自然数

And all that number crunching led the Sumerians to begin crude speculations about the nature of numbers and processes involving numbers—abstract mathematics.

所有的零碎的数字引导这苏美尔人开始了天然的对于自然数的思索和数字推理的抽象化。

This essay deduces the power sum formula of an arbitrary arithmetic progression step by step on the basis of Euler-Maclaurin formula,and then achieres the compute formula of natural number power sum -∑ni=1i m.

前n个自然数的方幂和，∑ni=1im是一个古老的难题。

Similarly, the addition of any three natural numbers must satisfy associative law:+c=a+, and then we can verify that for any n natural numbers, the results of adding is the same in any associative way which keep the order of numbers.

类似地任何三个自然数可以有结合律：+c=a+，进而可以证明任何n个数的任何结合方式进行加法所得的结果是一致的。

For any positive integer n, about the Smarandache LCM dual function SL*, the main purpose of the fifth Chapter is using the elementary method to study the calculating problem of a Dirichlet series involving the Smarandache LCM dual function SL* and the mean value distribution property of SL*, obtain an exact calculating formula and a sharper asymptotic formula for it.

对于给定的自然数n，关于Smarandache LCM对偶函数SL~*，本文第五章利用初等方法研究了一个包含SL~*的Dirichlet级数计算问题及SL~*的均值性质，分别给出了精确计算公式和一个较强的渐近公式。

Secondly, by using of the medium axiomatic set theory, a natural number system in MS is constructed, and it is proved that five axioms of Peano′s natural number system are theorems is MS.

其次，利用中介公理集合论MS的相关理论，构造了MS中的自然数系统，证明了Peano5条公理为MS中的定理。

In this paper, based on the analysis of the genetic algorithm on binary coding, a preliminary discussion is made on the genetic algorithm of natural number coding.

在分析基于二进制编码的遗传算法基础上，对基于自然数编码的遗传算法作了初步的探讨。

It is impossible Gua 0-63 from 64 natural that the number of complete binary form.

六四十卦正是从0到63这六十四个自然数的完整的二进制数形。

Departing from basic concepts of cardinal numbers and their numbering orders, the paper differentiated and analyzed some concepts easy to be confused related to binary system, pointing out that the introduced concepts of ordinals played an important role in the development of modern mathematics. As the foundation of SHAO Yong's mathematics, the primordial Yi , in the history of mathematics, exhibits the first specially defined ordinal system, in which the binary numbering method was used at the first time.

本文从自然数及其记数法的基本概念出发，辨析了与二进位制相关的几个容易混淆的概念，指出序数概念的引入在近代数学发展中有重要意义，作为邵雍数学学派的数学基础，先天易是数学史上第一个专门定义的序数体系，该序数体系在数学史上率先采用了二进位制记数法。

Through the analysis of the second power Cartesian product of natural number set N——N×N and the thirdpower Cartesian product of natural number set N——N×N×N,obtains the conclusion that they all have the bijective relation to natural number set N,it means that the set N×N and the set N×N×N are all countably infinite.

通过对自然数集合N的二次笛卡尔积运算———N×N和三次笛卡尔积运算———N×N×N的详细分析，得出了它们与自然数集合N之间都存在双射关系结论，即集合N×N和集合N×N×N都是可数无穷的。

The most simple mathematical induction is the most common is that n is all natural, an expression of the establishment.

中文摘要：数学归纳法是一种数学证明方法，是数学中关于自然数命题的主要证明方法，用于确定一个表达式在所有自然数范围内成立，或者用于确定一个表达式在一个无穷数列中成立。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。