Difference between revisions of "Natural number"

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The natural numbers, denoted by the set <math>\mathbb{N}</math>, is itself a subset of the integers <math>\mathbb{Z}</math>, which is a set of the reals, <math>\mathbb{R}</math>. The natural numbers can be defined as ''every integer greater than or equal to 1''. Don't confuse this with the whole numbers, starting at 0.
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A '''natural number''' is any positive [[integer]]: <math>\text{1, 2, 3, 4, 5, 6, 7,\dots}</math>. The set of '''natural numbers''', denoted <math>\mathbb{N}</math>, is a subset of the set of [[integer]]s, <math>\mathbb{Z}</math>. Some texts use <math>\mathbb{N}</math> to denote the set of [[positive integer]]s (sometimes called [[counting number]]s in elementary contexts), while others use it to represent the set of [[nonnegative]] integers (sometimes called [[whole number]]s).  In particular, <math>\mathbb{N}</math> usually includes zero in the contexts of [[set theory]] and [[abstract algebra | algebra]], but usually not in the contexts of [[number theory]]. When there is risk of confusion, mathematicians often resort to less ambiguous notations, such as <math>\mathbb{Z}_{\geq0}</math> and <math>\mathbb{Z}_0^+</math> for the set of non-negative integers, and <math>\mathbb{Z}_{>0}</math> and <math>\mathbb{Z}^+</math> for the set of positive integers.
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== See also ==
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* [[Induction]]
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* [[Well Ordering Principle|Well-ordering principle]]
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{{stub}}
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[[Category:Definition]]
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[[Category:Number theory]]

Latest revision as of 20:44, 13 March 2022

A natural number is any positive integer: $\text{1, 2, 3, 4, 5, 6, 7,\dots}$. The set of natural numbers, denoted $\mathbb{N}$, is a subset of the set of integers, $\mathbb{Z}$. Some texts use $\mathbb{N}$ to denote the set of positive integers (sometimes called counting numbers in elementary contexts), while others use it to represent the set of nonnegative integers (sometimes called whole numbers). In particular, $\mathbb{N}$ usually includes zero in the contexts of set theory and algebra, but usually not in the contexts of number theory. When there is risk of confusion, mathematicians often resort to less ambiguous notations, such as $\mathbb{Z}_{\geq0}$ and $\mathbb{Z}_0^+$ for the set of non-negative integers, and $\mathbb{Z}_{>0}$ and $\mathbb{Z}^+$ for the set of positive integers.

See also

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