View Full Version : My Mathematical Progress

NeFall

02-21-2008, 02:32 AM

It has been a couple of weeks since I posted "New Guy's Math Question" and I received a lot of great guidance. I wanted to tell you all, who care, where I am, and my course of study. Then I was hoping to ask some questions.

I chose to read a crash course book on Calculus I and II, I am hoping I will remember what I have forgotten and from there enroll in a Calculus III course this summer.

While reading the book it start brushing up on needed topics such as Geometry, Algebra, Functions, and Trigonometry. I have gone through it all pretty swiftly, but I notice something that has worried me, I am the worrying type.

A lot of this old stuff I do not really know how to inherently do, like factoring. I always relied on my TI-89 to factor. Like I understand how to do it and with brushing up my skills I remember sum of perfect squares and what not but when it it comes to just factoring a random expression such as 2x^3 + 2x + 4 I cannot do it without some trial and error, is that weird?

In the Trigonometry section I forgot all of the Unit Circle and a lot to do with Sine, Cosine, Tangent, Secant, etc Is this relied upon heavily in teh study of economics/finance at the PhD level?

Lastly, is it normal to forget this stuff and never really have it down like th back of your hand, meaning you can always look it up.

asianecon

02-21-2008, 02:54 AM

It has been a couple of weeks since I posted "New Guy's Math Question" and I received a lot of great guidance. I wanted to tell you all, who care, where I am, and my course of study. Then I was hoping to ask some questions.

I chose to read a crash course book on Calculus I and II, I am hoping I will remember what I have forgotten and from there enroll in a Calculus III course this summer.

While reading the book it start brushing up on needed topics such as Geometry, Algebra, Functions, and Trigonometry. I have gone through it all pretty swiftly, but I notice something that has worried me, I am the worrying type.

A lot of this old stuff I do not really know how to inherently do, like factoring. I always relied on my TI-89 to factor. Like I understand how to do it and with brushing up my skills I remember sum of perfect squares and what not but when it it comes to just factoring a random expression such as 2x^3 + 2x + 4 I cannot do it without some trial and error, is that weird?

In the Trigonometry section I forgot all of the Unit Circle and a lot to do with Sine, Cosine, Tangent, Secant, etc Is this relied upon heavily in teh study of economics/finance at the PhD level?

Lastly, is it normal to forget this stuff and never really have it down like th back of your hand, meaning you can always look it up.

While problem solving techniques are quite important, you shouldn't really stress too much especially on things as mundane as factoring algebraic expressions. Besides, you can always go back and use your TI. I often use Mathematica especially for research since you shouldn't be spending that much time on this. Just type out Factor[2x^3 + 2x + 4] and it spits out 2(1+x)(2-x+x^2).

However, keep in mind that one should be able to systematically (ie not resorting to formulae you've memorized) think about solutions to any problem since sometimes fancy calculators or symbolic software might not be able to do the job, eg I always find myself resorting to things like IVT or implicit function thm when looking at behavior of nasty functions.

Since I'm doing mostly applied theory (ie math modeling as opposed to pure theory), I find it useful to examine the problem numerically first and then attempt to solve/prove results analytically. For example in my short thesis, I wanted to see if firms price above marginal cost so given the model I formulated, I calibrated it using real data and the simulated the model to derive optimal prices. Upon seeing this, I played around with some of the parameters that I used (calibrate using a different companies' data) then derive prices again. Once convinced, I analytically proved the results for the general case.

israelecon

02-21-2008, 03:01 AM

it's totally normal to forget all those things. i'm a math major (and still in undergrad) i don't remember any trigonometric identities or anything like that. as for factoring, there are short cuts, but trial and error is basically the only way that will always work with expression like the one you gave. the point is that these are not the essential things in mathematics. math is all understanding, no remembering. if you need to differentiate and you don't remember the rules, you can look it up in a book, but if you need to prove something thats all understanding. i have math professors who can't differentiate a function without making 10 mistakes. mechanics is not what makes good mathematicians.

so basically, as long as you understand the math, i wouldn't be too worried that you don't remember all the technique.

polkaparty

02-21-2008, 03:25 AM

math is all understanding, no remembering.

I disagree. Of course you don't need to remember huge tables of trigonometric identities or integrals, but there are certain facts which you should know like the back of your hand: all the basic rules of derivatives, integration by parts, etc.

Furthermore, when you get to theory, you should know the basic definitions as well as you know the keys of on a keyboard.

i have math professors who can't differentiate a function without making 10 mistakes. mechanics is not what makes good mathematicians.

But can they recite important theorems in their line of work faster than you can say peter piper picked a peck of pickled peppers?

Israelecon is correct that mechanics is not what makes a good mathematician, but you're not going to get anywhere if you don't have the basics memorized.

Given that, the unit circle, factoring, etc., are not what I consider "the basics".

zappa24

02-21-2008, 04:17 AM

For Trig:

The important thing is the Pythagorean Theorem, slightly rewritten.

X^2 + Y^2 = R^2 (1)

Divide through by R^2 gives you

(X/R)^2 + (Y/R)^2 = 1 (2)

or

sin^2 (Theta) + cos^2(Theta) = 1 (3)

The Tan/Sec and Cot/Csc relationships can be derived by simply dividing (1) through by X^2 or Y^2.

Factoring:

The example given doesn't really have a nice way to factor. It comes down to guessing a possible x and plugging it in to see if the equation equals to 0. If it does, you have a root that can be factored out using synthetic division. If it doesn't, guess again (keeping in mind that you do have some idea which way to go after making a couple of wrong guesses.) OR you can use the graphing calculator to find one possible root and then factor it out using synthetic division. Finally, once you get it into a quadratic form, it's time for the quadratic formula (anything of the form

ax^(2d) + bx^d + c = 0

can be thought of as a quadratic form.

asianecon

02-21-2008, 04:43 AM

I disagree. Of course you don't need to remember huge tables of trigonometric identities or integrals, but there are certain facts which you should know like the back of your hand: all the basic rules of derivatives, integration by parts, etc.

Speaking of integration by parts (and other integration techniques), one needs to have a certain level of familiarity with these techniques so that you'll know what form and which solution you can use (akin to solving differential equations). However, memorizing these things isn't really that interesting. The way to make it interesting is to try to tell a story, build the "economics" behind some of these techniques. For example, you'll constantly use integration by parts in solving mechanism design models and the rationale for using this technique is that you want to simplify a dynamic problem by integrating out the state variable that creates dynamics (because what you do for a certain type [present date], affects other types [future dates]). After integrating by parts, you'll end up with essentially a static problem which you can solve through pointwise maximization. (Alternatively, solve the problem directly using control theory) Understanding why you use these techniques is as important as (or even more important than) knowing how to use them.

israelecon

02-21-2008, 09:22 AM

actually, you don't need to remember integration by parts at all. all you need to know is the product rule for derivatives. write out the product rule for differentiation take integrals on both sides move the appropriate integral to the other side of the equation and thats your formula for integration by parts.

that was my point about not having to remember if you understand.