Definition: Natural Numbers

Definition 1: A natural number is a positive “whole” number.  It is an element of the set \{1, 2, 3, 4, 5, 6, ...\}. //


  • Zero is not a natural number according to this definition (zero is not positive, it is only non-negative!).  Sometimes natural numbers are defined to include zero, but I will not do so here.  This is purely a matter of taste, and for whatever reason I don’t consider 0 as “natural” as the other ones.
  • This set of numbers is infinite—it starts at 1 and goes on forever.
  • 1.5, 2.56, and “\pi” (the oh-so-famous “Pi”, which is approximately 3.14) are not on this infinite list (only the whole numbers!).

There are lots of places that we study and/or use the natural numbers, and one of the coolest is in the proof the existence of infinitely many prime numbers.  We also rely heavily on natural numbers in counting both finite sets as well as infinite sets

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