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Difference between revisions of "Natural number"

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The set of '''natural numbers''', denoted <math>\mathbb{N}</math>, is a subset of the [[integer]]s <math>\mathbb{Z}</math>. Unfortunately, exactly which subset is not entirely clear: in some texts, <math>\mathbb{N}</math> is taken to be the set of [[counting number]]s ([[positive integer]]s), while in others it is taken to be the set of [[whole number]]s ([[nonnegative]] integers).  Because of this ambiguity, one should always be careful to define one's notation clearly.  Possible alternatives include<math>\mathbb{Z}_{\geq0}</math> for the non-negative integers and <math>\mathbb{Z}_{>0}</math> or <math>\mathbb{P}</math> for the positive integers (although <math>\mathbb{P}</math> is also sometimes used for the [[prime number]]s).
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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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Natural numbers are important in the link between the well-ordering principle and the principle of mathematical induction.
 
  
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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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[[Category:Definition]]
 
[[Category:Definition]]
 
[[Category:Number theory]]
 
[[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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