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

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

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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

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Whole number as nonnegative integer:

• Bourbaki, N. Elements of Mathematics: Theory of Sets]. Paris, France: Hermann, 1968. ISBN 3540225250.
• Halmos, P. R. Naive Set Theory. New York: Springer-Verlag, 1974. ISBN 0387900926.
• 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'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. ISBN 0395825172. Includes all three possibilities as definitions of "whole number."