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Natural Numbers — Definition, Examples & Table

Natural Numbers
Counting Numbers

The numbers used for counting. That is, the numbers 1, 2, 3, 4, etc.

 

Nested diagram showing number sets: natural numbers ⊂ whole numbers ⊂ integers ⊂ rationals ⊂ algebraic ⊂ reals; complex and...

 

 

See also

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

Key Formula

N={1,2,3,4,5,…}\mathbb{N} = \{1, 2, 3, 4, 5, \ldots\}
Where:
  • N\mathbb{N} = The standard symbol for the set of natural numbers
  • …\ldots = Indicates the pattern continues infinitely

Worked Example

Problem: Determine which of the following are natural numbers: 7, −3, 0, 4.5, 12.
Step 1: Recall that natural numbers are the positive counting numbers: 1, 2, 3, 4, ...
N={1,2,3,4,5,…}\mathbb{N} = \{1, 2, 3, 4, 5, \ldots\}
Step 2: Check 7: It is a positive whole number used for counting, so 7 is a natural number.
7∈N✓7 \in \mathbb{N} \quad \checkmark
Step 3: Check −3: It is negative, so it is not a natural number.
−3∉N-3 \notin \mathbb{N}
Step 4: Check 0: Zero is not used for counting objects (you don't start counting at zero), so 0 is not a natural number. It belongs to the whole numbers instead.
0∉N0 \notin \mathbb{N}
Step 5: Check 4.5 and 12: The value 4.5 is a decimal (not a whole counting number), so it is not natural. The value 12 is a positive counting number, so it is natural.
4.5∉N,12∈N4.5 \notin \mathbb{N}, \quad 12 \in \mathbb{N}
Answer: The natural numbers in the list are 7 and 12.

Another Example

This example shows a computational application of natural numbers rather than classification. It introduces the classic summation formula and demonstrates how natural numbers are used in arithmetic sequences.

Problem: Find the sum of the first 10 natural numbers.
Step 1: List the first 10 natural numbers.
1,2,3,4,5,6,7,8,9,101, 2, 3, 4, 5, 6, 7, 8, 9, 10
Step 2: Use the formula for the sum of the first n natural numbers, attributed to Gauss.
S=n(n+1)2S = \frac{n(n+1)}{2}
Step 3: Substitute n = 10 into the formula.
S=10×112=1102=55S = \frac{10 \times 11}{2} = \frac{110}{2} = 55
Answer: The sum of the first 10 natural numbers is 55.

Frequently Asked Questions

Is 0 a natural number?
This depends on the convention. In most U.S. math courses and the definition used here, 0 is NOT a natural number — natural numbers start at 1. However, some textbooks (especially in logic, set theory, and European traditions) define natural numbers to include 0. Always check which convention your course uses.
What is the difference between natural numbers and whole numbers?
Natural numbers are {1, 2, 3, 4, ...}, while whole numbers are {0, 1, 2, 3, 4, ...}. The only difference is that whole numbers include zero. Every natural number is a whole number, but 0 is a whole number that is not a natural number.
Are natural numbers infinite?
Yes. There is no largest natural number. For any natural number n, you can always find a larger one by computing n + 1. This means the set of natural numbers is countably infinite — it goes on forever, but you can still list its elements in order.

Natural Numbers vs. Whole Numbers

Natural NumbersWhole Numbers
DefinitionPositive counting numbers: {1, 2, 3, ...}Non-negative integers: {0, 1, 2, 3, ...}
Includes zero?NoYes
Symbolℕ𝕎 or ℕ₀
Smallest element10
Common useCounting objects (1st, 2nd, 3rd...)Counting including 'none' (0 items)

Why It Matters

Natural numbers appear constantly in algebra, number theory, and sequences. When you see summation notation like ∑k=1n\sum_{k=1}^{n}, the index kk runs through natural numbers. Understanding which numbers are natural helps you correctly interpret domain restrictions, identify valid inputs for formulas like factorials (n!n! requires n∈Nn \in \mathbb{N} or n=0n = 0), and classify numbers in set theory questions on exams.

Common Mistakes

Mistake: Including 0 as a natural number when the course defines natural numbers as starting at 1.
Correction: Check your textbook's convention. In most U.S. pre-algebra and algebra courses, natural numbers begin at 1. The set that includes 0 is called the whole numbers.
Mistake: Thinking that all positive numbers are natural numbers.
Correction: Numbers like 3.7, ½, and √2 are positive but not natural. Natural numbers must be positive AND whole (no fractions or decimals). Only 1, 2, 3, 4, ... qualify.

Related Terms