计算机问题求解一论题4-1 -数论基础 2017年03月20日
计算机问题求解 – 论题4-1 - 数论基础 2017年03月20日
问题1： 自然数有定义吗？
Number is the basis of modern mathematics.But what is number? What does it mean to say that多+多=1，.}=，and(-1)(-l)=1? We learn in school the mechanics of handling fractions and negative numbers,but for a real understanding of the number system we must go back to simpler elements.While the Greeks chose the geometrical con- cepts of point and line as the basis of their mathematics,it has become the modern guiding principle that all mathematical statements should be reducible ultiinately to statements about the nalural numbers, 1,2,3,...."God created the natural numbers;everything else is man's handiwork."In these words Leopold Kronecker (1823-1891) pointed out the safe ground on which the structure of mathematics can be built. Fortunately,the mathematician as such need not be concerned with the philosophical nature of the transition from collections of concrete objects to the abstract number concept.We shall therefore accept the natural numbers as given,togcther with the two fundamental opera- tions,addition and multiplication,by which they may be combined. from What is Mathematics,by Courant and Robbins
from What is Mathematics, by Courant and Robbins
用集合定义自然数 设a为集合，称U{d为a的后继，记为或at,或s(a)。 ·设A是集合，若A满足下列条件，称A为归纳集： 口0∈A 口Va(a∈A→叶∈A) ■自然数集合N是所有归纳集的交集。 口因此：N={0,{0},{0,{0},{0,{0),{0,{0},…} 口N的每一个元素称为一个自然数。 口0记为0,0+记为1,1+记为2,2+记为3，余此类推
用集合定义自然数 ◼ 设a为集合, 称a{a}为a的后继, 记为或a + , 或s(a)。 ◼ 设A是集合，若A满足下列条件，称A为归纳集： ❑ ØA ❑ a(aA→a+A) ◼ 自然数集合N是所有归纳集的交集。 ❑ 因此：N = { Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}}, … } ❑ N的每一个元素称为一个自然数。 ❑ Ø记为0，0 +记为1，1 +记为2，2 +记为3，余此类推
再具体一点 ■记号0表示：0 ■记号1表示0+：0U0}={0} ■记号2表示1+：{0狄U{0}={0,{0} ■记号3表示2*：{0，{0}U{0,{0}={0,{0,{0,{0} ■3U2=? 3∩2=？ ■2∈3？ 1∈3？ ■1c2?2c5? 自然数上的小于关系如何表达？
再具体一点 ◼ 记号0表示：Ø ◼ 记号1表示0 +： Ø{Ø}={Ø} ◼ 记号2表示1 +： {Ø}{{Ø}}={Ø,{Ø}} ◼ 记号3表示2 + : {Ø,{Ø}}{{Ø,{Ø}}}={Ø,{Ø}, {Ø,{Ø}}} ◼ 3∪2=? 3∩2=? ◼ 23? 13? ◼ 12? 25? 自然数上的小于关系如何表达？