Natural numbers (starting at 1)
Elementary
The natural numbers are the counting numbers $1, 2, 3, 4, \dots$
Undergraduate
The natural numbers are the set $\setdef{1, 2, 3, \dots}$ of positive integers, often written $\nats$, \(\mathbb{Z}_{> 0}\), or \(\nats_{+}\).
Examples
Examples
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$1$
Elementary
$1$ 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
Things that may be mistaken for examples.
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$0$
Elementary
$0$ 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.
Notes
This convention is common in number theory and analysis. Compare natural numbers starting at 0.
Distinctions involving this term
- 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 $1$.
- 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, “$\dots, -2, -1, 0, 1, 2, \dots$.