# Whole Number

A set of whole numbers.

A whole number is a member of the set {0,1, 2, 3, …}. It is either one of the positive integers (natural numbers) or zero.

The set of whole numbers is exactly equivalent to the set of non negative integers.

## Examples of Whole Numbers

Some examples of whole numbers are 4, 5, and 89,453,711.

Any fraction, decimal, negative number or imaginary number is not a whole number. For example, -5, 6.788, and 4/5 are not whole numbers.

## The Set of Whole Numbers

The set of whole numbers is closed under addition and multiplication. What that means is that for all whole numbers a and b, a + b = c and a b = d will also be whole numbers.

Just as 6 + 7 = 13 and 6 · 7 = 42 are whole numbers, the sum or product of any two whole numbers will also be whole.

The set of whole numbers isn’t closed under division and subtraction. For any whole number a, there is another whole number b such that a – b = c and a / b = d are not whole numbers.

Example: 7 and 9 are whole numbers, but 7 – 9 = -2, which is not a whole number. 4 and 5 are whole numbers, but 4/5 is not a whole number.

## Properties of Whole numbers

Whole numbers are commutative under addition and multiplication . For every a, b in the set of whole numbers, a + b = b + a and a · b = b a.

Whole numbers are associative under addition and multiplication. For every a, b, and c in the set of whole numbers, a(b·c) = (a·b) · c and (a + b) + c = a + (b + c).

The set of whole numbers includes an additive identity 0 so that a + 0 = a for every whole number a. It also includes a multiplicative identity 1 so that 1 · a = a.

## Another Definition of Whole Number

Sometimes the set of whole numbers is defined to exclude zero. Then it is equivalent to the set of natural numbers, and can be written as {1,2,3,4,5….}

## Sources

Extending the Natural Numbers to the Wholes

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