I programmed this loop onto a TI84 calculator for fun and it works quite well. Pseudocode for this looks like:

Input A

A/2 = B

While abs(A-B^2) > 0.01 do

A/B=C

(B+C)/2=B

else

Display B

Then I wondered if I could do something similar for other roots. I tried a fifth root program, but changed the “A/B = C” to “A/B^4=C” and tried to run the program, but it ended up in an infinite loop. Somehow the process was skipping over the fifth root I was looking for. I started with just to keep it simple. The algorithm started with large values but as it got closer to 2, it failed converge on the desired answer. Here’s the pseudocode for my first try:

Input A

(A+1)/2 = B

While abs(A-B^5) > 0.01 do

A/B^4=C

(B+C)/2=B

else

Display B

If you try this algorithm starting with:

32/2.3^4 = 1.1435

(2.3+1.1435)/2 = 1.72175

So far so good, 1.72175 is closer than 1.1435, but when we do 32/1.72175^4 we get 3.6414 which is farther away than 2.3 was. So I decided to take a weighted average and made the algorithm:

Input A

(A+1)/2 = B

While abs(A-B^5) > 0.01 do

A/B^4=C

(4*B+C)/5=B

else

Display B

And it worked like a charm! Then I extended this to other roots as:

Input A

Input R

(A+1)/2 = B

While abs(A-B^R) > 0.01 do

A/B^(R-1)=C

(R-1)*B+C)/R=B

else

Display B

And again – it works great! I haven’t figured out the problem with the original fifth root algorithm I tried, but I think that the A/B^4 introduces something that throws it off.

]]>In his review of of this collection, Gardner says:

In Princeton’s Fine Hall, Boas recalls, someone once posted a “Scale of Obviousness”:

If Wedderburn says it’s obvious, everybody in the room has seen it ten minutes ago.

If Bohnenblust says it’s obvious, it’s obvious.

If Bochner says it’s obvious, you can figure it out in half an hour.

If von Neumann says it’s obvious, you can prove it in three months if you’re a genius.

If Lefschetz says it’s obvious, it’s wrong.

What I love about this little bit of mathematical humor is that the idea of an “obvious” proposition in mathematics is so inherently subjective. In fact, in grad school (when I first read this) we had our own joke, which was that if we didn’t know how to justify some part of the proof we were working on, we should just write “clearly.” Or “the proof is left to the reader.”

On top of the humor is the historical nature of this little joke. The people mentioned are all well-known mid-century mathematicians from Princeton University faculty.

]]>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 in .

]]>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 and , 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…

I had fun doing this as an exercise in geometry rather than Calculus, but it was interesting afterwards to look back and think – yeah, of course these are parabolas, if you’re integrating linear functions!!

]]>The problem is posed with the following diagram:

What is the measure of ? That is, what is the value of ?

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:

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 and as , then and are both , and are both , which makes and is .

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…

His solution was primarily synthetic and not algebraic!

His first step was to draw line segment such that and are congruent, creating a small isosceles triangle in the interior of the larger one. Then and are both and is

Once this is done, we can add back in and look at triangle . Notice that and , so that is also , making isosceles with sides and congruent.

Now, if we draw , we know that , and are all congruent. With and congruent, then is isosceles. With vertex angle , then both and are as well, making equilateral with side congruent to , and .

Then, we add back into the diagram, is now as the original angle was and is .

So, with redrawn and , then in triangle , and , then must also be , making isosceles and sides and congruent.

But side is already congruent to in the equilateral triangle , so that means that and are congruent and is isosceles as well. Once we know this, the solution is trivial – with , then must be split equally between angles and , so and !

The reasoning here is somewhat tricky and convoluted, but I looked for a simple algebraic solution to this for hours and got nowhere.

In researching this problem I did run across a similar but different problem.

Chris Harrow, Mathematics Chair at the Hawken School in Cleveland, OH, has a math blog called CAS Musings. Here, he ponders a problem involving isosceles triangles with no angle measures given. This problem generates a system of 6 equation with 6 unknowns that he solves using Wolfram Alpha.

]]>Divide gently in a long division sauce pan:

Integrate briskly over a low flame:

Evaluate for 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

and make the trigonometric substitution so that then:

and

which, if we return to the original substitution , we see that

So, , which means that:

and since , then

Enjoy your pi!

]]>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.

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

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…

]]>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 , followed by the relationship I had used for the logistic with 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.

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