Tag Archives: number theory

Arithmetic of p-Adic Numbers

Anyone who has not touched upon the p-adic numbers might find the topic rather confusing. Truth be told, it is confusing. The arithmetic part of the process can be a bit challenging due to its abstraction. Assuming that the reader is not familiar with the p-adic numbers, we will consider an example to clarify the matter.

Let’s consider the equation $x+64=0$. Any one individual might tell us that the solution is $x=-64$ when we ask them to solve the equation and of course, they would be right. But we wish to know the 5-adic representation of $-64$. We follow the following procedure:

Solution to $x + 64 = 0$ $mod$ $5$ is $x = 1 = 1 * 5^0$

Solution to $x + 64 = 0$ $mod$ $5^2$ is $x = 11 = 1 * 5^0 + 2 * 5^1$

Solution to $x + 64 = 0$ $mod$ $5^3$ is $x = 61 = 1 * 50^0 + 2 * 5^1 + 2 * 5^2$

Solution to $x + 64 = 0$ $mod$ $5^4$ is $x = 561 = 1 * 50^0 + 2 * 5^1 + 2 * 5^2 + 4 * 5^3$

Solution to $x + 64 = 0$ $mod$ $5^5$ is $x = 3061 = 1 * 50^0 + 2 * 5^1 + 2 * 5^2 + 4 * 5^3 + 4 * 5^4$

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Solution to $x + 64 = 0$ $mod$ $5^n$ is $x = 1 *5^0 + 2 * 5^1 . . . 4 * 5^{n-1}$

For  $x + 64 = 0$ $mod$ $5$, we can instantly see that the solution is $x = 1$ since $1 + 64 = 65$ is evenly divisible by $5$. Notice that $x = 1$ can be written as $x = 1 * 5^0$. Now for $x + 64 = 0$ $mod$ $5^2$, we know that when we add $11$ to $64$ which gives us $75$, it will be divisible by $25$. Once again, notice that we can write $x=11$ as $11 = 1 * 5^0 + 2 * 5^1$. The same procedure applies from $x + 64 = 0$ $mod$ $5^3$ all the way up to $x + 64 = 0$ $mod$ $5^n$. In general, we take our $x$ value for $x + 64 = 0$ $mod$ $5^n$ and write the linear combination for it in the form of $x = 1 *5^0 + 2 * 5^1 . . . 4 * 5^{n-1}$.

Following the same procedure, we represent $-64$ in its 5-adic representation as $(1,2,2,4,4,…)$. Although quite interesting, the arithmetic for p-adic representation can be a tedious process. Now that we have seen an example, next we will consider p-adic representation of rational numbers and absolute value on $Q$.

Euclidean Algorithm for Complex Integers

We got into the Euclidean Algorithm at the beginning of this course and one of the questions I had at the end of the lab was how the algorithm applied to other numbers systems beyond the integers.  I was curious how it worked with polynomials and complex numbers.  Imagine my delight to discover this lab.

In this lab, we investigate the $Gaussian$ $Integers$, which take the form $a+bi$ and are named after Karl Friedrich Gauss who was the first person to discover the correct way to generalize the Euclidean Algorithm for these numbers.

Much like we did with the first Euclid lab, we will investigate using complex numbers in the division algorithm.  This lab also defines a new term.  In the complex plane, the distance a point is from the origin is called its $modulus$  and the $norm$ of a complex number $z$ is denoted by $N(z)$, which is the square of its modulus.

The Euclidean Algorithm is very fascinating and I’m excited to see how else it can be applied to other number systems.

Square root of 2 is irrational?

My name is Farighon. I have a deep interest in the field of Combinatorics and Number Theory although I do like to take time to study Complex Analysis when I do have some free time. Since the inception of my interest in mathematics, I have always been interested and fascinated by proofs for theorems, lemmas, and propositions. But I would be lying if I said that I understood each proof that I have come across. However, there is one proof that has made perfect sense. It is none other than the proof for the square root of 2.

The proof for the square root of 2 being irrational has been one of the primary interests of ancient mathematicians starting with the Babylonian’s. Then the ancient Indians. Although later on it was proven by a simple yet elegant proof that square root of 2 is irrational, the hunt for determining the square root of 2 to as many decimal places as possible is an ongoing task for mathematicians teamed up with Computer Scientists. After all, who can expect a mathematician to be ever satisfied when perfection is what shapes and disciplines them?

The article for the proof of $\sqrt(2)$ can be found at the following article for the curious reader: http://en.wikipedia.org/wiki/Square_root_of_2.