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Archive for the ‘Complex Analysis’ Category

I was remembering today a contest problem I encountered back in 1999-2000 about pirates and buried treasure.  At first, I couldn’t remember the problem, then when I found multiple on-line versions of it (physicsforums.com, mathpages.com, Bradley University, geometer.org, the mathfactor podcast, and University of Georgia), I couldn’t remember how I had solved it!  It apparently appears originally in George Gamow’s One, Two Three,…Infinity.

The essential features of the problem are that two pirates arrive on a desert island on which there are three prominent trees.  Choosing the most prominent tree as their starting place, the pirates walk first to one tree, counting their steps.  When they arrive at the tree, they turn 90 degrees to the right and walk the same number of steps and mark the spot.

Then they return to the starting place and walk to the other tree, again counting steps.  When they arrive at the second tree, they turn 90 degrees to the left and walk the same number of steps and mark that spot.  They then bury their treasure mid-way between the two marked spots.

Upon returning to the island some time later, they find that the most prominent tree that was their starting point is now gone.  Can they still find the treasure?

Once I was sure I had the right problem, I remembered finally that I had originally used a coordinate proof.

The comments in the Math Factor podcast contain a nice solution using coordinate geometry with a nifty ending that makes finding the treasure fairly simple.

The solution in Gamow’s original apparently uses complex numbers to solve it.

The University of Georgia site asks students to find four separate proofs using 1) complex numbers, 2) Euclidean geometry, 3) coordinate geometry and 4) vector algebra.

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Back in June of 2009, I wrote about the issues involved in graphing complex functions.  Since then, I’ve been showing this material to my students and discussing the relationship between graphing real-valued functions and graphing complex-valued functions.

As part of these talks, I’ve had to make explicit the difference between the Cartesian Plane and the Argand Diagram.  I’ve been doing this by showing the mapping of the points from the x-real number line to the y-real number line.  George Abdo and Paul Godfrey have a nice website that shows this process.

OK, then if we cross the x and y number lines to create the Cartesian Plane, we’ll see the picture of the graph that we’re used to.

Then, we have to consider the complex mapping.  In this situation, each x value is two-dimensional and is mapped to a two-dimensional y coordinate, like so:

(Click the picture for a better view, or check out the link to Prof. Abdo’s website)

But to show these together, as the Cartesian Plane does for real-valued functions would require 4 dimensions, which creates difficulty for human beings who normally have enough trouble with 3 dimensions.  What people have done instead is to take the x values from the complex plane and color them based on their corresponding y values.

Lawrence Crone at American University has some nice pictures showing this effect.

Andrew Bennett at Kansas State University has a nifty online complex graphing calculator on his website.  You can type in a formula and the software will show you a representation of the mapping.  I prefer the top view to see the roots, but the side view is interesting as well.

The graphing utility window is limited to complex x values a+bi in which a and b are both between 2 and -2, but this can be adjusted by right clicking on the window.

Something that shows up nicely on the complex grapher at the previous link are roots of unity – typing in z³-1 will show the cube roots of 1, both real and complex…

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…the COMPLEX PRIMES…

I’ve written about the complex or Gaussian primes in a previous post, but I realized that I had never put up a picture of the complex primes, which is what motivated me in the first place.

From Wikipedia:

Also, from the Dutch company Sanny de Zoete, here is a picture of linens made in the pattern of the complex primes:

 

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I posted about the Complex Numbers last summer, and also wrote about using Newton’s Method to find complex roots.

The New York Times math blog by Professor Steven Strogatz of Cornell University has an excellent post on complex numbers and Newton’s Method with some great graphics as well.

It ends with a wealth of sources on complex numbers, fractals and Newton’s Method and the interrelations among them.

The other posts from his blog are also well worth reading.

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Gaussian Primes

About 10 years ago, I latched onto the concept of Gaussian Primes.

A Complex (or Gaussian) Prime works much the same way that a real-valued prime does.  It has no divisors other itself and the unit element.

In the real numbers, the units are 1 and -1.  In the Complex number system, the units are 1, -1, i, –i.

In order to tell if a complex number (a+bi) is prime, simply compute a²+b².  If the result is a real-valued prime, then the original complex number (a+bi) is prime.  The proof of this is actually pretty simple and interesting.

Today, in Intermediate Algebra, we were caculating some multiplication problems involving complex numbers.

One of these was (5-3i)(1+i)=8+2i

I noticed that the answer was factorable as 2(4+i) and then realized that these were two different factor pairs for the complex number 8+2i.  That meant that I should be able to turn one factor pair into the other by shifting around the prime factors of the original number.

An example of this is that 24=6*4 and 24=3*8, so I can create one factor pair from the other by shifting a factor of 2.

In considering 6*4, if we look at the 6 as 3*2, then 6*4=(3*2)*4=3*(2*4)=3*8.

Pretty simple, in the real number system.

In the example we looked at in class today we said that

8+2i=(5-3i)(1+i)=2(4+i).

Now, I knew that 1+i is prime (because 1²+1²=2 which is prime) and I suspected that the 2 would factor into (1+i)(1-i), which would mean that 5+3i=(4+i)(1-i), which it does.

So the prime factorization of 8+2i is (1+i)(1-i)(4+i).

Each of these factors is prime and can be checked using the little trick of a²+b²=P.

Here’s a second problem, I’ll include the factorization below…

Find the prime factorization of 4+32i given that:

(-4+8i)(3-2i)=4+32i

I mentioned earlier that the proof for this is simple and interesting.  I’ll include it below as well… (more…)

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In graphing real valued functions, each x value chosen is a real number, and each corresponding y value is also a real number.

Because both the x and y values are one-dimensional real numbers, the relationship can be graphed on a plane, showing the x and y values together only requires TWO dimensions.

We can use the graph of the function relationship between x and y values to solve equations.

To solve the quadratic equation

0= x²+5x+3

we can graph the function

y=x²+5x+3

and look to see what value(s) of x make the y be zero.

When we graph the function we look for the x values where the graph crosses the x-axis.  This is because the value of y is zero along the x-axis.

graph3

So the values of x that make y be zero in the graph of (y=x²+5x+3) are approximately x≈-0.70 and x≈-4.31

Frequently, we see quadratic (parabolic) graphs that don’t intersect the x-axis at all:

graph1This is because there are no real values for x that make y be zero.  In the case of the graph above (y=x²+5x+9) all the x values we see along the x-axis make y a positive number.

Does this mean that there aren’t any x values that make y be zero?

No.  If we use the quadratic formula, we can find that there are complex-valued roots that are solutions of the equation

0=x²+5x+9

In this case, (x≈-2.5±1.658i) are the complex values of x that make y zero.

If we could see x values on the Complex Plane, then we would see these roots on the graph, but, as I mentioned in a previous post, this creates some difficulties.

The picture from Wikipedia that I posted recently is a graph of a different function from the ones above.

Even though it’s not the same graph, it does show one method to try to get around the difficulties of graphing relationships in the Complex Plane.

The picture

600px-Color_complex_plot

shows the Complex Plane of all x values.  The y values are interpreted by the color and intensity of each x value on the Complex Plane.  The roots (and asymptotes) for the function in this picture are indicated by the points or holes or peaks (however you want to think of them) that we see in the picture.

I haven’t done enough research with this method of graphing to know all the details of how it is colored and how to tell the roots from the asymptotes, but I find it both visually and mathematically beautiful.

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Analysis does not owe its really significant successes of the last century to any mysterious use of √(-1), but to the quite natural circumstance that one has infinitely more freedom of mathematical movement if he lets quantities vary in a plane instead of only on a line.

Leopold Kronecker, (1894) quoted in Remmert’s Theory of Complex Functions

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