The successor number refers to a number immediately following a natural number, for example, the successor number of 2 is 3, and the successor number of 4 is 5.0 is not the successor of any natural number. Every determined natural number has a determined successor.
Successor number refers to the number ofNatural numberThe next number, for example, the number of successors of 2 is 3, and the number of successors of 4 is 5.[1]
Peano axioms
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PianoThe five axioms of are described in a non formal way as follows:
① 0 is a natural number;
② Every determined natural number a has a determined successor number x ', and x' is also a natural number;
③ If b and c are the successors of the natural number a, then b=c;
④ 0 is not the successor of any natural number;
⑤ Let S be a subset of the set of natural numbers, and (i) 0 belongs to S;(2) If n belongs to S, then the number of successors of n also belongs to S.[2]
application
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Successors can be applied toMathematical inductionThe principle of mathematical induction is:
1) (i.e. initial number) The first is a natural number;
2) Every natural number is followed by a successor number, for example, the successor number of 2 is 3, the successor number of 3 is 4,The successor number of n is n+1, and so on until infinity.
So the proof of mathematical induction is very simple, and only needs to prove that 1) the initial number meets the conditions;2) If the natural number n satisfies the condition, then the natural number n+1 also satisfies the condition. It is enough to prove these two conditions.
The extended proof of mathematical induction refers to several practical cases: the first is that if n=k is true, it is true when n=k+1 is proved;The second is to prove that if n<=k is true, then n=k+1 is true;The third is that if proposition P (n) is in n=1,2,3,T, and for any natural number k, P (k), P (k+1), P (k+2),P (k+t-1) holds, where t is a constant, then P (n) holds for all natural numbers.
Research on Successor Function
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The second dispute over the development of number concept is whether children's number concept is an acquired performance of their innate ability, or the result of learning to count?Gelman and Gallister (1978) believed that the development of counting skills was dominated by inherent counting principles.Gelman and Galiistel once proposed the concept of successor function.The successor function refers to that if the base value of an numeral (quantity) n is n, and if the numeral p is immediately after n in the number sequence, then the base value (quantity) of p is n+1.Gelman and Galiistel believe that children are born with an understanding of this function.The researchers represented by Fuson believe that counting skills come from children's learning experience and are the result of acquired learning.
Some scholars divided the development level of number concept of 100 preschool children aged 2-5 years from the theoretical perspective of the level model of the number concept development of children, compared the understanding and mastery of subsequent functions of children at different levels, and discussed the development process of children's number concept.The results show that most children have reached the highest level of number concept development after the age of 4, that is, the cardinal principle level. Children at this level can map the directional and unit changes of subsequent functions to the numerals in the number sequence.However, most children aged 2 to 3 are still at the subset level. Compared with children at the cardinal principle level, children at this level have different understanding of the successor function.However, the development of subsequent functions is not complete or nonexistent, and children at the subset level also have knowledge of a small number of directionality and unitarity.[3]
