Tag Archives: polynomials

Repelling Numbers

Standard

An important fact in the theory of prime numbers is the Deuring-Heilbronn phenomenon, which roughly says that:

The zeros of L-functions repel each other.

Interestingly, Andrew Granville in his article for The Princeton Companion to Mathematics remarks that:

This phenomenon is akin to the fact that different algebraic numbers repel one another, part of the basis of the subject of Diophantine approximation.

I am amazed by this repelling relation between two different aspects of arithmetic (a.k.a. number theory). Since I have already discussed the post Colourful Complex Functions, wanted to share this picture of the algebraic numbers in the complex plane, made by David Moore based on earlier work by Stephen J. Brooks:

 

In this picture, the colour of a point indicates the degree of the polynomial of which it’s a root, where red represents the roots of linear polynomials, i.e. rational numbers,  green represents the roots of quadratic polynomials, blue represents the roots of cubic polynomials, yellow represents the roots of quartic polynomials, and so on.  Also, the size of a point decreases exponentially with the complexity of the simplest polynomial with integer coefficient of which it’s a root, where the complexity is the sum of the absolute values of the coefficients of that polynomial.

Moreover,  John Baez comments in his blog post that:

There are many patterns in this picture that call for an explanation! For example, look near the point i. Can you describe some of these patterns, formulate some conjectures about them, and prove some theorems? Maybe you can dream up a stronger version of Roth’s theorem, which says roughly that algebraic numbers tend to ‘repel’ rational numbers of low complexity.

To read more about complex plane plots of families of polynomials, see this write-up by John Baez. I will end this post with the following GIF from Reddit (click on it for details):

Arithmetic Operations

Standard

There are only 4 binary operations which we call “arithmetic operations”. These are:

  • Addition (+)
  • Subtractions (-)
  • Multiplication (×)
  • Division (÷)

Reading this fact, an obvious question is:

Why only four out of the infinitely many possible binary operations are said to be arithmetical?

Before presenting my attempt to answer this question, I would like to remind you that these are the operations you were taught when you learnt about numbers i.e. arithmetic.

In high school when \sqrt{2} is introduced, we are told that real numbers are of two types: “rational” and “irrational”. Then in college when \sqrt{-1} is introduced, we should be told that complex numbers are of two types: “algebraic” and “transcendental“.

As I have commented before, there are various number systems. And for each number system we have some valid arithmetical operations leading to a valid algebraic structure. So, only these 4 operations are entitled to be arithmetic operations because only these operations lead to valid algebraic numbers when operated on algebraic numbers.

Now this leads to another obvious question:

Why so much concerned about algebraic numbers?

To answer this question, we will have to look into the motivation for construction of various number systems like integers, rational, irrationals, complex numbers… The construction of these number systems has been motivated by our need to be able to solve polynomials of various degree (linear, quadratic, cubic…). And the Fundamental Theorem of Algebra says:

Every polynomial with rational coefficients and of degree n in variable x has n solutions in  complex number system.

But, here is a catch. The number of complex numbers which can’t satisfy any polynomial (called transcendental numbers) is much more than the number of complex numbers which can satisfy a polynomial equation (called algebraic numbers). And we wish to find solutions of a polynomial equation (ie.e algebraic numbers) in terms of sum, difference, product, division or m^{th} root of rational numbers (since coefficients were rational numbers). Therefore, sum, difference, product and division are only 4 possible arithmetic operations.

My previous statement may lead to a doubt that:

Why taking m^{th} root isn’t an arithmetic operation?

This is because it isn’t a binary operation to start with, since we have fixed m. Also, taking m^{th} root is allowed because of the multiplication property.

CAUTION: The reverse of m^{th} root is multiplying a number with itself m times and it is obviously allowed. But, this doesn’t make the binary operation of taking exponents, \alpha^{\beta} where \alpha and \beta are algebraic numbers, an arithmetic operation. For example, 2^{\sqrt{2}} is transcendental (called Gelfond–Schneider constant or Hilbert number) even though 2 and \sqrt{2} are algebraic.