# Whole Numbers – Definition and Examples in Math

In math, the whole numbers are the set of numbers that includes all of the natural numbers and zero. In other words, whole numbers are the counting numbers (1, 2, 3, …) together with 0. Here is everything you need to know about whole numbers, along with examples.

### What Are Whole Numbers?

Whole numbers are a set of numbers without decimals, fractions, or negative values. Another way of saying this is that the whole numbers are the set of non-negative integers. The symbol for the set is the capital letter W. The set of whole numbers is written as W = {0, 1, 2, 3, …}.

• A whole number has to be either 0 or else positive
• It contains no fractions.
• It has no decimal point.

### Examples of Numbers That Are and Are Not Whole Numbers

For example, the following are whole numbers:

• 0
• 33
• 7
• 1499932
• 2
• 13

The following are not whole numbers:

• -2
• 1/3
• 0.45
• 132.4
• 1/2
• -23
• 3
• Pi
• 3i
• 7 ⅚

### Whole Number Facts

Here are some facts about whole numbers:

• The smallest whole number is zero. (In contrast, the smallest natural number is 1.)
• The set of whole numbers is an infinite set. Although there is a smallest whole number, there is no largest one.
• All of the natural numbers or counting numbers are whole numbers.
• Every positive integer is a whole number.
• All of the whole numbers are both rational and real numbers.

### Properties

Here are some properties of the whole numbers:

• Closure: Adding two whole numbers gives another whole number. For example, 3 + 4 = 7.
• Closure: Multiplying two whole numbers gives another whole number. For example, 3 x 4 = 12
• Closure does not hold true for subtraction or division.
• Subtracting two whole numbers does not always give another whole number, but it does give an integer.
• Dividing two whole numbers does not always give another whole number. Assuming the number is not divided by 0 (denominator is not zero), it gives a rational number.
• Associative Property: a + (b + c) = (a + b) + c; 1 + (2 + 3) = (1 + 2) + 3
• Associative Property: a x (b x c) = (a x b) x c; 2 x (3 x 4) = (2 x 3) x 4
• Commutative Property: a + b + c = a + c + b = b + a + c and so on
• Commutative Property: a x b = b x a
• The commutative property does not hold true for subtraction or division.
• Distributive Property: a x (b + c) = (a x b) + (a x c)
• Additive Identity: a + 0 = a
• a x 0 = 0
• If a x b = 0, then a = 0 and/or b = 0
• a / 0 = undefined

### References

• Clapham, Christopher; Nicholson, James (2014). The Concise Oxford Dictionary of Mathematics (5th ed.). Oxford University Press. ISBN 978-0-19-967959-1
• Goldrei, Derek (1998). Classic Set Theory: A Guided Independent Study (1st ed.). Boca Raton, FL: Chapman & Hall/CRC. ISBN 978-0-412-60610-6.
• Kline, Morris (1990) . Mathematical Thought from Ancient to Modern Times. Oxford University Press. ISBN 0-19-506135-7.
• Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach (10th ed.). Wiley Global Education. ISBN 978-1-118-45744-3.