New to Math: Struggling with Spivak's Calculus

[l'Hôpital]

This Fall, I'll be an incoming freshman taking my school's equivalence to "Intro. to Proofs" class. However, I've been wanting to go ahead and start already, so I picked up Spivak's Calculus book, as many people suggested it to me as "real" Math. So far, the content itself is easy and understandable. His approach at the Delta-epsilon proofs was rather beautiful, if I do say so myself. However, when it comes to the problems, I just collapse. I'm able to some, but nowhere near enough to say "I've done enough for this section." It worries me. Am I not intelligent enough to be a mathematician? I mean, some of these proofs are doable, but then some I'm not able to understand them even after reading the answer manual. Granted, I'm not used spending so much time on problems (let's be honest, I used Stewart's for my Calculus knowledge and we all know those problems weren't exactly brain-busters) and perhaps I'm not spending adequate on time on them, but I can't help but feel like I should able to do them if they were in the problem set.

Should I be worrying that I'm struggling so? Is Spivak too much for someone who hasn't even taken their first proof class? How should one begin one's "mathematical enlightenment" for lack of better words? Am I just wasting my time? Or am I just worrying for nothing?

 

[snipez90]

First of all, I think you are prematurely judging whether or not you are fit to be a mathematician. Spivak's exercises, while difficult, do not necessarily reflect how well you can handle more advanced math courses. Learning the concepts well is more important than being able to come up with the trick that will easily lead you to a solution to a particular problem.

Having said that, if you want to become a better problem-solver, you have to spend more time thinking about the exercises. If you can't figure out the gist of a problem immediately, then the problem deserves more thought. You should not ever read an entire solution in the solution manual, but only enough to maybe provide you with the right first step. On the other hand, the problem may be that you haven't understood the material in the chapter well enough. Yes, Spivak is fairly easy to digest, but if you can't motivate every step of the proof on your own, then you won't be ready for most of the problems.

I think you should reread the chapters in Spivak and see if you can reproduce the proof on your own. Don't worry too much about the formal write-up, but make sure you can fill in the logical steps. Then see if the problems become clearer.

 

[Tac-Tics]

Something I've noticed with many proof-heavy math texts is that they like to stick real ballbuster problems after each chapter.

I'm pretty sure in a few of these cases, the theorem took months or years to prove originally. You, however, are expected to solve it in under a week, because you have all the necessary tools supplied to you in the preceding chapter.

Don't get too discouraged, though! As long as you can knock a few of them out, you're still in the clear. Being able to do every. single. problem. in a math text is a madman's pursuit. (It just happens that there are a fair number of less-than-sane mathematicians).

Make an honest attempt at each problem you can. After you work with a subject enough, you still might not be able to prove the theorem, but you can provide an outline for how it probably needs to be proved. You make medium-to-large leaps in logic whenever you get stuck, with confidence that a more careful or harder working person could go back and work out the annoying details.

Do you have any examples of problems you're having trouble with?

 

[l'Hôpital]

Here's one that I didn't quite understand.

For example, proving that the square root of 3 is irrational. Assume any number can be written in either as either 3n, 3n+1, or 3n+1.

Then, (3n + 1)^2 = (9n^2 + 6n + 1) = 3(3n^2 + 2n) + 1.

From this, they conclude that if a number k^2 is divisible by 3, so must k. The rest of the proof is simple, but that step confuses. How can you conclude such a thing from that statement?

 

[jgens]

I think you've made a typo, it should be: Assume any number can be written as $3n$, $3n + 1$, or $3n + 2$ where $n$ is some natural number.

We now have three cases:
(1) $(3n)^2 = 9n^2$, so if $k$ is divisible by three then clearly $k^2$ is also divisible by 3.

(2) $(3n + 1)^2 = 9n^2 + 6n + 1 = 3(3n^2 + 2n) + 1$, so if $k = 3n + 1$ then $k$ and $k^2$ are not divisible by 3.

(3) $(3n + 2)^2 = 9n^2 + 12n + 4 = 3(3n^2 + 4n + 1) + 1$, so if $k = 3n + 2$ then $k$ and $k^2$ are not divisible by 3.

This proves that if 3 divides $k^2$ then 3 also divides $k$.

Here's another way of looking at this: You know that $3$ is a prime number, so no product of primes unequal to $3$ will ever equal $3$. Now suppose that $3$ divides $k^2$ but not $k$. Do you see that you instantly run into a contradiction? This is a reason why that statement is simple.

 

[l'Hôpital]

Ah, that makes sense.

Spivak never broke it up into cases. It just gave the 2nd and 3rd case and just jumped into that conclusion, but that's make perfect sense.

How do you guys write with the fancy lettering?

 

[jgens]

Using LaTeX. If you click on the equations, it gives you the code that I used. You can find a reasonable tutorial here: http://en.wikibooks.org/wiki/LaTeX/Mathematics

 

[soandos]

Not sure if this works, but if does, i think its simpler:

if sqrt(a) is rational then
sqrt(a)=b/c a fraction in simplest form (c = 1 is the only time it is rational)
a = b^2/c^2
if a is an integer, there must be a common factor in b^2 and c^2 meaning that b/c could not have been in simplest form.
proof by contradiction.
does that work?
if a = 3, it holds

 

[union68]

Do you have a good grasp on logic? Have you been introduced to logical connectives, like "iff" and "if,... then" statements? Have you seen quantifiers? Truth tables? Do you understand direct and indirect methods of proof? Do you know that a false antecedent always guarantees the truth of an implication?

If you haven't seen this type of stuff, then don't give yourself too hard of a time. You'll cover this stuff in your "Intro to Proofs" class, and I'm pretty sure that you'll have a better handle on Spivak after. I think you just need more time to build up confidence with arguments, and I don't think Spivak is the place to start.

Specifically, using the epsilon/delta definition in proofs demands that you have a VERY solid grasp on the quantification of the variables and the exact meaning of the logical connective; not just the intuition behind it! That definition has three quantifiers on it -- it's not something I would give to somebody just starting out with proofs.

The jump from computational to theoretical is sometimes very difficult, but you'll get there. If I can do it, then you can...trust me!

 

[l'Hôpital]

I've seen "iff" but I only it means "if and only if" and I have no idea what the difference between that and just if means. xD.

I haven't seen any logical connectives or anything of the sort. I'm taking the equivalent of intro to proofs (called Sequences, Series and Foundations) this fall which, judging from the course descriptions, covers everything you mentioned.

And yeah...delta-epsilon...oh god, haha. I can do only the basic cases. Perhaps you're right. Either way, I'm just hoping to get some good stuff out of Spivak and perhaps be fully prepared for the class in the fall.

Thanks for the encouraging words, though. Really appreciated.

 

[Elucidus]

I'd recommend picking up a copy of How to Prove It: A Structured Approach by Daniel J. Velleman.

It nicely introduces all of the logical and set theoretic foundations one needs to explore different proof techniques and then uses these techniques to work out problems from function theory, number theory, relations, and cardinality. Nice book.

Don't worry about the $\delta , \varepsilon$ proofs; everyone struggles with these. Fortunately only a few are needed to bootstrap more sophisticated methods in the lower level courses. By the time they become more commonplace, hopefully one has had more exposure and experience with proof techniques.

--Elucidus

 

[Tac-Tics]

Specifically, using the epsilon/delta definition in proofs demands that you have a VERY solid grasp on the quantification of the variables and the exact meaning of the logical connective

This is very true. To make matters worse, the epsilon-delta definition is usually provided in an English-language way that confuses it even further -- using the word "whenever" as a left-implication, and leaving the outermost universal quantification on the second variable implicit (which is sometimes appropriate).

I've seen "iff" but I only it means "if and only if" and I have no idea what the difference between that and just if means. xD.

"If" is a logical implication. It's only one way, though! If a number is divisible by 6, then it's even. That's a true statement. If you "turn around" an if statement, it's not necessarily true. "If a number is even, then it's divisible by 6" clearly isn't.

If you want to "flip" an if statement around, the best you can do is negate both sides. This is called the contrapositive. It's the basis of proof by contradiction. "If a number ISN'T even, then it ISN'T divisible by 6".

If an if statement really DOES work both ways, the two statements are logically equivalent and we sometimes say "if and only if" or "iff". "A number is congruent to 0 mod 2 iff it's even". You can flip these around all you want.

I'd recommend picking up a copy of How to Prove It: A Structured Approach by Daniel J. Velleman.

I've browsed through this book at the book store and I really like it. I wish I had it when I was learning.

 

[l'Hôpital]

Hm...Yay! How to Prove It: A Structured Approach is at the local library. I'll have to get it. So should I read that before reading Spivak? Amusingly enough, Spivak gets easier once you enter the calculus portion, but it could be because I've seen a lot of this derivative and integration things. Either way, I'll read them both. The more I keep reading in some of these threads, the more I feel behind everyone else. It seems as if everybody started Calculus with Spivak, Apostol and whatnot and I got stuck with Stewart : /.

Either way, thanks everybody for everything.

 

[union68]

This is very true. To make matters worse, the epsilon-delta definition is usually provided in an English-language way that confuses it even further -- using the word "whenever" as a left-implication, and leaving the outermost universal quantification on the second variable implicit (which is sometimes appropriate).

Exactly. Learn the definition as

$$\lim_{x\to a}f \left(x\right) = L \quad\textnormal{means}\quad \forall \epsilon >0, \exists \delta>0, \forall x \left( 0 < \left|x-a\right| < \delta \Rightarrow \left| f\left(x\right) - L \right| < \epsilon \right)$$

and later on you'll understand why mathematicians are always harping about rigor and precision!

 

[Tac-Tics]

$$\lim_{x\to a}f \left(x\right) = L \quad\textnormal{means}\quad \forall \epsilon >0, \exists \delta>0, \forall x \left( 0 < \left|x-a\right| < \delta \Rightarrow \left| f\left(x\right) - L \right| < \epsilon \right)$$

I agree. This is the clearest way to write it out.

There are still subtleties to what the qualifiers mean and how they work. For the epsilon-delta, keep in mind that since the delta quantifier exists inside the scope of epsilon's scope, delta may be defined in terms of epsilon. Another way to say it is epsilon is a constant with respect to delta.

 

[union68]

Hm...Yay! How to Prove It: A Structured Approach is at the local library. I'll have to get it. So should I read that before reading Spivak? Amusingly enough, Spivak gets easier once you enter the calculus portion, but it could be because I've seen a lot of this derivative and integration things. Either way, I'll read them both. The more I keep reading in some of these threads, the more I feel behind everyone else. It seems as if everybody started Calculus with Spivak, Apostol and whatnot and I got stuck with Stewart : /.

Either way, thanks everybody for everything.

Don't sweat it, man. I was in the same boat you were, and I'm actually probably not that far ahead of you. I'm an applied mathematics major at an engineering school. Translation: my intro calculus classes had NO theory (the engineers would likely have revolted if there was some).

You're self-studying Spivak, which means you have curiosity and initiative. Those are VERY important traits. Don't sell yourself short.

Do you know who Terence Tao is? If you were to look up "child-prodigy" in the dictionary, his picture would be right there. He's now a prof at UCLA...read this:

http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

 

[jasonRF]

Do you have a good grasp on logic? Have you been introduced to logical connectives, like "iff" and "if,... then" statements? Have you seen quantifiers? Truth tables? Do you understand direct and indirect methods of proof? Do you know that a false antecedent always guarantees the truth of an implication?

If you haven't seen this type of stuff, then don't give yourself too hard of a time. You'll cover this stuff in your "Intro to Proofs" class, and I'm pretty sure that you'll have a better handle on Spivak after. I think you just need more time to build up confidence with arguments, and I don't think Spivak is the place to start.

Specifically, using the epsilon/delta definition in proofs demands that you have a VERY solid grasp on the quantification of the variables and the exact meaning of the logical connective; not just the intuition behind it! That definition has three quantifiers on it -- it's not something I would give to somebody just starting out with proofs.

The jump from computational to theoretical is sometimes very difficult, but you'll get there. If I can do it, then you can...trust me!

I second Union68. I am a practicing engineer, with a lot of experience doing analytical
"analysis" (in a loose sense of the word). The past year or so I have been trying to self-learn some more theoretical math, mainly just for fun, but partially so that my eyes don't glaze over when I read some of the more high-brow engineering literature.

Anyway, a lot of books were too intimidating for me, and I had to struggle too much and would give up very quickly. Reading a book that explicitly taught about logic, quantifiers, relations, sets, and structures of proofs has helped a bunch! The other folks here know better than I what books to recommend on such topics; I found one that worked for me but it is uninspiring. I have made a lot of progress, and you can too! If you take the proofs class next semester and work hard at it, I would bet that by this time next year you will be amazed at your progress, and Spivak will be more fun!

Also, remember that your profs and TAs have been struggling with the material for years, and that is how we all learn. They are not necessarily a lot smarter or more gifted than you, just because it looks like this stuff is easy for them. I know that when I TA'd upper division electrodynamics I had taken at least 5 courses at the same or higher level - yes the problems for that class were pretty easy for me, but only because I had spent hundreds of hours solving electrodynamics problems and struggling with the material on my own. It was HARD for me the first time through, that is for sure!

You have great initiative - more than I did before I started school. You can do this!

jason

 

[Elucidus]

Hm...Yay! How to Prove It: A Structured Approach is at the local library. I'll have to get it. So should I read that before reading Spivak? Amusingly enough, Spivak gets easier once you enter the calculus portion, but it could be because I've seen a lot of this derivative and integration things. Either way, I'll read them both. The more I keep reading in some of these threads, the more I feel behind everyone else. It seems as if everybody started Calculus with Spivak, Apostol and whatnot and I got stuck with Stewart : /.

Either way, thanks everybody for everything.

I have not read Spivak, so I cannot comment on its utility, but I do instruct from Stewart's (our college is shifting from Essential Calculus to the sixth edition Calculus) and I have found it servicable, not scintilating, but functional and complete enough to expose freshmen to the fundamentals of differential and integral calculus.

I have always found that texts that focus on exercises that can only be solved using "tricks" or clever solution methods tend not to prepare students for the reality that many times things just don't behave in a friendly or elegant way and brute force is the appropriate approach.

However, a strong enough exposure to the theory behind how things work (and just as importantly under what conditions it doesn't) can be invaluable in adapting techniques to new situations.

Not to demean anyone, but I have run into more than a few students who've been shown the "plug-and-play" recipe methods for too long who get lost when faced with something they haven't encountered yet. I suspect a little theory can help in a pinch.

The other thing I encounter is the impression that the proofs or derivations should somehow be obvious. Not so. It took many mathematicians often decades to solve some of the theorems that are sometimes offhandedly summarized beginning with "obviously ..." The impossibility of cubing the circle took more than a dozen centuries!

So if you aren't getting things at first, do not despair; many luminaries wracked their brains trying to solve these things too (many of then unsuccessfully). We are learning from their trials - whether successful or not.

I have found that true understanding of how everything works comes gradually, some gaps being filled in only after subsequent work causes things to fall into place. Trying to explain it to someone else is often a telling test of how well one grasps something as one will often be asked unexpected questions.

Good luck and enjoy the journey.

--Elucidus

EDIT: That should be "squaring the circle."

 

[Edgardo]

Have a look at the thread How to write Math Proofs.