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Mathematical ramblings

Note: (written October 24 2016) This page links to a bunch of essays I wrote a decade or so ago. And they're still worth reading, in my opinion. But, if you want to see my more recent content, my blog is where it's happening. And now, back to the old content.

When I first came across the following quote, it resonated with me.

"If you understand something in only one way, then you do not really understand it at all. This is because if something goes wrong you get stuck with a thought that just sits in your mind with nowhere to go. The secret of what anything means to us depends on how we have connected it to all the other things we know. This is why, when someone learns "by rote," we say that they do not really understand. However, if you have several different representations, when one approach fails you can try another. Of course, making too many indiscriminate connections will turn a mind to mush. But well-connected representations let you turn ideas around in your mind, to envision things from many perspectives, until you find one that works for you. And that is what we mean by thinking!" Marvin Minsky

In the following essays and notes exploring various mathematical topics, I'll be guided by the above quote.


3 times 5 equals 15 Different interpretations in different contexts
The identity 3*5=15 is connected to a surprising variety of mathematics: various algebraic identities, Fibonacci numbers, the Golden Ratio, Mersenne primes, sums of squares, complex numbers, and quaternions.
Table proofs
I'm always on the lookout for ways of visualizing mathematical results. In this essay I'm going to restrict myself to pictures. Furthermore, I'm going to restrict myself to pictures that arise from html tables and can be displayed by internet browsers: plain old fashioned static html with no javascript or images. I like to think of it as an html analogue of ascii art: maybe it should be called "table art".
There are infinitely many primes
For quite a while, I knew of only one proof of the "infinitude of primes", namely Euclid's proof. Over the years, I've come across others, and I've noticed that several of them are really relying on deriving a contradiction from a particular consequence of assuming that there are finitely many primes. That consequence, and various ways of disproving it, are the subject of this essay.
Fibonacci numbers revisited There are alternatives to induction
Properties of Fibonacci numbers are often proved by induction. Although this results in technically correct proofs, I find that proofs by induction usually give me very little insight. Over the years I've found various alternative definitions of the Fibonacci numbers that I can try out when I'm trying to understand a result. I recently realized that the sequence of Fibonacci numbers can be viewed as the projection of a two dimensional geometric sequence, and I'm writing about it here.
10, 10^10, 10^10^10 … Infinity Getting a feel for large numbers
This is about some ways I've come up with to get a gut feeling for the behavior of numbers of the form 101010n, where n is in the range from 1 to 10. These numbers don't come up in the physical sciences or economics, so you probably haven't had a need to become familiar with them. Some of their properties may surprise you.
Solving the Cubic a x3 + b x2 + c x + d = 0
This is my attempt to explain Cardano's technique for solving the cubic. Instead of theory followed by examples, it's examples followed by theory.
An algebraic identity a2-b2 = (a-b)(a+b)
Here's an identity that should be an old friend to anyone who's taken algebra. It's another one of those identities that has many roles in many contexts. I've barely scratched the surface.
A trigonometric identity cos2(t)+sin2(t) = 1
Here's an identity that most people first see in their trigonometry classes. If they continue with their mathematical studies, they're sure to see it again in various contexts. It has surprisingly many interpretations, some of which I discovered as I started writing.


Why are 3d graphics programmers using quaternions?
This is a short note on a simple question regarding an application of quaternions to 3d computer graphics.
Visualizing the Hopf Fibration.
In this essay I experiment with animated anaglyphs as a way to help visualize 3d phenomena. I'm very interested in your feedback.

Mailing List

I've started a yahoo group, Understanding Mathematics, at, which is intended to allow participants to explore ways of improving our understanding of mathematics. Also, whenever I add, or significantly modify, an essay, I will post a message to the group. If you're interested in this kind of thing, please consider joining .

August 5 2007 Last Updated

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