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# 计算机代写|计算方法代写Algorithmic Methods代写|CSCI256 Numbers

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## 计算机代写|计算方法代写Algorithmic Methods代写|The Real Numbers

In this book we assume the following number systems as known:
$\mathbb{N}={1,2,3,4, \ldots} \quad$ the set of natural numbers;
$\mathbb{N}_{0}=\mathbb{N} \cup{0} \quad$ the set of natural numbers including zero;
$\mathbb{Z}={\ldots,-3,-2,-1,0,1,2,3, \ldots} \quad$ the set of integers;
$\mathbb{Q}=\left{\frac{k}{n} ; k \in \mathbb{Z}\right.$ and $\left.n \in \mathbb{N}\right} \quad$ the set of rational numbers.
Two rational numbers $\frac{k}{n}$ and $\frac{\ell}{m}$ are equal if and only if $k m=\ell n$. Further an integer $k \in \mathbb{Z}$ can be identified with the fraction $\frac{k}{1} \in \mathbb{Q}$. Consequently, the inclusions $\mathbb{N} \subset$ $\mathbb{Z} \subset \mathbb{Q}$ are true.

Let $M$ and $N$ be arbitrary sets. A mapping from $M$ to $N$ is a rule which assigns to each element in $M$ exactly one element in $N .{ }^{1}$ A mapping is called bijective, if for each element $n \in N$ there exists exactly one element in $M$ which is assigned to $n$.
Definition 1.1 Two sets $M$ and $N$ have the same cardinality if there exists a bijective mapping between these sets. A set $M$ is called countably infinite if it has the same cardinality as $\mathbb{N}$.

The sets $\mathbb{N}, \mathbb{Z}$ and $\mathbb{Q}$ have the same cardinality and in this sense are equally large. All three sets have an infinite number of elements which can be enumerated. Each enumeration represents a bijective mapping to $\mathbb{N}$. The countability of $\mathbb{Z}$ can be seen from the representation $\mathbb{Z}={0,1,-1,2,-2,3,-3, \ldots}$. To prove the countability of $\mathbb{Q}$, Cantor’s ${ }^{2}$ diagonal method is being used:
$$\begin{array}{llllllll} \frac{1}{1} & \rightarrow & \frac{2}{1} & & \frac{3}{1} & \rightarrow & \frac{4}{1} & \cdots \ & \swarrow & & \nearrow & & \swarrow & & \ \frac{1}{2} & & \frac{2}{2} & & \frac{3}{2} & & \frac{4}{2} & \cdots \ \downarrow & \nearrow & & \swarrow & & & & \ \frac{1}{3} & & \frac{2}{3} & & \frac{3}{3} & & \frac{4}{3} & \cdots \ & \swarrow & & & & & & \ \frac{1}{4} & & \frac{2}{4} & & \frac{3}{4} & & \frac{4}{4} & \cdots \ \vdots & & \vdots & & \vdots & & \vdots & \ & & & & & & & \end{array}$$
The enumeration is carried out in direction of the arrows, where each rational number is only counted at its first appearance. In this way the countability of all positive rational number (and therefore all rational numbers) is proven.

To visualise the rational numbers we use a line, which can be pictured as an infinitely long ruler, on which an arbitrary point is labelled as zero. The integers are marked equidistantly starting from zero. Likewise each rational number is allocated a specific place on the real line according to its size, see Fig. 1.1.

However, the real line also contains points which do not correspond to rational numbers. (We say that $\mathbb{Q}$ is not complete.) For instance, the length of the diagonal $d$ in the unit square (see Fig. 1.2) can be measured with a ruler. Yet, the Pythagoreans already knew that $d^{2}=2$, but that $d=\sqrt{2}$ is not a rational number.

## 计算机代写|计算方法代写Algorithmic Methods代写|Order Relation and Arithmetic on R

In the following we write real numbers (uniquely) as decimals with an infinite number of decimal places, for example, we write $0.2999 \ldots$ instead of $0.3$.

Definition $1.8$ (Order relation) Let $a=a_{0} \cdot a_{1} a_{2} \ldots$ and $b=b_{0} \cdot b_{1} b_{2} \ldots$ be nonnegative real numbers in decimal form, i.e. $a_{0}, b_{0} \in \mathbb{N}{0}$. (a) One says that $a$ is less than or equal to $b$ (and writes $a \leq b$ ), if $a=b$ or if there is an index $j \in \mathbb{N}{0}$ such that $a_{j}<b_{j}$ and $a_{i}=b_{i}$ for $i=0, \ldots, j-1$.
(b) Furthermore one stipulates that always $-a \leq b$ and sets $-a \leq-b$ whenever $b \leq a$

This definition extends the known orders of $\mathbb{N}$ and $\mathbb{Q}$ to $\mathbb{R}$. The interpretation of the order relation $\leq$ on the real line is as follows: $a \leq b$ holds true, if $a$ is to the left of $b$ on the real line, or $a=b$.

## 计算机代写|计算方法代写ALGORITHMIC METHODS代写|THE REAL NUMBERS

$\mathbb{N}=1,2,3,4, \ldots$ 自然数集；
$\mathbb{N}_{0}=\mathbb{N} \cup 0$ 包括零的自然数集;
$\mathbb{Z}=\ldots,-3,-2,-1,0,1,2,3, \ldots$ 整数集合;
$\lfloor$ mathbb ${Q}=\backslash$ left ${$ frac ${k}{n} ; k \backslash$ in $\backslash$ mathbb ${Z} \backslash$ right.\$和$\$\backslash$ left.n $\backslash$ in \mathbb ${N} \backslash$ right $} \backslash q u a d$ 有理数的集合。

## 计算机代写|计算方法代写ALGORITHMIC METHODS代写|ORDER RELATION AND ARITHMETIC ON R

$b$ 此外，一项规定总是 $-a \leq b$ 并设置 $-a \leq-b$ 每当 $b \leq a$

## 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。