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Rolfe’s Math Blog

Lecture notes

For those of you who are a bit more technically minded, I'm keeping some lecture notes for an Inverse Problems course at UW here. I may also add some discussions of papers I'm reading, etc. I didn't want to flood this blog with them.

Of course, it wouldn't hurt to get something on this blog.  I'm thinking about it.  Just busy.

Posted in Uncategorized

A new toy

The boys and I have been talking a lot about number theory lately -- GCD's, LCM's and primes -- and one thing that keeps coming up is the Sieve of Eratosthenes.  They love the sieve, and this is how we have been doing it:

  1. I write a list of numbers on a grid (e.g. the numbers 1 to 105 on a 7-by 15 grid)
  2. They circle "2" with a color, then mark out all of the even numbers with that color.
  3. The circle "3" with a different color, and mark out every third number with the new color.
  4. Continue like this for 5, 7, 11, etc.  Always circle the next number that hasn't been colored with a new color.

As you do this, you will find interesting patterns that speed up the coloring and teach you a bit about numbers.  It is interesting to try grids with different widths and find new patterns too.

Eventually I got tired of writing these grids out by hand, so I wrote a little web page to do it for me.  It colors the grid too -- try it!

Posted in Education, Number Theory, Tools

More ways to make squares

When we were building squares from triangles, we found a few interesting identities. One of the more striking ones showed how we could make a square from two triangles:

\[T_n + T_{n-1} = n^2\]

This opens the door to a little bit more fun. Remember that

\[T_n = 1 + 2 + 3 + ... + n\]

So we know

\[n^2 = T_n + T_{n-1} = (1 + 2 + 3 + ... + n) + (1 + 2 + 3 + ... + n-1)\]

Let's arrange the terms on top of each other, like this

1 + 2 + 3 + 4 ... + n
  + 1 + 2 + 3 ... + n - 1

Now if we group the terms that are on top of each other, we se we can write

\[ n^2 = T_n + T_{n-1} = 1 + (2 + 1) + (3 + 2) + (4 + 3) + ... + (n + n - 1)\]


\[ n^2 = 1 + 3 + 5 + 7 + ... + (2n - 1)\]

And we see that the n-th square number is just the sum of the first n odd numbers!

Now in pictures

There are other ways to see this. Read more »

Posted in Arithmetic, Figurate Numbers, Geometry

Triangles, Rectangles, and Squares

In my last post about triangle numbers, I claimed we could find all sorts of arithmetic identities just by moving our triangles around and sticking them together to build new shapes.  Here we will get our first taste of how this works.

We all know that we can split a square into two equal triangles by drawing a line down the diagonal:

Breaking a square into two triangels

Breaking a square into two triangles

When we try this for the "blocky" triangles we used to build our triangle numbers, it doesn't quite work.  Have a look at what happens for the "4-triangle":

A rectangle from two equal triangles

A rectangle from two equal triangles

We get a 4-by-5 rectangle rather than a square. Read more »

Posted in Arithmetic, Education, Figurate Numbers

Triangle Numbers

Figurate numbers can add a bit of fun to otherwise dry Arithmetic practice.  Triangle numbers are my current favorites.  What is a triangle number?  Well, here are the first few:

1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21

So the n-th triangle number is just the sum of the first n numbers. Why are they called triangle numbers? Have a look at this picture:

The first 6 triangle numbers

The first 6 triangle numbers

In some future posts, all available in this blog's Figurate Numbers category, I'll show how we can put triangles and squares together to build new triangles and squares.  These pictures will translate directly to equations, or Arithmetic problems.  We'll see how some tedious looking problem sets are really telling us stories about shapes.

Some other reasons triangle numbers are important

I like the fact that triangle numbers allow us to play some fun games with numbers, and provide a simple example of how Arithmetic can represent seemingly unrelated concepts like "shape".  They are also valuable because they give us the simplest example of an arithmetic series, and once you know how to compute triangle numbers, it is pretty easy to figure out how to add any arithmetic progression (again, I'll go into the details in a later post).

Once we know how to sum these series, we'll be able to move into elementary Calculus and evaluate our first integrals.  Along the way, we'll also find a way to test whether a number is a triangle number and develop a formula that tells us which triangle number it is.

These numbers give us a nice setting to introduce some real Mathematical thought in a fairly elementary way.  This is a theme I plan to come back to repeatedly on this blog.

Posted in Arithmetic, Education, Figurate Numbers

Pythagorean Puzzles

As I was playing around with logo the other day I remembered about one of my favorite little proofs of the Pythagorean Theorem. I was going to throw together a little program to draw the picture, but when I thought about it a little bit, I realized that it would be more fun to turn this proof into a puzzle my kids could play with.

In case your Geometry is a little rusty, the Pythagorean Theorem tells you how to compute the length of one side of a right triangle when you know the length of the other two. If we have a right triangle like this one:

right triangle

then the Pythagorean Theorem states that \(a^2 + b^2 = c^2\).

Here is one way to see that it is true. Let's take four of these triangles and arrange them like this:

Spiral square

Now we have made a large square, with sides of length \(c\), out of four of our triangles. We just have a little square "hole" in the middle. Read more »

Posted in Geometry

A Zeta Function Primer

With the last couple of posts under our belt, we're ready to have a peek at something a little more exciting: the Riemann \(\zeta\)-function and it's relationship to the prime numbers. This is at the heart of one of the most famous unsolved Mathematics problems around, the Riemann Hypothesis.

No, we won't take on the Riemann Hypothesis, but I will try to convince you that understanding how the \(\zeta\)-function behaves is really the same as understanding prime numbers and how they are distributed. We'll start this by proving a specific result, namely

\[\sum_{p \in \textbf{primes}}\frac{1}{p} = \infty\]

In other words, the sum of the reciprocals of the primes diverges.

Here I'll give the simplest proof I can think of, relying only on a little bit of Algebra II. If anyone can do better, please let me know! I'm not aiming for rigor here, although I'm happy to provide rigor on request.
Read more »

Posted in Analysis, Infinite Series, Number Theory

Harmonic Series

Before we can start looking at the Riemann \(\zeta\)-function and the prime numbers, we need to add a couple more tools to our tool kit.  In particular, we need to understand harmonic series and some related sums.

The harmonic series

Here is a sum that shows up frequently in Mathematics:

\[\sum_{n = 1}^{\infty} \frac{1}{n}\]

This is called the harmonic seriesRead more »

Posted in Analysis, Infinite Series

Geometric Series

Infinite series come up frequently in Analysis and Number Theory and you'll need to be adroit at managing them to make progress in these fields. Luckily there are a few simple concepts that will take you far if you leverage them well. In this series of posts, I'll talk about a few of these ideas and use them to help us peer into the cutting edge of Mathematics: we'll see why the Riemann Zeta function has every thing to do with the distribution of the prime numbers.

If that sounds like it's beyond the scope of your Calculus class, don't worry. All you need are good Algebra skills and a bit of motivation.

Let's start by looking at one of the simplest types of series around, the Geometric Series. Read more »

Posted in Algebra, Analysis, Infinite Series
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