Back to Algebra (and Problem Solving in math)

[johnfisch]

Hello all,

So far, I have taken the calculus sequence, introductory differential equations, and some proof-based courses. I'll be moving into the introductory analysis courses (what my institution calls "advanced calculus," which is just before real analysis at my school) in the fall, but before that I would like to return to the very foundation of calculus and differential equations: Algebra. Most of us know that algebraic manipulations in integral calculus and differential equations are (usually) more difficult that the integral and differential "operations," simply because many of these manipulations require stokes of genius, mathematical maturity (and perhaps some of theses insightful moments are based on mood and good timing). Just look at how to solve the integral for sec(x) and you know what I mean. I'm certain that I would have never come up with such a manipulation. However I'm also certain that these people also have a superior understanding of algebra. Not just in a calculation sense, but almost an intuitive sense of just how numbers interact with each other.

Alongside my school's required intro analysis textbook (Rosenlicht's Introduction to Analysis), I will be reading Spivak's calculus. So what I'm looking to do before the fall is to get an book whose primary goal is to teach strategies and general problem-solving principles, perhaps not just limited to algebra or even math at all. For example, I am reading Gelfand's "Algebra" text. Those Russians make mathematics so crystal clear. I also read Gelfand's Trig book, and I instantly saw great improvements in my integral calculus skills. Gelfand does not focus on rote computation like most of the textbooks in the general American math curricula , but rather on properties functions and problem solving skills. Perhaps I should also think about getting a book in basic number theory.

I hope you more experienced mathematicians can help me out here when you were hungry-for-understanding math major. Thanks!
-John

 

[Simon Bridge]

Welcome to PF;

Most of us know that algebraic manipulations in integral calculus and differential equations are (usually) more difficult that the integral and differential "operations," simply because many of these manipulations require stokes of genius, mathematical maturity (and perhaps some of theses insightful moments are based on mood and good timing). Just look at how to solve the integral for sec(x) and you know what I mean.

The difficulty in that example, though, is not with algebra. The algebra is just multiply by 1 and some division.

The proof of the relation: $\frac{d}{dx}\big[\ln|\sec x + \tan x | +c\big]=\sec x$ is not intuitive and what you are taught is actually the result of lots of mathematicians working on the problem over a long time ... so it is very polished and elegant. What you don't get taught is why anyone would suspect you should try such a substitution as $u=\sec x + \tan x$ in the first place. But that's not algebra.

It is possible that the substitution was intuited by someone having a strong feel for how different numbers are related to each other - but it is also possible that the substitution was arrived at through a painful process of working through lots of dead ends. More likely it was a mixture of using results already worked out and lots of painful dead ends with a little bit of inspiration close to the end. Take a look at how Fermat's Last Theorem got proved for eg.

I'm certain that I would have never come up with such a manipulation.

Have you tried?

One approach is to notice that $\sec x = 1/\cos x$ which is just geometry.
Since $\int dx/x = \ln|x|+c$ we may suspect that the solution in this case has form $\ln|f(x)|$ where f(x) will have some trig functions. This narrows down the hunt. You could also use the Euler relations. When you write those out it kinda suggests trying to simplify the denominator. The more experienced mathematicians here may have some other techniques that would lead more naturally to discovering the modern proof.

I don't think there's a way that does not require some serious trial and error though.
It's mostly lots and lots of practice.

The example I normally use is the proof of $\sum_{n=1}^N n = N(N+1)/2$
... the modern elegant proof taught in school is unlikely to be how the relationship was originally discovered - but it could be discovered starting from the common schoolyard trick of changing the order the sum is done

i.e. for 1+2+3+4+5+6+7+8+9=? you do (1+9=10)+(2+8=10)+(3+7=10)+(4+6=10)+5=45
Then you experiment to see how the method works for bigger numbers... but you can already see the start of the relation.

Gelfand does not focus on rote computation like most of the textbooks in the general American math curricula , but rather on properties functions and problem solving skills. Perhaps I should also think about getting a book in basic number theory.

This sounds very useful.
There is an emphasis in a lot of countries on an algorithmic approach to math which emphasizes identifying the type of equation/problem and then applying the appropriate step-by-step formula to get to the solution.
The result is a lot of rote memorization followed by practice examples.
The advantage is that it makes sitting and setting exams easier.
The disadvantage is that the fun part of maths - the exploration of relationships - tends to be discovered by students almost incidentally to the course.

Emphasizing problem solving itself - which is solving the meta-problem - will be heavier going at first but can pay off in the long run. Especially if you have trouble with rote learning anyway.

It looks to me like you are book-heavy right now.
I'd suggest you hold some funds back for books but don't spend them until you are more familiar with your requirements - then you can target your resources better for your needs.

 

[johnfisch]

Very interesting and helpful. Thanks! :D

 

[Simon Bridge]

No worries. Enjoy.