Author: John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

How I Impressed My Wife: Index

Some husbands try to impress their wives by lifting extremely heavy objects or other extraordinary feats of physical prowess.

That will never happen in the Quintanilla household in a million years.

But she was impressed that I broke an impasse in her research and resolved a discrepancy between Mathematica 4 and Mathematica 8 by finding the following integral by hand in less than an hour:

\displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}

Yes, I married well indeed.

In this post, I collect the posts that I wrote last summer regarding various ways of computing this integral.

Part 1: Introduction
Part 2a, 2b, 2c, 2d, 2e, 2f: Changing the endpoints of integration, multiplying top and bottom by \sec^2 x, and the substitution u = \tan x.
Part 3a, 3b, 3c, 3d, 3e, 3f, 3g, 3h, 3i: Double-angle trig identity, combination into a single trig function, changing the endpoints of integration, and the magic substitution u = \tan \theta/2.
Part 4a, 4b, 4c, 4d, 4e, 4f, 4g, 4h: Double-angle trig identity, combination into a single trig function, changing the endpoints of integration, and contour integration using the unit circle
Part 5a, 5b, 5c, 5d, 5e, 5f, 5g, 5h, 5i, 5j: Independence of the parameter a, the magic substitution u = \tan \theta/2, and partial fractions.
Part 6a, 6b, 6c, 6d, 6e, 6f, 6g:Independence of the parameter a, the magic substitution u = \tan \theta/2, and contour integration using the real line and an expanding semicircle.
Part 7: Concluding thoughts… and ways that should work that I haven’t completely figured out yet.

The Inspection Paradox

From http://allendowney.blogspot.com/2015/08/the-inspection-paradox-is-everywhere.html:

Airlines complain that they are losing money because too many flights are nearly empty.  At the same time passengers complain that flying is miserable because planes are too full.  They could both be right.  When a flight is nearly empty, only a few passengers enjoy the extra space.  But when a flight is full, many passengers feel the crunch.
Once you notice the inspection paradox, you see it everywhere.  Does it seem like you can never get a taxi when you need one?  Part of the problem is that when there is a surplus of taxis, only a few customers enjoy it.  When there is a shortage, many people feel the pain.
This article gives multiple examples (including computations) of the Inspection Paradox (also known as the Friendship Paradox), meaning that even the simple concept of “average” can be a little elusive. See also http://www.technologyreview.com/view/523566/how-the-friendship-paradox-makes-your-friends-better-than-you-are/

Is 2i less than 3i? (Index)

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on why it doesn’t make sense to say one complex number is less than another complex number (using the usual definition of “less than”).

Part 1: A sketch of a direct proof based on the order axioms proving that i is not an element of a number system that has the usual definition of inequality.

Part 2: An indirect proof.

Part 3: Discussion about the lexicographic ordering, which almost works.

Part 4: Two other partial orderings which almost work.

 

Predicate Logic and Popular Culture (Part 31): The Godfather

 

Let p be the proposition “I took the gun,” and let q be the proposition “I took the cannoli.” Translate the logical statement

\lnot p \land q.

Obviously, this is an allusion to one of the great lines in The Godfather.

Even though this is a simple example, it actually serves a pedagogical purpose (when I first introduce students to propositional logic) by illustrating two important points.

First, there is an order of precedence with \lnot and \land. Specifically, \lnot p \land q means (\lnot p) \land q (“I did not take the gun, and I took the cannoli”) and not \lnot (p \land q) (“It is false that I took both the gun and the cannoli”).

Second, the actual line from The Godfather is not a proposition because both “Leave the gun” and “Take the cannoli” are commands. By contrast, a proposition must be a declarative sentence that is either true or false. That’s why I had to slightly modify the words to “I took the cannoli” instead of “Take the cannoli.”

green line

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

green lineI’m afraid that I found plenty more examples from popular culture to illustrate predicate logic, but a month of posts on this topic is probably enough for now. I’ll return to this topic again at some point in the future.

Predicate Logic and Popular Culture (Part 30): The Platters

 

Let R(x) be the proposition “x can make all this the world seem right,” and let B(x) “x can make the darkness bright.” Translate the logical statement

R(\hbox{you}) \land B(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot (R(x) \lor B(x))),

where the domain is all people.

The clunky way of translating this into English is, “You can make all this world seem right, you can make the darkness bright, and everyone else can neither make all this world seem right nor make the darkness bright.” Of course, this is the sentiment expressed by the first two lines of this classic by the Platters.

green line

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 29): Grease

 

Let W(x) be the proposition “I want x.” Translate the logical statement

W(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot W(x)),

where the domain is all people.

The clunky way of translating this into English is, “I want you, and I don’t want anyone who isn’t you.” But it sounds a lot better when John Travolta and Olivia Newton-John sing it.

For professional mathematicians (as opposed to students first learning predicate logic), the more compact way of writing this would be

W(\hbox{you}) \land \exists! x W(x).

green line

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.