Tag Archives: stackexchange

Are the calculus MOOCs any good: After week 1

This is a continuation of a previous post.

I’ve been following the two Coursera calculus MOOCs: the elementary introductory to calculus being taught by Dr. Fowler of Ohio State University, and a course designed around Taylor expansions taught by Dr. Ghrist of UPenn, meant to be taken after an introductory calculus course. I’ve completed the ‘first week’ of Dr. Fowler’s course (there are 15 total), and the ‘first unit’ of Dr. Ghrist’s course (there are 5 total), and I have a few things to say – after the fold.

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Posted in Mathematics, MOOC, SE | Tagged , , , , , , , , | 4 Comments

From the Exchange: Is it unheard of to like math but hate proofs?

A flurry of activity at Math.Stackexchange was just enough to rouse me from my blogging slumber. Last week, the following question was posted.

I have enjoyed math throughout my years of education (now a first year math student in a post-secondary institute) and have done well–relative to the amount of work I put in–and concepts learned were applicable and straight-to-the-point. I understand that to major in any subject past high school means to dive deeper into the unknown void of knowledge and learn the “in’s and out’s” of said major, but I really hate proofs–to say the least.

I can do and understand Calculus, for one reason is because I find it interesting how you can take a physics problem and solve it mathematically. It’s applications to real life are unlimited and the simplicity of that idea strikes a burning curiosity inside, so I have come to realize that I will take my Calculus knowledge to it’s extent. Additionally, I find Linear Algebra to be a little more tedious and “Alien-like”, contrary to popular belief, but still do-able nonetheless. Computer Programming and Statistics are also interesting enough to enjoy the work and succeed to my own desire. Finally, Problems, Proofs and Conjectures–that class is absolutely dreadful.

Before I touch upon my struggle in this course, let me briefly establish my understanding of life thus far in my journey and my future plans: not everything in life is sought after, sometimes you come across small sections in your current chapter in which you must conquer in order to accomplish the greater goal. I intend to complete my undergraduate degree and become a math teacher at a high school. This career path is a smart choice, I think, seeing as how math teachers are in demand, and all the elder math teachers just put the students to sleep (might as well bring warm milk and cookies too). Now on that notion and humour aside, let us return to Problems, Proofs and Conjectures class.

Believe me, I am not trying to butcher pure math in any way, because it definitely requires a skill to be successful without ripping your hair out. Maybe my brain is wired to see things differently (most likely the case), but I just do not understand the importance of learning these tools and techniques for proving theorems, and propositions or lemmas, or whatever they are formally labelled as, and how they will be beneficial to us in real life. For example, when will I ever need to break out a white board and formally write the proof to show the N x N is countable? I mean, let’s face it, I doubt the job market is in dire need for pure mathematicians to sit down and prove more theorems (I’m sure most of them have already been proven anyways). The only real aspiring career path of a pure mathematician, in my opinion, is to obtain a PHd and earn title of Professor (which would be mighty cool), but you really have to want it to get it–not for me.

Before I get caught up in this rant, to sum everything up, I find it very difficult to study and understand proofs because I do not understand it’s importance. It would really bring peace and definitely decrease my stress levels if one much more wise than myself would elaborate on the importance of proofs in mathematics as a post-secondary education. More simply, do we really need to learn it? Should my decision to pursue math be revised? Perhaps the answer will motivate me to embrace this struggle.

I happened to be the first to respond (the original question and answer can be found here, and I’m a bit fond of my answer. So I reproduce it below.

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Posted in Mathematics, SE | Tagged , , , , , | 2 Comments

Visiting Gaussian Quadrature

I frequent math.stackexchange these days (if you haven’t heard of it, you should go check it out), and every once in a while I get stunned by a solution or a thoughtful question. As I took my Numerical Analysis Class my last semester as an undergrad (last semester, woo hoo!), I remember coming up against Gaussian Quadrature estimates for integration. It’s a very cool thing, but the system of equations to solve seems very challenging – in fact, it feels like one must use a numerical approximation method to solve them. While I don’t have any big qualms with numerical solutions, I much prefer exact solutions. Here is the best method I’ve seen in solving these (this is for 3 points, but we see how it could be used for 1,2, and 4 points as well), and all credit must be given to the user Aryabhatta at Math SE, from this post.

The task is easy to state: solve the following system:

$latex \begin{aligned}
a + b + c &= m_0 \\
ax + by + cz &= m_1 \\
ax^2 + by^2 + cz^2 &= m_2 \\
ax^3 + by^3 + cz^3 &= m_3 \\
ax^4 + by^4 + cz^4 &= m_4 \\
ax^5 + by^5 + cz^5 &= m_5
\end{aligned}$

We are to solve for x, y, z, a, b, and c; the m are given. This is unfortunately nonlinear. And when I first came across such a nonlinear system, I barely recognized the fact that it would be so annoying to solve. It would seem that for too many years, the solutions the most of the questions that I’ve had to come across were too pretty to demand such ‘vulgar’ attempts to solve them. Anyhow, one could use iterative methods to arrive at a solution. Or one could use the Golub-Welsch Algorithm (which I also discovered at Math SE). One could use resultants, which I did in my class. Or one could be witty.

Let’s introduce three new variables. Let x, y, and z be the roots of $latex t^3 + pt^2 + qt + r$. Then we have

$latex \begin{aligned}
x^3 + px^2 + qx + r = 0\\
y^3 + py^2 + qy + r = 0\\
z^3 + pz^2 + qz + r = 0
\end{aligned}$

Multiply equation (1) by $latex a$, equation (2) by $latex b$, and equation (3) by $latex c$ and add. Then we get

$latex m_3 + pm_2 + qm_1 + rm_0 = 0 $

Multiply equation (1) by $latex ax$, equation (2) by $latex by$, and equation (3) by $latex cz$ and add. Then we get

$latex m_4 + pm_3 + qm_2 + rm_1 = 0 $

Finally (you might have guessed it) multiply equation (1) by $latex ax^2$, equation (2) by $latex by^2$, and equation (3) by $latex cz^2$ and add. Then we get

$latex m_5 + pm_4 + qm_3 + rm_2 = 0 $

Now (4),(5), (6) is just a set of 3 linear equations in terms of the variables p, q, r. Solving them yields our cubic. We can then solve the cubic (perhaps using Cardano’s formula, etc.) for x, y, and z. And once we know x, y, and z we have only a linear system to solve to find the weights a, b, and c. That’s way cool!

Posted in Expository, Math.NA, Mathematics | Tagged , , , , | 4 Comments