Whole number

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The whole numbers are the nonnegative integers (0, 1, 2, 3, ...)

The set of all whole numbers is represented by the symbol <math>\mathbb{W}</math> = {0, 1, 2, 3, ...} <p>Algebraically, the elements of <math>\mathbb{W}</math> 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, ...)

To remove ambiguity from mathematical terminology, those uses are now discouraged.

References

Whole number as nonnegative integer:

• 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.

Whole number as positive integer:

• 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.)

Whole number as integer:

• 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."