Math 2.0

example

The natural numbers under addition are not a group

Example

Let N={0,1,2,… }\mathbb{N} = \{0, 1, 2, \dots\} be the natural numbers with addition. Addition is associative, and 00 is an identity element. The element 11 has no inverse, because 1+m≥11 + m \ge 1 for each m∈Nm \in \mathbb{N}, so 1+m≠01 + m \ne 0. Thus (N,+)(\mathbb{N}, +) is not a group. It is a commutative monoid.

Remarks

If we add the negative integers, we get the group (Z,+)(\mathbb{Z}, +).

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Binary operation, definition, no evidence levelBinary operationdefinitionNatural numbers, object, no evidence levelNatural numbersobjectIdentity element, definition, no evidence levelIdentity elementdefinitionInverse element, definition, no evidence levelInverse elementdefinitionThe natural numbers under addition are not a group, example, no evidence levelThe natural numbers u…example
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Sources

  1. Web https://en.wikipedia.org/wiki/Natural_number, Addition

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