Whole numbers vs. natural numbers
“Whole numbers” and “natural numbers” are used synonymously by some others to indicate all nonnegative integers, $0, 1, 2, 3, \dots$. “Whole numbers” is the usual name in primary schools, and “natural numbers” (or “nonnegative integers”) is the usual name in mathematics. “Natural numbers” is, however, used by some authors to indicate the non-negative integers, $1, 2, 3, \dots$ and “whole numbers” is used to indicate all integers, “$\dots, -2, -1, 0, 1, 2, \dots$.
Definitions
The whole numbers are $0, 1, 2, 3, \dots$, the numbers with no fractional or decimal part.
Non-examples
-
$2.5$
$2.5$ is not a whole number or a natural number, because it has a fractional part.
The natural numbers are the numbers $0, 1, 2, 3, \dots$ that can be used to answer questions about “how many?” objects there are.
The natural numbers are the set $\setdef{0, 1, 2, 3, \dots}$ of nonnegative integers, often written $\nats$, \(\nats_0\), or \(\mathbb{Z}_{\geq 0}\).
The natural numbers are the smallest set that contains $0 = \emptyset$ and contains $n \cup \setdef{n}$ whenever it contains $n$ (the von Neumann construction). Each natural number is the set of all smaller natural numbers.
Non-examples
The natural numbers are the counting numbers $1, 2, 3, 4, \dots$
The natural numbers are the set $\setdef{1, 2, 3, \dots}$ of positive integers, often written $\nats$, \(\mathbb{Z}_{> 0}\), or \(\nats_{+}\).
Examples
-
$1$
$1$ is a natural number under both conventions: it is the first natural number when the natural numbers start at $1$, and the second when they start at $0$.
Non-examples
-
$0$
$0$ is a natural number when the natural numbers start at $0$, but not when they start at $1$. It is the only number on which the two conventions disagree.
Examples
| Example | whole numbers | natural numbers (from 0) | natural numbers (from 1) | Why |
|---|---|---|---|---|
| $0$ | ✓ | ✓ | ✗ | |
Missing example: seven | ||||
Missing example: minus_three | ||||
| $2.5$ | ✗ | ✗ | – | |
Notes
- The two names are synonyms only when the natural numbers start at $0$. Many authors start them at $1$; see natural numbers: starting at 0 or at 1.
- “Whole numbers” also has rival meanings: some authors use it for all integers, or for the positive integers. In technical writing, “nonnegative integers” or \(\mathbb{Z}_{\geq 0}\) is unambiguous.