Rings: Natural numbers and integers

Abstract

The positive whole numbers are undoubtedly the oldest and most primitive objects of all mathematics. They formed, and still form, the basis from which all other mathematics sprang. Zero appeared upon the mathematical scene later. The natural numbers are so much the genesis of all mathematics that the 19th century mathematician Leopold Kronecker was led to say that God created the natural numbers and man created everything else in mathematics. The axiomization presented in this chapter of the natural numbers is credited to Giuseppe Peano, another of the great 19th century mathematicians who reworked and solidified the foundations of our number systems. The whole numbers as a key and insight to the nature of the universe were recognized by the Pythagoreans in the 6th century B.C. When Pythagoras said, “All is number, ” he meant the positive integers.