Mathwords logoMathwords

Whole Numbers — Definition, Examples & Properties

Whole Numbers
Nonnegative Integers

The numbers 0, 1, 2, 3, 4, 5, etc.

 

Nested diagram showing number sets: natural numbers ⊂ whole numbers ⊂ integers ⊂ rationals ⊂ reals ⊂ complex numbers, with...

 

 

See also

Natural numbers, integers, rational numbers, algebraic numbers, real numbers, imaginary numbers, complex numbers

Key Formula

W={0, 1, 2, 3, 4, 5, … }\mathbb{W} = \{0,\, 1,\, 2,\, 3,\, 4,\, 5,\, \dots\}
Where:
  • W\mathbb{W} = The set of whole numbers (some textbooks use W without the double-struck font)
  • 0,1,2,3,…0, 1, 2, 3, \dots = All nonnegative integers, starting from zero and continuing without end

Worked Example

Problem: From the list {−3, 0, 2.5, 4, 7, −1, ¾, 10}, identify all the whole numbers.
Step 1: Recall the definition: whole numbers are 0, 1, 2, 3, 4, … — nonnegative and with no fractional or decimal part.
W={0,1,2,3,4,… }\mathbb{W} = \{0, 1, 2, 3, 4, \dots\}
Step 2: Eliminate negative numbers. Both −3 and −1 are negative, so they are not whole numbers.
−3∉W,−1∉W-3 \notin \mathbb{W}, \quad -1 \notin \mathbb{W}
Step 3: Eliminate fractions and decimals. The values 2.5 and ¾ are not integers, so they are not whole numbers.
2.5∉W,34∉W2.5 \notin \mathbb{W}, \quad \tfrac{3}{4} \notin \mathbb{W}
Step 4: Check the remaining values: 0, 4, 7, and 10 are all nonnegative integers.
0,4,7,10∈W0, 4, 7, 10 \in \mathbb{W}
Answer: The whole numbers in the list are 0, 4, 7, and 10.

Another Example

This example tests whether the result of an operation is a whole number, rather than simply picking whole numbers from a list. It highlights that whole numbers are not closed under division.

Problem: Is the result of 15 ÷ 4 a whole number? What about 15 ÷ 5?
Step 1: Compute 15 ÷ 4.
15÷4=3.7515 \div 4 = 3.75
Step 2: Since 3.75 has a decimal part, it is not a whole number.
3.75∉W3.75 \notin \mathbb{W}
Step 3: Compute 15 ÷ 5.
15÷5=315 \div 5 = 3
Step 4: The result 3 is a nonnegative integer with no fractional part, so it is a whole number.
3∈W3 \in \mathbb{W}
Answer: 15 ÷ 4 = 3.75 is not a whole number, but 15 ÷ 5 = 3 is a whole number.

Frequently Asked Questions

Is 0 a whole number?
Yes. Zero is included in the set of whole numbers. The whole numbers are defined as the nonnegative integers: 0, 1, 2, 3, … This is actually the key difference between whole numbers and natural numbers in most conventions.
What is the difference between whole numbers and natural numbers?
The most common convention is that natural numbers start at 1 (the counting numbers), while whole numbers start at 0. In other words, whole numbers = natural numbers ∪ {0}. Be aware that some textbooks define natural numbers to include 0 as well, so always check which convention your course uses.
Are negative numbers whole numbers?
No. Whole numbers are nonnegative, meaning they are zero or positive. If you need to include negative values as well, you are working with the full set of integers: {…, −2, −1, 0, 1, 2, …}.

Whole Numbers vs. Natural Numbers

Whole NumbersNatural Numbers
DefinitionNonnegative integers: 0, 1, 2, 3, …Positive integers (counting numbers): 1, 2, 3, … (most common convention)
Includes zero?YesNo (in most conventions)
Smallest element01
Notation𝕎 or Wℕ or N
Typical useCounting objects when zero is possible (e.g., number of absences)Counting objects when at least one exists (e.g., page numbers)

Why It Matters

Whole numbers appear constantly in everyday life — counting items, measuring discrete quantities, and indexing positions. In algebra and number theory, knowing which number set you are working with determines which operations are valid; for example, subtraction of two whole numbers can leave the set (5 − 8 = −3), which matters when solving equations restricted to whole-number solutions. Standardized tests and math courses frequently ask you to classify numbers into sets, so understanding where whole numbers fit in the number-system hierarchy is essential.

Common Mistakes

Mistake: Forgetting that 0 is a whole number.
Correction: The set of whole numbers explicitly begins at 0. If you exclude 0, you have the natural numbers (under the most common convention), not the whole numbers.
Mistake: Including negative integers as whole numbers.
Correction: Whole numbers are nonnegative only. The set that includes both positive and negative integers (and zero) is called the integers, denoted ℤ.

Related Terms