don't have much experience in studying math but I recently worked through Spivak. I really think the book is a great place to become comfortable with many pre-calc topics. In my case I finally understood induction and could remember the properties of logs after I had read Calculus. Anyway I really don't think anything beyond basic algebra and some trig identities is needed at all.

 

[G037H3]

I looked at it a bit...I saw the notation and I was like "what o.o".

 

[fluxions]

I looked at it a bit...I saw the notation and I was like "what o.o".

Spivak, like the author of any decent mathematical text, introduces and carefully defines the meaning of all the notation that he uses. In other words, there is no reason to avoid the text because you aren't already familiar with all the notation contained therein.

Also, I think you are wasting your time reading "Elements of Algebra, Elements, La Geometrie, Principles of Mathematics, Archimedes, Geometry Revisited" (cf. post #11) as a prelude to Spivak. All that is required is a decent command of high school algebra, and perhaps some feeling for what it means to prove a statement.

 

[G037H3]

Spivak, like the author of any decent mathematical text, introduces and carefully defines the meaning of all the notation that he uses. In other words, there is no reason to avoid the text because you aren't already familiar with all the notation contained therein.

Also, I think you are wasting your time reading "Elements of Algebra, Elements, La Geometrie, Principles of Mathematics, Archimedes, Geometry Revisited" (cf. post #11) as a prelude to Spivak. All that is required is a decent command of high school algebra, and perhaps some feeling for what it means to prove a statement.

Think of the other material as developing solid understanding of patterns and space. >_> I'm very abstract/intuitionist, so developing things in a 'left brain' way isn't very interesting to me. =) I may start Spivak before finishing all of those things (probably before starting Archimedes and Geometry Revisited), but I feel that to bring my intellectual powers into play, I have to develop a proper intuition for numerical/object relationships. I'll learn things from these books that others don't learn. ;D

 

[mathwonk]

i think you have enough advice to make your choices. (Still I offer more.) i agree you could probably begin spivak now, but since your goal seems to be to explore a bit more, you should do that.

In that vein, Principles of mathematics is a good book to help bridge the gap from high school to college math. it contains small introductions to several different topics, logic, set theory, circuit theory and switching, groups, fields, complex numbers, analytic geometry, calculus, probability, as I recall, but not much depth on any of them. Just the logic section of this book gave me the ability to begin a spivak style calculus course in college without totally floundering. I think I gave my copy away last month when moving out of office or I would offer it to you.

The books you are already reading have the advantage of being written by masters, like Euclid, and Euler. And geometry revisited is by the great geometer coxeter. If you can read and understand any of those books you have the stamina and smarts to read any contemporary math textbook for college students, even spivak.

If you want a book that surveys a wide range of mathematical topics, "What is mathematics?" by Courant and Robbins is more substantial and has both more breadth and more depth probably than Principles of Math.

 

[G037H3]

I intend on ordering Principles of Mathematics once I finish Elements of Algebra in a week or two. I want to develop different strands of mathematics because it's the same way I approach other subjects; as I am a polymath in the humanities, I wish to be a polymath in mathematics/physics. I know that in modern times that hyperspecialization takes place, but I feel that if I organize my education properly, I can develop knowledge that has relevance to every branch of mathematics. I don't mean to the degree of Euler, thinking of everything as math, but more like Poincare or Gauss. Number theory/analysis, and probability, I suppose (based on what I know now about different branches, i.e. not much).

I looked at Spivak again (I'm a member of a Google Documents group that has hundreds of books; unfortunately the copy of Spivak is somewhat poorly scanned, and some pages are illegible), and it seemed to be much less daunting than the first time I laid eyes on it. I did notice that I need to learn trigonometric identities though. >_>

I remembered seeing What is Mathematics? (blue cover, right?), and not looking it over that closely, but since you brought it up I will actively search for it.

Since I'm not in university, I don't have anyone to ask questions that elude my immediate intuition and reasoning, so would asking them in this forum be okay? I don't mean this subforum specifically, but I don't know. =/

http://www.youtube.com/watch?v=IUBPH6vp5Uo"

 

[mathwonk]

questions on calculus should go into the calculus forum section, etc...

by the way the right way to approach trig identities is via the exponental function. probably euler treats this in some way.

anyway, the power series approach, which euler takes, quickly shows that if you use complex as well as real numbers, then e^(ix) = cos(x) + i sin(x).

Then the straightforward and plausible fact that e(a+b) = (e^a).(e^b), immediately yields, by equating real and imaginary parts, that

sin(x+y) = sin(x)cos(y) + cos(x)sin(y), and cos(x+y) = cos(x)cos(y) - sin(x)sin(y).

these are the basic trig identities. E.g. the one for tan(x+y) = sin(x+y)/cons(x+y) comes from dividing these.