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
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.
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:
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
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
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
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.
I've started a yahoo group,
http://groups.yahoo.com/group/understandingmath/, 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
| August 5 2007 || Last Updated |
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