Natural numbers: starting at 0 or at 1
Whether the natural numbers include $0$ depends on the author. Set theory, logic, and computer science usually include it, while number theory and analysis often start at $1$.
Definitions
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
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$1$
$1$ 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
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$0$
$0$ 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.
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
Examples
Avoiding the ambiguity
When it matters whether $0$ is included, say so explicitly. Unambiguous notation includes \(\mathbb{Z}_{\geq 0}\) (or \(\nats_0\)) for $\setdef{0, 1, 2, \dots}$ and \(\mathbb{Z}_{> 0}\) (or \(\nats_{+}\)) for $\setdef{1, 2, 3, \dots}$.
Natural numbers that start at $0$ are also called whole numbers, mostly in schools.