Textbook
In this article, the natural numbers are $\mathbb N=\{1,2,3,\ldots\}$. They specify positive whole-number counts. A separate notation is $\mathbb N_0=\{0,1,2,\ldots\}$ for the set including zero. Some books include zero in $\mathbb N$ itself, so the convention must be specified before a formula is used.
Natural numbers can index measurements such as $x_1,x_2,x_3$. A subscript selects an element; it is not a multiplier. The indexing here starts at one. An index of zero requires a different stated convention. Every natural number is an integer, but negative integers are not natural numbers.
The sum and product of positive natural numbers are natural numbers again. Subtraction can leave the chosen set: for instance, $2-3=-1$. Every natural number $n$ has a successor $n+1$, so there is no greatest natural number. A finite count can still have a final value: a bounded list and the entire set of natural numbers are different objects.
Definitions and further reading: Mathematics for Computer Science.
In plain language
Natural numbers count whole objects: apples give answers 1, 2, 3, while half an apple describes a portion rather than a count. Counting can continue after any natural number; the apples run out before the numbers do. An empty basket requires a convention about zero, which some definitions include among the natural numbers and others exclude. Mathematics has no passport office to settle the matter, so the convention must be stated.
Example
The illustrative values 3, 0, β1, and $1/2$ are tested first against the chosen zero-free $\mathbb N$, followed by the explicit extension to $\mathbb N_0$. This does not admit the other rejected values.
Natural Numbers: Counting Discrete Objects
Positive counting
Three is allowed as a count of whole objects.
Step 1 of 3: Positive counting
Read all steps
Positive counting
Three is allowed as a count of whole objects.
- State
- 3included0β11/2
Chosen convention
Zero, the negative value, and the fraction are outside the chosen N.
- State
- 3included0excludedβ1excluded1/2excluded
Counting with zero
Nβ admits zero, but still rejects β1 and one half.
- State
- 3included0includedβ1excluded1/2excluded
Moving to $\mathbb N_0$ changes only the decision for zero among the displayed candidates. It neither admits $-1$ as a count nor removes the fractional part of $1/2$. If a sample may be empty, its size counter needs zero. If the task specifies an element index in a nonempty one-based enumeration, the first allowed index is 1. These tasks use related numbers but distinct declared ranges.
Related concepts: Integers.