Posted on Categories:Mathematical Methods, 数学代写, 数学物理方法

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## 数学代写|数学物理方法代写Mathematical Methods代考|Real numbers

Most of the present chapter will be already familiar to those who have studied a good modern book on calculus, and it is not intended to compete with standard works on pure mathematics. We think, however, that some discussion here is not out of place, for several reasons. First, the latter works for the most part do not emphasize why the refined arguments that they give have any relevance to physics, and physicists therefore tend to believe that they are irrelevant. Secondly, they are liable to be so long that a physicist can hardly be blamed if he decides that he has not the time to work through them. Thirdly, the attention to very peculiar functions has led the subject to be regarded as the pathology of functions. The reply is that every function, except an absolute constant, is peculiar somewhere, and that by studying where a function is peculiar we can arrive at constructive results about it that would be very hard to obtain otherwise. But we are entitled to regard ourselves as general practitioners and to restrict ourselves to the kinds of peculiarities that occur in physics; rare diseases may be handed over for treatment to a specialist, in this case a professional pure mathematician.

The nature of the problem was foreshadowed in a theorem of Euclid that the ratio of the hypotenuse to one side of an isosceles right-angled triangle is not equal to any rational fraction. Euclid, it must be remembered, made no use of what we should now call numerical measures of physical magnitudes. When he said that two lines were equal he meant that one could be placed on the other so that the two ends of one coincided with the two ends of the other; this is the direct physical comparison and does not require any numerical description of the lengths. When he said that the square on the hypotenuse was twice that on a side he meant that it could be cut into pieces and that the pieces could then be put together so as to make the square on the side twice over. He was working throughout with the quantities themselves, not with the numbers that we choose to associate with them in measurement with regard to any special unit. The use of numbers for this purpose is a choice of a language. What Euclid’s theorem showed was that the language of rational numbers was incapable of describing simultaneously the lengths of the side and the hypotenuse of a triangle that could easily be drawn by the rules of his geometry.

## 数学代写|数学物理方法代写Mathematical Methods代考|Nests of intervals: Dedekind section

Dedekind section. The fundamental property of real numbers is that they can be approximated to as closely as we please by rational numbers. When we say that
$$\sqrt{ } 2=1.414 \ldots,$$
we assert the following set of propositions: (1) 2 is between $1^{2}$ and $2^{2}$; (2) 2 is between $1 \cdot 4^{2}$ and $1 \cdot 5^{2}$; (3) 2 is between $1 \cdot 41^{2}$ and $1 \cdot 42^{2}$; (4) 2 is between $1 \cdot 414^{2}$ and $1 \cdot 415^{2}$; and so on to any desired accuracy. At each stage this process can be regarded as separating the decimals, to a given number of places, into two classes, those whose squares are respectively greater or less than 2 . At stage 3 , for instance, the squares of $1 \cdot 414,1 \cdot 413,1 \cdot 412$ are less than 2 , those of $1.415,1 \cdot 416,1.417$ greater than 2 . We say nothing at this stage about the fractions $1 \cdot 4141,1 \cdot 4142, \ldots, 1 \cdot 4149$; but at the next stage we say that 2 lies between the squares of $1 \cdot 4142$ and $1 \cdot 4143$. By taking a sufficient number of decimals we can make the unconsidered interval as small as we like, since we divide it by 10 at each step. Thus any decimal with a finite number of places will ultimately be classified according as its square is less or greater than 2. Now this process determines a unique infinite decimal, which we can take to be $\sqrt{ } 2$, and it can be regarded as the limit approached by the successive approximations from either side.

This process, which is capable of great extension, is an example of the definition of a

• Hence the name ‘irrational numbers’.

real number by a nest of rationals. We take a succession of rationals $\left{a_{n}\right}$ and another succession $\left{b_{n}\right}$, satisfying the following conditions:
(i) $a_{n+1} \geqslant a_{n}$
(ii) $b_{n+1} \leqslant b_{n}$,
(iii) $a_{n} \leqslant b_{n}$,
for all $n$, and
(iv) Given any positive rational number $\epsilon$, a number $N$ can be found such that $b_{n}-a_{n}<\epsilon$ for every $n>N$.

# 数学物理方法代写

## 数学代写|数学物理方法代写Mathematical Methods代想| Nests of intervals: Dedekind section

• 因此得名”无理数”。
实数由一窝有理数组成。我们采取一系列理性缺少或无法识别 \left 的分隔符 和另一个继承 缺少或无法识别 \left 的分隔符，满足以下条件:
(i) $a_{n+1} \geqslant a_{n}$
(二) $b_{n+1} \leqslant b_{n}$,
(三) $a_{n} \leqslant b_{n}$ ，
面向所有人 $n$, 和
(iv) 给定任何正有理数 $\epsilon_{r}$ 一个数字 $N$ 可以这样找到 $b_{n}-a_{n}<\epsilon$ 对于每个 $n>N$.

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。