Whole Numbers

The Set of all whole numbers is represented by the symbol $\backslash mathbb\{W\}$ = {0, 1, 2, 3, ...}

Algebraically , the elements of $\backslash mathbb\{W\}$ form a Commutative Monoid under addition (with Identity Element zero), and under multiplication (with identity element one).


ASIDE

Unfortunately, this term is used by various authors to mean:

• the Positive Integers (1, 2, 3, ...)


• all Integers (..., -3, -2, -1, 0, 1, 2, 3, ...)

• Bourbaki, N. ''Elements of Mathematics: Theory of Sets'' . Paris, France: Hermann, 1968.


• Halmos, P. R. ''Naive Set Theory'' . New York: Springer-Verlag, 1974.


• Wu, H. ''Chapter 1: Whole Numbers.'' University of California at Berkeley, 2002. "Notice that we include 0 among the whole numbers."


• The Math Forum, in explaining real numbers, describes "whole number" as "0, 1, 2, 3, ..." .


• Simmons, B. MathWords presents the whole numbers as "0, 1, 2, 3, ..." in a Venn Diagram of common numeric Domains .

• Weisstein, Eric W. "Whole Number." From MathWorld—A Wolfram Web Resource . (Weisstein's primary definition is as positive integer. However, he acknowledges other definitions of "whole number," and is the source of the reference to Bourbaki and Halmos above.)

• Beardon, Alan F., Professor in Complex Analysis at the University of Cambridge: "of course a whole number can be negative..."


• The ''American Heritage Dictionary of the English Language'' , 4th edition, includes all three possibilities as definitions of "whole number."