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Back around 2000, I found a copy of Neal Koblitz’s text A Course in Number Theory and Cryptography at the Borders bookstore in Bangor, Maine.  I only worked my way through the first chapter, but was fascinated with these ideas.  I found Professor Koblitz’s website which, at the time, had a tutorial section on finite fields and elliptic curve cryptography (this may have been on the Certicom website, I can’t remember now).  I moved on to other forms of digital cryptography, like the Diffie-Hellman Key Exchange and RSA Cryptosystem, but always appreciated Prof. Koblitz’s work.  Recently, we dressed up for Halloween as a number and I chose to be the number 4.  As part of my costume, I drew the addition table for the Galois Field of order 4GF(4)=^{GF(2)[x]}/_{x^2+x+1}, and did a lot of thinking that week about the element a, which was defined as the root of the equation 0=x^2+x+1 in GF(2)

This past week, I decided to look at the mathematics behind Bitcoin and blockchain, and lo and behold, it is Finite Fields and Elliptic Curve Cryptography – I don’t know why it took me so long to find this out, but now I’m excited about these topics.  I am a little skeptical about the current “Bitcoin bubble.”  I’m not sure that these valuations are sustainable, but from everything I’ve read, the blockchain algorithm behind Bitcoin is revolutionary and the mathematics is “supercool.”

Here’s a graph of the equation y=x+1 in GF(2).

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I was perusing some old AP Calculus exams recently and ran across an interesting problem.  The free-response questions are an interesting bunch.  I won’t analyze or critique them too much except to say that they tend to be kind of the same, without much variety.

The question I was really drawn to presents the graph of a derivative function and asks a series of questions about the maximum/minimum values and points of inflection of the underlying function.  It says that if the graph below is f(x) and g(x)=\int_2^xf(x)\;dx, then etc, etc.

The graph of the derivative looks like this:

The test questions based on the graph aren’t all that interesting, but I got really interested in wanting to see the original function.  I suppose you can integrate the piecewise derivative graph and use the identified points to build a piecewise function, but I did this geometrically, since these are all triangles.  Really I was just interested in what the original function looked like – which will appear after the jump for those of you who want to think about this for a minute…

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Isosceles Problems

One of my students gave me a problem last fall that was very interesting.

The problem is posed with the following diagram:

isosceles

What is the measure of \angle CDA?  That is, what is the value of x?

I’ve given a similar problem in my trigonometry class for the past few years, except that version of the problem has a side length included and the triangle is not isosceles.

A pdf of this problem is linked below:

jun_7_mth_112_river_problem

Working from my experience with the other version of this problem, I began to write in values for the various unlabeled angles in the diagram – if we label the intersection of \overline{AD} and \overline{BC} as K, then \angle CKD and \angle AKB are both 70^{\circ}, \angle CKA and \angle DKB are both 110^{\circ}, which makes \angle KCA 50^{\circ} and \angle ADB is 40^{\circ}.

I added in new variables and created a system of four equations with four unknowns, but it was a dependent system.

The solution for this problem that was devised by the student who gave it to me is after the jump…

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

Divide gently in a long division sauce pan:

\frac{1}{1+x^2}=1-x^2+x^4-x^6+x^8-x^{10}...

Integrate briskly over a low flame:

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

=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}-\frac{x^{11}}{11}+...

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

=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}...

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

and

\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:

\tan^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}-\frac{x^{11}}{11}+...

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

\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}...

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:

“DON’T MISTAKE THE MAP FOR THE TERRITORY.”

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.

Proto-Calculus

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…