natural number (original) (raw)

The first few natural numbers are:

• •
  0:=∅
• •
  1:=0′={0}={∅}
• •
  2:=1′={0,1}={∅,{∅}}
• •
  3:=2′={0,1,2}={∅,{∅},{∅,{∅}}}

Note that the set 0 has zero elements, the set 1 has one element, the set 2 has two elements, etc. Informally, the set n is the set consisting of the n elements 0,1,…,n-1, and n is both a subset of ℕ and an element of ℕ.

In some contexts (most notably, in number theory), it is more convenient to exclude 0 from the set of natural numbers, so that ℕ={1,2,3,…}. When it is not explicitly specified, one must determine from context whether 0 is being considered a natural number or not.

Addition of natural numbers is defined inductively as follows:

• •
  a+0:=a for all a∈ℕ
• •
  a+b′:=(a+b)′ for all a,b∈ℕ

Multiplication of natural numbers is defined inductively as follows:

• •
  a⋅0:=0 for all a∈ℕ
• •
  a⋅b′:=(a⋅b)+a for all a,b∈ℕ

The natural numbers form a monoid under either addition or multiplication. There is an ordering relation on the natural numbers, defined by: a≤b if a⊆b.