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Algebraic Number

An algebraic number is defined as the root of a nonzero polynomial equation. In general, algebraic numbers are complex numbers, however algebraic numbers can also be algebraic real numbers. In other words, natural numbers, whole numbers, integers, rational numbers and algebraic irrational numbers can also be alegbraic numbers.

Natural number

Natural numbers, 1,2,3,... are the numbers used for counting. These numbers are generated by a successor function, S(n)=n+1 and number 0 is naturally the empty quantity used in counting. By setting a+0=a and a+S(b)=S(a+b) for all a,b, if 1 is defined as S(0), then a+1=a+S(0)=S(a+0)=S(a). In other words, the set N= {0,1,2,...} of natural numbers including number 0 are the closure set of the set {0} under the successor operation, +1. The properties of the set of succerssor operation are 0∈S, and n+1∈S when n∈S. The set of  natural numbers N is therefore a subset of set S, that is N⊆S. The properties of N by induction are

1. If 0∈S, and if n+1∈S when n∈S, then N⊆S.
2. If 0∈S, and if n+1∈S when 0,1,2,...,n∈S, then N⊆S.
3. If T⊆N is nonempty, then T has a least member, n:n∈T where T is the closure set of the set {n} under the successor operation +1.

From property 1 and assuming property 2 is true. Considering a subset S'={n:0,...,n∈S} of S that is S'⊆S.

• 0∈S'⇒  0∈S, by property 1
• n∈S'⇒  n∈S, by property 1 for all n in S'
• n∈S'⇒  0,...n∈S, by property 1
  • n∈S⇒  n+1∈S, by the assumption of property 2
  • n+1∈S⇒  0,...,n+1∈S, by property 2
• n∈S'⇒n+1∈S' by considering number n+1 in S as element of S'
• n+1∈S'⇒N⊆S' by property 1
• N⊆S'⇒N⊆S by set property S'⊆S

 From property 2 and assuming property 3 is true. Considering a subset T⊆N and let S'=N-T.

• If T has no least member ⇒0∈S' since number 0 cannot be the least member of T
• 0∈S'⇒0∉T by the assumption S'=N-T
• 0,...,n∈S'⇒0,...,n∉T by property 2 and assumption S'=N-T
• n+1∉T⇒n+1∈S' Since n+1 cannot be the least member of T
• n+1∈S'⇒ N⊆S' by property 2
• N⊆S' ⇒ T is empty. since T⊆N and  S'=N-T
• If T has no least member then T=∅⇒T has a least member n then T is nonempty and n∈T 

In other words,

• If T has an least member n ⇒n∈T
• n∈T⇒n∉S'
• n+1∈T⇒n +1∉S' by the successor operation +1
• T in nonempty

From property 3 and assume property 1 is true. Suppose 0∈S, and n+1∈S when n∈S. Let T=N-S.

• 0∈S'⇒0∉T
  •  0<n∈T⇒n∉S' assuming T has a least member n
  • n∉S⇒n-1∉S by assumption of property 1
  • n-1∉S⇒ n-1∈T
  • n-1<n ⇒ n is not the least member of T, therefore T do not has a least member
•  T has no least member⇒T is empty by property 3
• T=N-S=empty⇒N⊆S.

Addition Function

The addition function can be based on the successor function. The definition of m+n are

• n=0: m+0=m for all m∈N
• n=k+1: m+(k+1)=(m+k)+1 for all m,k∈N