The whole numbers are the nonnegative integers (0, 1, 2, 3, ...) A negative number is a number that is less than zero, such as −3. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

The set of all whole numbers is represented by the symbol = {0, 1, 2, 3, ...} In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

Algebraically, the elements of form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

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.^{[citation needed]} In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

See also

• Natural number

In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

References

Whole number as nonnegative integer:

• Bourbaki, N. Elements of Mathematics: Theory of Sets]. Paris, France: Hermann, 1968. ISBN 3-540-22525-0.
• Halmos, P. R. Naive Set Theory. New York: Springer-Verlag, 1974. ISBN 0-387-90092-6.
• 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 an Euler diagram of common numeric domains.

Whole number as positive integer: A Venn diagram shows all possible intersections. ... In mathematics, the domain of a function is the set of all input values to the function. ...

• Weisstein, Eric W., Whole Number at MathWorld. (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: MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

• 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 0-395-82517-2. Includes all three possibilities as definitions of "whole number."