Number Theory
Draft for Information Only Content Algebraic Number
Algebraic NumberAn 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 numberNatural 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
From property 1 and assuming property 2 is true. Considering a subset S'={n:0,...,n∈S} of S that is S'⊆S.
From property 2 and assuming property 3 is true. Considering a subset T⊆N and let S'=NT.
In other words,
From property 3 and assume property 1 is true. Suppose 0∈S, and n+1∈S when n∈S. Let T=NS.
Addition FunctionThe addition function can be based on the successor function. The definition of m+n are
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