Difference between revisions of "Natural number"

Revision as of 12:13, 26 January 2008

The set of natural numbers, denoted $\mathbb{N}$ , is a subset of the set of integers, $\mathbb{Z}$ . Unfortunately, exactly which subset is not entirely clear: in some texts, $\mathbb{N}$ is taken to be the set of positive integers (sometimes called counting numbers in elementary contexts), while in others it is taken to be 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.

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-The set of '''natural numbers''', denoted <math>\mathbb{N}</math>, is the set most conveniently associated with the notion of "counting".
+The set of '''natural numbers''', denoted <math>\mathbb{N}</math>, is a subset of the set of [[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 [[positive integer]]s (sometimes called [[counting number]]s in elementary contexts), while in others it is taken to be 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.
-==Definition==
+{{stub}}
-Let <math>\mathcal{F}</math> be the set of all [[Successor set |successor sets]] <math>S</math>.
-The set of Natural Numbers <math>\mathbb{N}</math> is defined as
+== See Also ==
-<math>\mathbb{N}=\bigcap_{S\in\mathcal{F}} S</math>
-Note that as <math>1\in S</math> <math>\forall S\in\mathcal{F}</math>, <math>\mathbb{N}</math> is non-empty.
+* [[Induction]]
+* [[Well-ordering principle]]
-==Common Usage==
-According to this definition, <math>\mathbb{N}</math> is the set <math>\{1,2,3,\ldots\}</math> (Which is also called the set of [[counting number]]s or [[positive integer]])s. Unfortunately, in some texts, <math>\mathbb{N}</math> is taken to be the set of [[whole number]]s or [[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).
-Natural numbers are important in the link between the well-ordering principle and the principle of mathematical induction.
-==Mathematical Induction==
-'''Mathematical Induction''' is an extremely useful
-tool for problems regarding Natural Numbers.
-Statement:
-Let <math>S\subset \mathbb{N}</math>
-Let (i)<math>1\in S</math>
-Let (ii)<math>\forall n\in S</math>; <math>n+1\in S</math>
-Then <math>S</math> is the set of natural numbers, or <math>S=\mathbb{N}</math>
-==Well-Ordering Principle==
-The '''Well-Ordering Principle''' states that every subset of <math>\mathbb{N}</math> has a least element.
 [[Category:Definition]]
 [[Category:Number theory]]

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Difference between revisions of "Natural number"

Revision as of 12:13, 26 January 2008

See Also