Sunday, May 3, 2020

The Natural Numbers

A number, as a concept, is a surprisingly ambiguous term.  There are whole numbers, fractions, negative numbers, real numbers, irrational numbers, imaginary numbers, and transcendental numbers.  It may not be intuitively obvious what they all have in common.  There are also mathematical operations that are not defined and are probably not numbers at all such as $\dfrac{1}{0}$ (see here) or $0^0$ (see here).  The concept of infinity is another difficult to nail down concept.  Is it a number?  It turns out that there are different types of infinity.  For example, the number of counting numbers (1, 2, 3, ...) is less than the number of points in an infinitely divisible line (see here).  I will tackle these more advanced topics later

Historically, the concept of number began with the activities of counting and measuring.  Counting led to the whole numbers which did not include 0 at the very beginning and measuring led to simple fractions (more complicated fractions require a more complicated number system).  

The most important historical breakthroughs which we pretty much take for granted today are:
In mathematics, the set of positive whole numbers and zero is called the Natural Numbers and is represented by N;.    When combined with the set of negative numbers, it becomes the sets of Integers which is represented by Z. (Why Z, it comes from the German word Zahlen for 'numbers')

The study of the properties of Integers is called number theory and this will be the primary focus of this blog.    

In my view, the most surprising and deepest insight in all of mathematics is that natural numbers, despite their appearances of being the simplest of mathematical objects, contain some of the most challenging and important problems known to mathematics.  

These challenges have led to the development of public key cryptography which today powers the global economy.  Innovations such as block chain, which build on top of the complexity of mathematical problems, will further transform our economic, social, and political institutions that depend on hierarchy and centralized authority.  

One popular definition of Mathematics is "the study of patterns" (see here for an interesting overview).  Many of the most surprising and profound patterns can be found in Number Theory.

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