Whole number
Whole numbers are non-negative numbers including 0 that do not have a fractional or decimal component.
Whole number definition
Whole numbers are numbers that are not fractions, decimals, or negative. They can also be defined as a subset of the integers, or even in terms of counting numbers: Whole numbers are all the positive integers as well as 0; they can also be defined as the set of cardinal numbers, also referred to as counting numbers, including 0.
Set of whole numbers
The set of whole numbers is commonly represented by the whole number symbol W and can be written as follows:
{0, 1, 2, 3, 4, 5, 6, ...}
Whole numbers on a number line
Whole numbers can also be expressed using a number line:
Note that only the points on the number line that have labeled values are whole numbers. Depending on the definitions used, whole numbers may be synonymous with natural numbers.
Smallest and largest whole number
The smallest whole number is 0. There is no largest whole number because the set of whole numbers continues towards infinity.
Difference between whole numbers and natural numbers
The key difference between the set of whole numbers and natural numbers is that the set of whole numbers includes 0 while the set of natural numbers does not. Essentially, every natural number is a whole number and every whole number except for 0 is a natural number. We can see this when comparing the two sets. The set of whole numbers is,
{0, 1, 2, 3, 4, 5, 6, ...},
and the set of natural numbers is:
{1, 2, 3, 4, 5, 6, ...}
In some cases, the set of natural numbers is defined as including 0. Under that definition, there would not be a difference between natural numbers and whole numbers, but this definition is less common. The table below is a summary of the key differences between the sets of numbers:
| Whole numbers | Natural numbers |
|---|---|
| Includes 0 | Doesn't include 0 |
| {0, 1, 2, 3, 4, 5, 6, ...} | {1, 2, 3, 4, 5, 6, ...} |
| Smallest number is 0 | Smallest number is 1 |
Whole numbers and integers
Integers include negative values. Aside from this difference, the set of whole numbers and integers include the same values. This means that a whole number is always an integer, but an integer is not always a whole number.
| The set of whole numbers: | {0, 1, 2, 3, 4, 5, 6, ...} |
| The set of integers: | {..., -3, -2, -1, 0, 1, 2, 3, ...} |
If we took the absolute value of any integer, we would get a whole number.
Can whole numbers be negative
Whole numbers cannot be negative. They are defined as the set of positive integers including 0.
Is 0 a whole number?
Yes 0 is a whole number. This is simply by definition. It is included in the set of whole numbers as part of the definition.
Properties of whole numbers
Below are some properties of whole numbers.
Closure property of whole numbers
The closure property states that given two whole numbers, a and b,
a × b → whole number
a + b → whole number
In other words, if we add or multiply two whole numbers, the result will be a whole number.
Commutative property of whole numbers
The commutative property states that the order in which we add or multiply whole numbers doesn't matter; the result will be the same. Given two whole numbers, a and b,
a × b = b × a
a + b = b + a
Additive identity of whole numbers
The additive identity is 0. If 0 is added to any whole number, the result is the same whole number. Given a whole number a,
a + 0 = 0 + a
Thus, 0 is referred to as the additive identity of whole numbers.
Multiplicative identity of whole numbers
The multiplicative identity of whole numbers is 1. If 1 is multiplied by any whole number, the result is the same whole number. Given a whole number a,
a × 1 = a
Thus, 1 is referred to as the multiplicative identity of whole numbers.
Associative property of whole numbers
The associative property of whole numbers states that whole numbers being added or multiplied can be grouped in any order; the result will be the same. Given three whole numbers a, b, and c,
(a × b) × c = a × (b × c)
(a + b) + c = a + (b + c)
Distributive property of whole numbers
The distributive property states that multiplication of a whole number is distributed over the sum or difference of the whole numbers. Given three whole numbers a, b, and c,
a × (b + c) = (a × b) + (a × c)