THE NATURAL NUMBERS, THE WELL-ORDERING PRINCIPLE, AND THE INTEGERS

Natural numbers were developed for counting objects. The ancient Egyptians invented a number system that started with a symbol for number 1. This symbol represented one object that is counted. Two similar symbols were used to represent two objects counted. There will be nine of this symbol if there are nine objects counted.

Another symbol distinct from the previous symbol was used to represent ten objects counted.  

Ancient Egyptian Numerals (  http://en.wikipedia.org/wiki/Egyptian_numerals)

Later on the Babylonians considered zero (0) as a number.

Much later the set of natural numbers is considered to be the set consisting of the counting numbers 1,2,3, …, and so on including the number 0.

An important property of the set natural numbers is the “Well-Ordering Principle” which states that every non-empty subset of the natural numbers has a least element.

Suppose we have a subset S = {x, y, z} of three natural numbers.

Then by the well-ordering principle,  S has a smallest element, say x. Then x < y and x < z.

Let us consider the subset T = {y, z}. Again, by the well-ordering principle, T has a least element. Let us assume that it is y. Then y < z.  

Therefore, we have x < y < z. This means that the elements x, y and z of S are well-ordered.

When the number 3 is subtracted from 5, the difference which is equal to 2 is a natural number. However, if the number 5 is subtracted from 3, the difference is not a natural number. Hence, the negative of a non-zero natural number is invented and we have the set of integers.

 The set of integers were developed to include the natural numbers and their negatives. Note that the negative of 0 is still 0.

An interesting property about the set of integers is the “Division Algorithm”which states that “for any two integers a and b with b > 0, there exist unique integers q and r such that  a = bq + r where r is greater than or equal to 0 but less than b.”

 Repeated applications of the Division Algorithm will give us the greatest common divisor of two integers and the k-ary expansion of a natural number. 

by Felix P. Muga II
Associate Professor, Mathematics Department, Ateneo de Manila University
Senior Fellow, Center for People Empowerment in Governance

The Mathematics of Kits

Next topic:
Immediate - "A Rational Number Expressed as an Integer or as Sum of an Integer and a Positive Rational Number Less Than 1"
Main Objective: "Converting a Rational Number into a Sequence of Kits"