Ingredient: \frac{1}{1+x^2}

Divide gently in a long division sauce pan:


Integrate briskly over a low flame:

\int (1-x^2+x^4-x^6+x^8-x^{10}...)dx=


Evaluate for x=1 and let stand at room temperature for 1000 terms for accuracy to three decimal places.


Add lemon zest to taste.

But this doesn’t look like a pi! or taste like a pi!

or a quarter pi!

Ah, but if we return to the original integral

\int \frac{1}{1+x^2}dx

and make the trigonometric substitution x=tan \theta so that dx=sec^2 \theta d \theta then:

\int \frac{1}{1+x^2}dx=\int \frac{1}{1+tan^2 \theta}sec^2 \theta d \theta


\int \frac{1}{1+tan^2 \theta}sec^2 \theta d \theta=\int \frac{1}{sec^2 \theta}sec^2 \theta d \theta

=\int d \theta=\theta

which, if we return to the original substitution x=tan \theta, we see that \tan^{-1} x=\theta

So, \int \frac{1}{1+x^2}dx=\tan^{-1}x, which means that:


and since \tan^{-1}(1)=\frac{\pi}{4}, then


Enjoy your pi!

There was an interesting article in the January 31, 2016 New York Times about the proliferation of fake locksmiths on Google.

I was really interested in this because I ran into this problem about four years ago when I locked my keys in my car (while it was running!) in Seaside.  I went into the office of the hotel where I was parked and used their internet connection to search for a locksmith in Seaside.  The one I called immediately started asking for personal information rather than the address where the car was.

I asked them where they were and it turned out they were in Salem.  It wasn’t clear to me how they were going to help me from Salem, so I hung up and found a phone book.  I called a local locksmith in Seaside listed in the (paper) yellow pages.  They showed up in ten minutes and got in my car for $20.  Problem solved (for me).

This is why I was so interested to see this article in the NY Times about locksmith internet scams.  Apparently, the call centers that come up in a locksmith search farm the jobs out to independent contractors who often bait and switch by charging far more than the original quoted price.  They also are typically short-term temps who don’t care about their reputation.

Some of these scammers go so far as to create fake digital storefronts that show up on Google maps as if they were an actual local business.

The moral of all this is:


In other words don’t think that, because something exists in a mediated form, that it will necessarily exist in the physical world.  This problem has essentially no effect on Google’s revenue, so they have almost no interest in fixing or monitoring the problem.  The internet is a wonderful tool, but be aware of the REAL physical local businesses in your area and support them in the real world.


The New York Times has an interesting article about Mesopotamian mathematics.  Two-thousand-year-old clay tablets excavated in the late 19th century appear to show astronomical calculations based on the movement of the planet Jupiter.  Babylonian astronomers created a velocity-time graph and appear to have calculated the distance traveled as the area under the curve.

The original article by Mathieu Ossendrijver is posted at Science magazine.

About fifteen years ago, when the WorldWideWeb was still text-based, I came across some of the writings of Professor Richard Mitchell (1929-2002).  Mitchell was a Professor of English and Classics at Glassboro State College in New Jersey, now known as Rowan University.  Although he was a specialist in grammar, literature and the humanities in general, he had a tremendous appreciation for mathematics and a deep and penetrating understanding of what constitutes mathematics and what it’s good for.  These short excerpts below come from his two essays, “The Uses of Audacity” and “Wise Choices in Peoria.”


From “The Uses of Audacity”

Algebra is a world of principle, and a dramatic revelation of the power of principle. In fact, algebra, and even algebra alone, could provide a true and sufficient education out of which to understand the worth of living by principle…

…[Y]ou will have it in your mind that you can know something–truly know it, and not just believe it, or be informed of it–and maybe, since that is so, you can truly know something else. It’s interesting to wonder what such a something else might be.

…You will find that algebra shows you some truths. The first great truth is that there can be something real, and complete, and harmonious, and even, in some strange way, absolutely perfect right in your own mind, and made by you alone. You will see that you have a wonderful freedom not mentioned in the Bill of Rights, the freedom to decide what your mind will contain and how it will work.  You don’t have to copy the rest of the world.

Algebra tells sad truths too. Where there is no balance, there is no truth. What is equal is equal, and between the equal and the unequal there is no conference table, no convenient compromise. In this terrible law there is a hinting question for all of life…

Algebra will show you the inexorable, the endless and permanent chain of consequence, the dark thread of necessity that brought you to a wrong answer because of a tiny little mistake back in the second line. I know how unfair that seems, and how scary that what seems unfair is nevertheless justice. Is life like that too, as all of nature seems to be? How then shall we live? What are the laws of the algebra of our living, and where do they exist, where created? Who can show us how to learn them?

It takes some serious living to see the truth hidden in algebra…


From “Wise Choices in Peoria”

[Some people]…assume that things like geometry and the multiplication table are taught in schools only out of tradition, and they are easily seduced into believing that such arts are useless to those who aren’t going to make some money from them.

But in fact the mathematical arts are the best studies in which to learn certain truths that are essential to the making of wise choices. It is in mathematics that we most readily see that the permanent relationship between principle and necessity is not subject to appeal, that every particular is a local manifestation of some universal, that there is a demonstrable difference between what we believe and what we know, and that experience can never do the work of logic. It is in mathematical studies that a child … can have his first inkling of Justice and Truth…

I never studied the logistic equation as a student.  I first encountered this relationship as an instructor in one of the College Algebra textbooks I was reviewing and/or teaching from and was intrigued by the application of this “growth with constraints” model to a natural resource.  In researching applying the logistic model to natural resource consumption, I immediately ran into M. King Hubbard’s work on Peak Oil.

Then, when I was teaching integral calculus last winter, we began a unit on separable differential equations.  I was poking around looking for good application problems that would utilize separable ODEs and ran into the fundamental population differential relationship \frac{dP}{dt}=kP, followed by the relationship I had used for the logistic \frac{dP}{dt}=kP(1-\frac{P}{N}) with N defined as the “carrying capacity” or maximum growth for the population.

We went through the procedures for solving each of these ODEs (as well as the continuous mixing problems which follow a similar pattern) and then we moved on.

This year when I was teaching this topic again I was reminded of a paper my thesis advisor had given to me back in 2003 about the application of differential equations to modeling fish populations.  I was intrigued by the profusion of models that could be generated by changing the constraints for a given relationship.

Since I didn’t teach integral calculus for another 10 years after I read that paper, I had essentially forgotten most of the equations, formulas and relationships that generated the graphs that had stuck with me.  This year, while covering the separable ODEs with their applications, I began to look into the application of these relationships to fish populations and population in general.

I found two great resources that go through the set up of these relationships in a very clear manner, and each of them includes wonderful graphs showing the multiple solutions that result when the same differential relationship is solved with different initial conditions.

The opening section of Robert Borelli and Courtney Coleman’s Differential Equations: A Modeling Approach can be read here.

A student project from James Madison University written by Bailey Steinworth, Yuhui Wang and Xing Zhang can be read here.

Education Reform

The Education Reform movement has been ubiquitous nationwide since at least 2010.  The Common Core standards and the testing that goes along with them is a part of this movement, as are charter schools, vouchers, Teach for America, VAM methods of evaluating teachers, MOOCs and a variety of other disruptive innovations in education.  I just read an article by Jill Lepore in the New Yorker from June 2014.  It was linked from the website of Diane Ravitch who has been blogging about Education Reform for quite some time now.

The paragraphs that are quoted on Diane Ravitch’s web page and which I think really get at the issue are as follows:

…Innovation and disruption are ideas that originated in the arena of business but which have since been applied to arenas whose values and goals are remote from the values and goals of business. People aren’t disk drives. Public schools, colleges and universities, churches, museums, and many hospitals, all of which have been subjected to disruptive innovation, have revenues and expenses and infrastructures, but they aren’t industries in the same way that manufacturers of hard-disk drives or truck engines or drygoods are industries. Journalism isn’t an industry in that sense, either.

Doctors have obligations to their patients, teachers to their students, pastors to their congregations, curators to the public, and journalists to their readers—obligations that lie outside the realm of earnings, and are fundamentally different from the obligations that a business executive has to employees, partners, and investors…

At the University of Amsterdam, staff and faculty have joined the students in protests against recent actions taken by the institution.  Jerome Roos of the European University Institute connects the events in Amsterdam to a world wide crisis in higher education:

Structurally underfunded, severely over-financialized and profoundly undemocratic, universities everywhere are increasingly abandoning their most crucial social functions of yore — to produce high-quality research and educate the next generation of skilled, conscious citizens — and devolving ever more into quasi-private companies run by an utterly detached managerial elite.

To make matters worse, these managers…are actually being paid six-sum figures to push around insane amounts of pointless paperwork, forcing destructive workloads and unrealistic expectations onto increasingly precarious staff, treating students like simple-minded consumers and impersonal statistics, and putting immense pressure on highly talented researchers to spew out mind-numbing amounts of nonsensical garbage just to meet rigid quantitative publication quotas….

The protesters at UvA thus find themselves at the front-line of what is essentially a global fightback against the commodification of higher education and the steady reduction of knowledge and learning to an increasingly unaffordable consumer good. In many countries, this neoliberal logic has resulted in dramatic tuition hikes and budget cuts, combined with the metastization of a culture of top-down managerialism, creeping bureaucratization and the systematic precarization of academic labor — with all the attendant consequences of rising student indebtedness, the proliferation of work floor bullying, and deepening anxiety, depression and burnout among university staff.


Yesterday in Pre-Calculus class we were discussing combinatorics and came across a series of questions on how many ways can a best 2 out of 3 sets tennis match be won and how many ways can a best 3 out of 5 tennis match be won.

For a 2 out of 3 scenario there are six possibilities (AA, ABA, BAA, BB, BAB, ABB) and for the 3 out of 5 scenario there are 20 possibilities (AAA, BAAA, ABAA, AABA, BBAAA, BABAA, BAABA, ABBAA, ABABA, AABBA and ten more in which B wins).

The question then came up of how to generalize these results.  The way I approached this was to consider how many ways one of the players could win and then double the answer to include the scenarios in which the other won.

One of the issues in this type of problem is that once the winner has won the required number of games the match ends, so that a best 2 out of 3 match could never end AAB, for instance.  Another issue is that each of the possibilities of winning in a different number of sets should be considered separately.  That is, winning in the minimum number of sets, then winning after having lost one set and so forth.

Taking all of these issues into account for a competition in which the winner must win \frac{n+1}{2} out of n leads to a general formula of 2*\displaystyle \sum_{k=0}^{\frac{n-1}{2}}{_\frac{n+2k-1}{2}}C_{k}.

Or – if the winner must win g out of 2g-1 then the formula would be 2*\displaystyle \sum_{k=0}^{g-1}{_{g+k-1}}C_{k}.