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:
- Invention of number symbols (typically, the earliest form of writing that comes after use of tally scratches and tally scratch groupings) (For context, see history of writing and history of ancient numeral systems)
- Discovery of zero (For context, see history of zero)
- Positional Notation which allows all numbers to be represented by a finite set of digits (For context, see here)
- Negative numbers (For context, see history of negative numbers)
- Irrational numbers which showed that not all measurements are ratios of whole numbers (For context, see history of irrational numbers)
- The Decimal System (For context, see here. Not all societies used Base 10. The Ancient Babylonians for example used a Sexagesimal System which is Base 60).
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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