College Math Teaching

March 29, 2013

The Quadratic Formula: case study in misunderstanding its meaning (and a moral)

Filed under: basic algebra, editorial, elementary mathematics, mathematical ability — collegemathteaching @ 8:29 pm

I admit that I never dreamed that something as innocent as this picture (a friend tagged me on Facebook) would lead to a sort-of heated argument.

qf

Of course this is the famous quadratic formula; it gives the roots to the following equation: ax^2+bx+c = 0 with a, b, c complex numbers and \sqrt{w} interpreted as the principle solution to (\sqrt{w})^2 = w . In fact this works in any field in which the square root is defined.
This formula is just a trivial consequence of completing the square: assume that a \ne 0 then
a (x^2 + \frac{b}{a}  +\frac{c}{a}) = 0 which implies a (x^2 + \frac{b}{a} + \frac{b^2}{4a^2} +\frac{c}{a}-\frac{b^2}{4a^2}) = 0 which implies (x+\frac{b}{2a})^2 = \frac{b^2}{4a^2}-\frac{c}{a} which implies x + \frac{b}{2a} = \pm \sqrt{\frac{b^2}{4a^2}-\frac{c}{a}} which implies x = -\frac{b}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} which is the formula.

But one of my friend’s “Facebook Friends” said:

I’ve never actually used the quadratic equation, i just relied on algebraic reasoning

That is a curious statement; my friend remarked that the quadratic formula WAS “algebraic reasoning.” I was curious as to what the comment meant so I posted

Ok, solve 2x^2 + 13x – 17 = 0 without using the quadratic formula OR completing the square (the two are actually the same thing).”

Then came the following response:

when you say 2x^2, do you mean (2x)^2? if so then it is one. it really is easy. you are talking to someone who took ap classes from an MIT grad without using a calculator. and jason, the quadratic formula is an example of algebraic reasoning, which is to say that there are other methods. I was not trying to imply that the quadratic formula is different from algebraic reasoning. math is the language of logic, so i usually relied on my own means to find the answer, although my means can be convoluted at times.

Evidently this individual didn’t understand the significance of my response. This is evident later:

“You really should reserve that for yourself. If it is a quadratic equation, it will almost always have more than one answer, which is outlined by the +/- part of the quadratic equation. Is 1 a possible answer? yes. Is it the only answer? no. I am referring to the the equation i provided btw. Did I assert that 1 was the ONLY answer? no.”
[…]
“I know, that is why i said it wasnt the one I was referring to the whole time. Why else would I ask to clarify? I already know that if I were to go with 2x^2, it would produce an answer with a decimal. That is because if you follow the quadratic equation you will notice that the number in the square root is 305, and the root of the 305 is pretty messy.”

See where the confusion is? Evidently he (yes, the friend and his Facebook friend is a male) did not understand that, while the quadratic formula (or the completing the square process) yields ALL possible solutions for every true quadratic (a \ne 0 ) that in no way means that one can’t, at times, guess a solution or, at times, find an easy factorization. So if you want to solve the general quadratic and find all solutions, you need this formula or the completing the square process.

Of course, in the complex coefficients case, the answer is frequently ugly.

Side notes: there is a formula for the solution to a cubic (very messy) and for the degree 4 polynomial. However, it is impossible to find a general formula to solve the degree 5 polynomial; this is a reason to learn some Galois Theory from abstract algebra!

The other fields: in general the quadratic cannot be solved if the field is, say, an integer of odd prime order, unless one extends the field by adjoining \sqrt{p-1} where p is the prime in question. This is a good reason to learn some number theory.

Moral
Often, students will put the time and effort into understanding a concept if they know WHY it is important. However, they don’t always appreciate what a formula like the quadratic does. One doesn’t always have to use it, but it
1. Provides a method of obtaining ALL solutions that is guaranteed to work in every case (where a \ne 0 )
2. Proves that, in fact, the solutions always exist and what kind they are (real or complex).

These points are not obvious to every beginner, even some who consider themselves to be “bright” and talented. Such self perceptions are the topic of a different post.

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March 25, 2013

Mary Ellen Rudin Tribute (1924-2013): Shelling a triangulated disk or ball.

Filed under: advanced mathematics, editorial, famous mathematicians, topology — Tags: , — collegemathteaching @ 1:29 am

Back in January 1991, I attended the American Mathematical Society meeting in San Francisco. I was there mostly to absorb the talks, and to help myself with the job search. It was there where I made my first contact with my current university.

One evening, I attended some point set talks. Since this was not my area, I kept my mouth shut; I didn’t ask many questions. In one talk, someone raised a question about a possible “bisection” type process in a certain case. I thought “that question makes no sense because there is no metric” but figured that I had missed something.

Then in the other corner of the room, a white haired old lady said something to the effect “that question makes no sense because there is no mention of a metric on this space.” 🙂

Then I realized who this was: Mary Ellen Rudin.

Because she was known for topology and because she was a student at the University of Texas (Ph. D. in 1949), I had heard stories about her and her interactions with R. H. Bing. It turns out that her (late) husband was also famous, although in analysis (yes, he is THAT Rudin; the one associated with the analysis books that we studied for our Ph. D. comprehensive examinations.)

I also remember reading her paper on shellings; it was the first one that I tackled as a graduate student. The paper itself is terse (and very brief) but is not overly technical; it just requires time and effort to understand. You don’t have to spend half a decade learning gauge theory. 🙂 Here is a workthrough of her paper.

So, in her honor, I’ll talk about this topic. I’ll keep it as basic as possible and try not to lie too much. 🙂
(“lie” as in, leaving out some details that a rigorous presentation would cover).

Shellings: what are they?
First of all, let’s discuss what topology is about. Two objects X, Y are said to be topologically equivalent if there exists a function f:X \rightarrow Y where f is a bijection (one to one and onto) and bicontinuous; that is f is continuous and its inverse function f^{-1} is also continuous. Note: I am using “continuous” in a topological sense, but for the spaces we are talking about, the beginner can imagine that X, Y are subsets of R^n for some n \ge 1 and that “continuous” is the “calculus” definition of continuous. This is NOT always true in general, but it works for the kind of objects we discuss here. If two objects are topologically equivalent, we say that they are “homeomorphic” and call f the “homeomorphism”.

Now geometric topology restricts itself to the study of certain nice objects, and often we require that f be differentiable (“differentiable topology”) or take polygonal objects to polygonal objects (“piecewise linear topology”, denoted by p. l.). We will work with polygonal maps, but one can make the translation to differentiable maps (“smooth category”).

Now consider a disk in the plane; the calculus student might think of the unit disk: \{(x,y) | x^2 + y^2 \le 1 \} .
In the p. l. category these objects are represented by triangles. It is an exercise to see that a traditional “round” disk is homeomorphic to a triangle (with interior).

Now consider a polygonal disk which is the finite union of triangles put together “in a nice way” (e. g., two triangles touch along a common face or exactly at a vertex, every interior point has an open set in the disk that surrounds it and the boundary consists of a finite number of line segments; faces of the triangles.). This crude illustration shows a “triangulated p. l. disk”

(figures: click to see larger)

001

One technical problem is this: it is clear that the whole disk is topologically equivalent to any one of its triangles. However it is often useful to see WHAT homeomorphism gives us the equivalence we want and to be able to describe this homeomorphism in a finite number of steps. Homeomorphisms which can be described “triangle by triangle” are especially useful. Now look at the labeled disk on the left. It is clear that the whole disk is homeomorphic to the disk that has all of the triangles EXCEPT for the first one. True, the process of simply removing triangle 1 is not a homeomorphism, but there is a homeomorphism that starts with the whole disk and, in effects, “pushes” triangle 1 into the larger disk

shelling 001

In spirit, this is a bit like the one dimensional homeomorphism f: [0,1] \times [0,1] \rightarrow [0,1] \times [0,\frac{1}{2}] given by f(x,y) =(x, \frac{1}{2}y) which compresses the unit square into a rectangle of half its original height.

So, in the disk on the left hand side of the first figure, we can find a series of homeomorphisms which, in step by step fashion, takes the whole disk to the triangle labeled 10. f_1 has the effect of taking the original disk and mapping it to the disk with triangle 1 missing. f_2 then starts with this modified disk and has the effect of removing triangle 2, and so on. One can check out that f_1 \circ f_2 \circ ....\circ f_9 is a homeomorphism from the original disk to the triangle labeled 10.

So, we an think about these maps as a labeling of this triangulated p. l. disk and such a labeling is called a “shelling” of the p. l. disk with the triangle 10 saved for last. More precisely: a shelling is a labeling of a triangulated p. l. disk with N triangles \Delta_i so that, for ALL 1 \le k \le N, \cup^{N}_{i=k} \Delta_i is a p. l. disk.

Note: not just any labeling produces a shelling! For example, if you look at the first figure and look at the disk on the right, the removal of the triangle labeled “1” leaves us with an annulus (ring with a hole) and NOT a disk; hence there is no homeomorphism that does that!

So here is a question: does every 2 dimensional p. l. disk have a shelling? The answer: “yes” and not only that: we can choose, in advance, the ending triangle. There is always a path to the last triangle (though we’ve seen that we do NOT have our choice of starting triangle).

To see a proof, see Bing’s book The Geometric Topology of 3-manifolds:

bing2dshell

This is an outline of the proof. It is fun to try it yourself.

Ok, good enough. But the follow on question:
if we now try to extend this “shellable 3-balls”: if we have a 3-dimensional triangulated (into tetrahedra) ball (homeomorphic to \{(x,y,z)| x^2+y^2+z^2 \le 1 \} ) can you always shell it?

The answer is NO. That is the subject of Mary Ellen Rudin’s paper that I talked about at the start. However there are easier counterexamples, and I’ll close with my favorite, which is suggested by this figure:

knotted_edge

Here is an idea: start with a cube (homeomorphic to a ball) and break this into a bunch of small cubes. Then start at the top of the cube and start removing a cube in the middle of the top face of the cube. Then work your way down, removing small cubes so as to leave a “knotted tunnel” from the top face of the cube to the bottom face. Of course what is left is NOT a ball due to this hole; you have “added genus”. Now we need to make this a ball, so we add the last “removed cube” back (“plugging the hole”); hence the “cube with knotted tunnel” is now a cube with an indentation in the form of an “almost hole”; hence it remains homeomorphic to a ball.

Now divide the remaining small cubes into tetrahedra to obtain a triangulated 3-ball.

002

It turns out that this ball cannot be shelled. Here is why:

1. If one removes the “plug” by shelling before ALL of the cubes above the plane of the plug cube have been removed (by removing tetrahedra) one changes the genus from zero (a ball) to non-zero (not a ball).

2. If one tries to remove all of the cubes in the plane above plug (by removing the tetrahedra), this has the effect of taking the knotted tunnel to the sides of the cube via a homeomorphism of space; this is impossible if the tunnel truly is knotted! (that is why having an honest to goodness knot is essential).

So the existence of knots in 3-space really complicates things. Hence “knot theory” remains an active field of research in mathematics.

March 18, 2013

Odds and the transitive property

Filed under: media, movies, popular mathematics, probability — Tags: — collegemathteaching @ 9:51 pm

I got this from Mano Singham’s blog: he is a physics professor who mostly writes about social issues. But on occasion he writes about physics and mathematics, as he does here. In this post, he talks about the transitive property.

Most students are familiar with this property; roughly speaking it says that if one has a partially ordered set and a \le b and b \le c then a \le c . Those who have studied the real numbers might be tempted to greet this concept with a shrug. However in more complicated cases, the transitive property simply doesn’t hold, even when it makes sense to order things. Here is an example: consider the following sets of dice:

dice

What we have going here: Red beats green 4 out of 6 times. Green beats blue 4 out of 6 times. Blue beats red 4 out of 6 times. All the colored dice tie the “normal” die. Yet, the means of the numbers are all the same.

Note: that this can happen is probably not a surprise to sports fans; for example, in boxing: Ken Norton beat Muhammed Ali (the first time), George Foreman destroyed Ken Norton and, Ali beat Foreman in a classic. Of course things like this happen in sports like basketball but when team doesn’t always play its best or its worst.

But this dice example works so beautifully because this “impossibility of the dice obeying a transitive ordering relation is theoretically impossible, by design.

Movies
Since the wife has been gone on a trip, I’ve watched some old movies at night. One of them was the Cincinnati Kid, which features this classic scene:

Basically, the Kid has a full house, but ends up losing to a straight flush. Yes, the odds of the ten cards (in stud poker) ending up in “one hand a full house, the other a straight flush” are extremely remote. I haven’t done the calculations but this assertion seems plausible:

Holden states that the chances of both such hands appearing in one deal are “a laughable” 332,220,508,619 to 1 (more than 332 billion to 1 against) and goes on: “If these two played 50 hands of stud an hour, eight hours a day, five days a week, the situation would arise about once every 443 years.”

But there is one remark from this Wikipedia article that seems interesting:

The unlikely nature of the final hand is discussed by Anthony Holden in his book Big Deal: A Year as a Professional Poker Player, “the odds against any full house losing to any straight flush, in a two-handed game, are 45,102,781 to 1,”

I haven’t done the calculation but that seems plausible. But, here is the real point to the final scene: the Kid knows that he has a full house but The Man is showing 8, 9, 10, Q of diamonds. He knows that the only “down” card that can beat him is the J of diamonds but he knows that he has 3 10’s, 2 A’s. So there are, to his knowledge, 52 - 9 = 43 cards out, and only 1 that can beat him. So the Kid’s probability of winning is \frac{42}{43} which are pretty strong odds, but they are not of the “million to one” variety.

March 12, 2013

Giving a Job Interview Talk at a Undergraduate oriented but “research required” university

Filed under: academia, advanced mathematics, editorial — Tags: — collegemathteaching @ 7:26 pm

My background: I got tenure many years ago and have been at a regional “undergraduate oriented” university for over 20 years. I’ve been to scores of interview talks. So, I’ll give some unsolicited advice about these.

Note: what I say doesn’t apply to those who are interviewing for a research oriented job (say, Big Ten or Pac Ten university) nor does it apply for “no research required” colleges and universities.

Our place has a research requirement; you won’t get tenure without a few scholarly articles. But on the other hand; we are realistic; you won’t publish in The Annals of Mathematics if you come here. You’ll spend most of your time with teaching duties (typically 11-12 hours per semester) and some time with committee stuff. So, here are my suggestions:

1. Remember that there might be undergraduate students there. So you might actually DEFINE terms like GL_n(C), GL_n and “Simple Lie Algebra” at the start, even though there is no need to do so if you are giving a talk to researchers. You might even give an elementary example of such a beast.

2. Even if you think that your talk is “general”, remember that people like me had our graduate courses 20-25 YEARS ago! So, yes, I remember learning about the “Lie Bracket” but I last saw that a long time ago. And since then, I’ve published, but in a fairly narrow area. My “absorb things quickly” part of my brain is flabby, out of shape, and calcified by 20 years of convincing students that \int \frac{dx}{x^2 + 1} \ne ln|x^2 + 1| + C .

A few elementary examples might be a good “hook” to draw people into your talk.

My recommendation:
The first recommendation is to ASK your contact about the type of talk that they want. Beyond that:

1. First third of the talk; give motivational examples, easy examples; pretend that you are trying to get FIRST YEAR graduate students interested in talking a FIRST COURSE in your area.
2. Second third of the talk: talk to the general mathematicians. We have our Ph. D.s and publications, but many of our research areas are kind of narrow. Our ring theorist might not recall off of the top of his/her head what a “Levi-Civita Connection” is. Our geometric topologist might have forgotten what the Radon–Nikodym theorem says. Our analyst might not remember what a “flat module” is. So “elementary reminders” (the kind you wouldn’t give if you were speaking to a research university faculty department) might be nice.
3. In the last third of the talk, go ahead and speak to the expert/experts in the audience; now is the time to show off. It is ok to lose the rest of us here.

Remember what we are looking for: someone who will explain things well to undergraduates, someone who is in touch with their class (audience in this case) and someone who won’t “die on the vine” due to disinterest in their subject.

March 5, 2013

Math in the News (or: here is a nice source of exercises)

I am writing a paper and am through with the mathematics part. Now I have to organize, put in figures and, in general, make it readable. Or, in other words, the “fun” part is over. 🙂

So, I’ll go ahead and post some media articles which demonstrate mathematical or statistical concepts:

Topology (knot theory)

As far as what is going on:

After a century of studying their tangled mathematics, physicists can tie almost anything into knots, including their own shoelaces and invisible underwater whirlpools. At least, they can now thanks to a little help from a 3D printer and some inspiration from the animal kingdom.

Physicists had long believed that a vortex could be twisted into a knot, even though they’d never seen one in nature or the even in the lab. Determined to finally create a knotted vortex loop of their very own, physicists at the University of Chicago designed a wing that resembles a delicately twisted ribbon and brought it to life using a 3D printer.

After submerging their masterpiece in water and using electricity to create tiny bubbles around it, the researchers yanked the wing forward, leaving a similarly shaped vortex in its wake. Centripetal force drew the bubbles into the center of the vortex, revealing its otherwise invisible, knotted structure and allowing the scientists to see how it moved through the fluid—an idea they hit on while watching YouTube videos of dolphins playing with bubble rings.

By sweeping a sheet of laser light across the bubble-illuminated vortex and snapping pictures with a high-speed camera, they were able to create the first 3D animations of how these elusive knots behave, they report today in Nature Physics. It turns out that most of them elegantly unravel within a few hundred milliseconds, like the trefoil-knotted vortex in the video above. […]

Note: the trefoil is the simplest of all of the non-trivial (really knotted) knots in that its projection has the fewest number of crossings, or in that it can be made with the fewest number of straight sticks.

I do have one quibble though: shoelaces are NOT knotted…unless the tips are glued together to make the lace a complete “circuit”. There ARE arcs in space that are knotted:

WildArc

This arc can never be “straightened out” into a nice simple arc because of its bad behavior near the end points. Note: some arcs which have an “infinite number of stitches” CAN be straightened out. For example if you take an arc and tie an infinite number of shrinking trefoil knots in it and let those trefoil knots shrink toward an endpoint, the resulting arc can be straightened out into a straight one. Seeing this is kind of fun; it involves the use of the “lamp cord trick”

Screen shot 2013-03-05 at 5.02.43 PM

(this is from R. H. Bing’s book The Geometric Topology of 3-Manifolds; the book is chock full of gems like this.)

Social Issues
It is my intent to stay a-political here. But there are such things as numbers and statistics and ways of interpreting such things. So, here are some examples:

Welfare
From here:

My testimony will amplify and support the following points:

A complete picture of time on welfare requires an understanding of two seemingly contradictory facts: the majority of families who ever use welfare do so for relatively short periods of time, but the majority of the current caseload will eventually receive welfare for relatively long periods of time.

It is a good mental exercise to see how this statement could be true (and it is); I invite you to try to figure this out BEFORE clicking on the link. It is a fun exercise though the “answer” will be obvious to some readers.

Speaking of Welfare: there is a debate on whether drug testing welfare recipients is a good idea or not. It turns out that, at least in terms of money saved/spent: it was a money losing proposition for the State of Florida, even when one factors in those who walked away prior to the drug tests. This data might make a good example. Also, there is the idea of a false positive: assuming that the statistic of, say, 3 percent of those on welfare use illegal drugs, how accurate (in terms of false positives) does a test have to be in order to have, say, a 90 percent predictive value? That is, how low does the probability of a false positive have to be for one to be 90 percent sure that someone has used drugs, given that they got a positive drug test?

Lastly: Social Security You sometimes hear: life expectancy was 62 when Social Security started. Well, given that working people pay into it, what are the key data points we need in order to determine what changes should be made? Note: what caused a shorter life expectancy and how does that effect: the percent of workers paying into it and the time that a worker draws from it? Think about these questions and then read what the Social Security office says. There are some interesting “conditional expectation” problems to be generated here.

March 3, 2013

Mathematics, Statistics, Physics

Filed under: applications of calculus, media, news, physics, probability, science, statistics — collegemathteaching @ 11:00 pm

This is a fun little post about the interplay between physics, mathematics and statistics (Brownian Motion)

Here is a teaser video:

The article itself has a nice animation showing the effects of a Poisson process: one will get some statistical clumping in areas rather than uniform spreading.

Treat yourself to the whole article; it is entertaining.

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