计算机代写|计算机图形学作业代写computer graphics代考|Trigonometry

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Units of Angular Measurement

The measurement of angles is at the heart of trigonometry, and today two units of angular measurement are part of modern mathematics: degrees and radians. The degree (or sexagesimal) unit of measure derives from defining one complete rotation as $360^{\circ}$. Each degree divides into $60 \mathrm{~min}$, and each minute divides into $60 \mathrm{~s}$. The number 60 has survived from Mesopotamian days and appears rather incongruous when used alongside today’s decimal system – nevertheless, it is still convenient to work with degrees even though the radian is a natural feature of mathematics.

The radian of angular measure does not depend upon any arbitrary constant, and is often defined as the angle created by a circular arc whose length is equal to the circle’s radius. And because the perimeter of a circle is $2 \pi r, 2 \pi$ rad correspond to one complete rotation. As $360^{\circ}$ corresponds to $2 \pi \mathrm{rad}, 1 \mathrm{rad}$ equals $180^{\circ} / \pi$, which is approximately $57.3^{\circ}$. The following relationships between radians and degrees are

worth remembering:
$\frac{\pi}{2}[\mathrm{rad}] \equiv 90^{\circ}, \quad \pi[\mathrm{rad}] \equiv 180^{\circ}$
$\frac{3 \pi}{2}[\mathrm{rad}] \equiv 270^{\circ}, \quad 2 \pi[\mathrm{rad}] \equiv 360^{\circ} .$
To convert $x^{\circ}$ to radians:
$$\frac{\pi x^{\circ}}{180}[\mathrm{rad}]$$
To convert $x[\mathrm{rad}]$ to degrees:
$$\frac{180 x}{\pi} \text { [degrees]. }$$
For those readers wishing to know the background to radians we need to use power series. We start with the power series for $\mathrm{e}^{\theta}, \sin \theta$ and $\cos \theta$ :
\begin{aligned} \mathrm{e}^{\theta} &=1+\frac{\theta^{1}}{1 !}+\frac{\theta^{2}}{2 !}+\frac{\theta^{3}}{3 !}+\frac{\theta^{4}}{4 !}+\frac{\theta^{5}}{5 !}+\frac{\theta^{6}}{6 !}+\frac{\theta^{7}}{7 !}+\frac{\theta^{8}}{8 !}+\frac{\theta^{9}}{9 !}+\cdots \ \sin \theta &=\theta-\frac{\theta^{3}}{3 !}+\frac{\theta^{5}}{5 !}-\frac{\theta^{7}}{7 !}+\frac{\theta^{9}}{9 !}+\cdots \ \cos \theta &=1-\frac{\theta^{2}}{2 !}+\frac{\theta^{4}}{4 !}-\frac{\theta^{6}}{6 !}+\frac{\theta^{8}}{8 !}+\cdots \end{aligned}
Euler proved that these three power series are related, and when $\theta=\pi, \sin \theta=0$, and $\cos \theta=-1$. Figure $4.1$ shows curves of the sine power series for $3,5,7$ and 9 terms, and when $\theta=2 \pi$, the graph reaches zero.

计算机代写|计算机图形学作业代写computer graphics代考|The Trigonometric Ratios

Ancient civilisations knew that triangles-whatever their size-possessed some inherent properties, especially the ratios of sides and their associated angles. This means that if these ratios are known in advance, problems involving triangles with unknown lengths and angles, can be discovered using these ratios.

Figure $4.2$ shows a point $P$ with coordinates (base, height), on a unit-radius circle rotated through an angle $\theta$. As $P$ is rotated, it moves into the 2 nd quadrant, 3rd quadrant, 4th quadrant and returns back to the first quadrant. During the rotation, the sign of height and base change as follows:
$$\begin{array}{ll} \text { 1st quadrant: } & \text { height }(+) \text {, base }(+) \ \text { 2nd quadrant: } & \text { height }(+) \text {, base }(-) \ \text { 3rd quadrant: } & \text { height }(-) \text {, base }(-) \ \text { 4th quadrant: } & \text { height }(-) \text {, base }(+) . \end{array}$$
Figures $4.3$ and $4.4$ plot the changing values of height and base over the four quadrants, respectively. When radius $=1$, the curves vary between 1 and $-1$. In the context of triangles, the sides are labelled as follows:
\begin{aligned} \text { hypotenuse } &=\text { radius } \ \text { opposite } &=\text { height } \ \text { adjacent } &=\text { base. } \end{aligned}
Thus, using the right-angle triangle shown in Fig. 4.5, the trigonometric ratios: sine, cosine and tangent are defined as
$$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}, \quad \cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}, \quad \tan \theta=\frac{\text { opposite }}{\text { adjacent }} .$$

计算机代写|计算机图形学作业代写computer graphics代考|Inverse Trigonometric Ratios

The functions $\sin \theta, \cos \theta, \tan \theta, \csc \theta, \sec \theta$ and $\cot \theta$ provide different ratios for the angle $\theta$, and the inverse trigonometric functions convert a ratio back into an angle. These are arcsin, arccos, arctan, arccsc, arcsec and arccot, and are sometimes written as $\sin ^{-1}, \cos ^{-1}, \tan ^{-1}, \csc ^{-1}, \sec ^{-1}$ and $\cot ^{-1}$. For example, $\sin 30^{\circ}=0.5$, therefore, $\arcsin 0.5=30^{\circ}$. Consequently, the domain for arcsin is the range for sin:
$[-1,1]$
and the range for arcsin is the domain for sin:
$$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$
as shown in Fig. 4.8. Similarly, the domain for arccos is the range for cos:
$$[-1,1]$$
and the range for arccos is the domain for $\cos$ :
$$[0, \pi]$$
as shown in Fig. 4.9.

计算机代写|计算机图形学作业代写computer graphics代考|Units of Angular Measurement

3圆周率2[r一种d]≡270∘,2圆周率[r一种d]≡360∘.

180X圆周率 [度]。

计算机代写|计算机图形学作业代写computer graphics代考|The Trigonometric Ratios

第一象限：  高度 (+)， 根据 (+)  第二象限：  高度 (+)， 根据 (−)  第三象限：  高度 (−)， 根据 (−)  第四象限：  高度 (−)， 根据 (+).

斜边 = 半径   对面的 = 高度   邻近的 = 根据。

计算机代写|计算机图形学作业代写computer graphics代考|Inverse Trigonometric Ratios

[−1,1]
arcsin 的范围是 sin 的域：
[−圆周率2,圆周率2]

[−1,1]
arccos 的范围是因 :
[0,圆周率]

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考|Function Domains and Ranges

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Function Domains and Ranges

The following descriptions of domains and ranges only apply to functions with one independent variable: $f(x)$.
Returning to the above function:
$$y=f(x)=3 x^{2}+2 x+4$$
the independent variable $x$, can take on any value from $-\infty$ to $\infty$, which is called the domain of the function. In this case, the domain of $f(x)$ is the set of real numbers

$\mathbb{R}$. The notation used for intervals, is also used for domains, which in this case is
$$]-\infty, \infty[$$
and is open, as there are no precise values for $-\infty$ and $\infty$.
As the independent variable takes on different values from its domain, so the dependent variable, $y$ or $f(x)$, takes on different values from its range. Therefore, if the domain of the linear function $f(x)=3 x+4$ is $[-4,4]$, the range of $f(x)$ is calculated by finding $f(-4)$ and $f(4)$ :
$$\begin{gathered} f(-4)=-12+4=-8 \ f(4)=12+4=16 \end{gathered}$$
and the range is $[-4,4]$.
Although calculating the range of linear functions is simple, other types of functions require a knowledge of calculus.
The domain of $\log x$ is
] $0, \infty[$
which is open, because $x \neq 0$. Whereas, the range of $\log x$ is
$$]-\infty, \infty[.$$
The domain of $\sqrt{x}$ is
$[0, \infty[$
which is half-open, because $\sqrt{0}=0$, and oo has no precise value. Similarly, the range of $\sqrt{x}$ is
$[0, \infty[.$
Sometimes, a function is sensitive to one specific number. For example, in the function
$$y=f(x)=\frac{1}{x-1}$$
when $x=1$, there is a divide by zero, which is meaningless. Consequently, the domain of $f(x)$ is the set of real numbers $\mathbb{R}$, apart from $1 .$

计算机代写|计算机图形学作业代写computer graphics代考|Odd and Even Functions

An odd function satisfies the condition:
$$f(-x)=-f(x)$$

where $x$ is located in a valid domain. Consequently, the graph of an odd function is symmetrical relative to the origin. For example, $\sin (\theta)$ is odd because
$$\sin (-\theta)=-\sin \theta$$
as illustrated in Fig. 3.6. Other odd functions include:
\begin{aligned} &f(x)=a x \ &f(x)=a x^{3} \end{aligned}
An even function satisfies the condition:
$$f(-x)=f(x)$$
where $x$ is located in a valid domain. Consequently, the graph of an even function is symmetrical relative to the $f(x)$ axis. For example, $\cos \theta$ is even because
$$\cos (-\theta)=\cos \theta$$
as illustrated in Fig. 3.7. Other even functions include:
\begin{aligned} &f(x)=a x^{2} \ &f(x)=a x^{4} \end{aligned}

计算机代写|计算机图形学作业代写computer graphics代考|Power Functions

Functions of the form $f(x)=x^{n}$ are called power functions of degree $n$ and are either odd or even. If $n$ is an odd natural number, then the power function is odd, else if $n$ is an even natural number, then the power function is even.

The above description of algebra should be sufficient for the reader to understand the following chapters. However, one should remember that this is only the beginning of a very complex subject.

计算机代写|计算机图形学作业代写computer graphics代考|Function Domains and Ranges

R. 用于区间的符号也用于域，在这种情况下是
]−∞,∞[

F(−4)=−12+4=−8 F(4)=12+4=16

]0,∞[

]−∞,∞[.

[0,∞[

[0,∞[.

计算机代写|计算机图形学作业代写computer graphics代考|Odd and Even Functions

F(−X)=−F(X)

F(X)=一种X F(X)=一种X3

F(−X)=F(X)

F(X)=一种X2 F(X)=一种X4

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考|Explicit and Implicit Equations

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Explicit and Implicit Equations

The equation
$$y=3 x^{2}+2 x+4$$
associates the value of $y$ with different values of $x$. The directness of the equation: ‘ $y=$ ‘, is why it is called an explicit equation, and their explicit nature is extremely useful. However, simply by rearranging the terms, creates an implicit equation:
$$4=y-3 x^{2}-2 x$$
which implies that certain values of $x$ and $y$ combine to produce the result 4 . Another implicit form is
$$0=y-3 x^{2}-2 x-4$$
which means the same thing, but expresses the relationship in a slightly different way.

An implicit equation can be turned into an explicit equation using algebra. For example, the implicit equation
$$4 x+2 y=12$$
has the explicit form:
$$y=6-2 x$$
where it is clear what $y$ equals.

计算机代写|计算机图形学作业代写computer graphics代考|Function Notation

The explicit equation
$$y=3 x^{2}+2 x+4$$
tells us that the value of $y$ depends on the value of $x$, and not the other way around. For example, when $x=1, y=9$; and when $x=2, y=20$. As $y$ depends upon the value of $x$, it is called the dependent variable; and as $x$ is independent of $y$, it is called the independent variable.
We can also say that $y$ is a function of $x$, which can be written as
$$y=f(x)$$
where the letter ‘ $f$ ‘ is the name of the function, and the independent variable is enclosed in brackets. We could have also written $y=g(x), y=h(x)$, etc.

Eventually, we have to identify the nature of the function, which in this case is
$$f(x)=3 x^{2}+2 x+4$$
Nothing prevents us from writing
$$y=f(x)=3 x^{2}+2 x+4$$
which means: $y$ equals the value of the function $f(x)$, which is determined by the independent variable $x$ using the expression $3 x^{2}+2 x+4$.

An equation may involve more than one independent variable, such as the volume of a cylinder:
$$V=\pi r^{2} h$$
where $r$ is the radius, and $h$, the height, and is written:
$$V(r, h)=\pi r^{2} h .$$

计算机代写|计算机图形学作业代写computer graphics代考|Intervals

An interval is a continuous range of numerical values associated with a variable, which can include or exclude the upper and lower values. For example, a variable such as $x$ is often subject to inequalities like $x \geq a$ and $x \leq b$, which can also be written as
$$a \leq x \leq b$$
and implies that $x$ is located in the closed interval $[a, b]$, where the square brackets indicate that the interval includes $a$ and $b$. For example,
$$1 \leq x \leq 10$$
means that $x$ is located in the closed interval [1, 10], which includes 1 and 10 .
When the boundaries of the interval are not included, then we would state $x>a$ and $x<b$, which is written
$$a<x<b$$
and means that $x$ is located in the open interval ]a, $b[$, where the reverse square brackets indicate that the interval excludes $a$ and $b$. For example,
$$1<x<10$$
means that $x$ is located in the open interval ]1, 10[, which excludes 1 and 10 .

4=是−3X2−2X

0=是−3X2−2X−4

4X+2是=12

F(X)=3X2+2X+4

1≤X≤10

1<X<10

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考|Laws of Indices

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Laws of Indices

The laws of indices are expressed as follows:
\begin{aligned} a^{m} \times a^{n} &=a^{m+n} \ \frac{a^{m}}{a^{n}} &=a^{m-n} \ \left(a^{m}\right)^{n} &=a^{m n} \end{aligned}
and are verified using some simple examples:
\begin{aligned} 2^{3} \times 2^{2} &=2^{5}=32 \ \frac{2^{4}}{2^{2}} &=2^{2}=4 \ \left(2^{2}\right)^{3} &=2^{6}=64 \end{aligned}
From the above laws, it is evident that
\begin{aligned} a^{0} &=1 \ a^{-p} &=\frac{1}{a^{p}} \ a^{\frac{1}{4}} &=\sqrt[q]{a} \ a^{\frac{2}{q}} &=\sqrt[4]{a^{p}} . \end{aligned}

计算机代写|计算机图形学作业代写computer graphics代考|Logarithms

Two people are associated with the invention of logarithms: the Scottish theologian and mathematician John Napier (1550-1617), and the Swiss clockmaker and mathematician Joost Bürgi (1552-1632). Both men were frustrated by the time they spent multiplying numbers together, and both realised that multiplication could be replaced by addition using logarithms. Logarithms exploit the addition and subtraction of indices shown above, and are always associated with a base. For example, if $a^{x}=n$, then $\log _{a} n=x$, where $a$ is the base. Where no base is indicated, it is assumed to be 10 . Two examples bring the idea to life:
$$\begin{array}{rll} 10^{2}=100 & \text { then } & \log 100=2 \ 10^{3}=1000 & \text { then } & \log 1000=3 \end{array}$$
which is interpreted as ‘ 10 has to be raised to the power (index) 2 to equal 100.’ The log operation finds the power of the base for a given number. Thus a multiplication

is translated into an addition using logs. Figure $3.3$ shows the graph of $\log x$, up to $x=100$, where we see that $\log 20 \approx 1.3$ and $\log 50 \approx 1.7$. Therefore, given suitable software, logarithm tables, or a calculator with a log function, we can compute the product $20 \times 50$ as follows:
\begin{aligned} \log (20 \times 50)=\log 20+\log 50 & \approx 1.3+1.7=3 \ 10^{3} &=1000 \end{aligned}
In general, the two bases used in calculators and software are 10 and $e=2.718281846 \ldots$. To distinguish one type of logarithm from the other, a logarithm to the base 10 is written as log, and a natural logarithm to the base $e$ is written $\ln$.
Figure $3.4$ shows the graph of $\ln x$, up to $x=100$, where we see that $\ln 20 \approx 3$ and $\ln 50 \approx 3.9$. Therefore, given suitable software, a set of natural logarithm tables or a calculator with a $\ln$ function, we can compute the product $20 \times 50$ as follows:
\begin{aligned} \ln (20 \times 50)=\ln 20+\ln 50 & \approx 3+3.9=6.9 \ e^{6.9} & \approx 1000 \end{aligned}

计算机代写|计算机图形学作业代写computer graphics代考|Further Notation

All sorts of symbols are used to stand in for natural language expressions; here are some examples:
$<$ less than $>$ greater than
$\leq$ less than or equal to
$\geq$ greater than or equal to
$\approx$ approximately equal to
$\equiv$ equivalent to

• not equal to
$|x|$ absolute value of $x$.
For example, $0 \leq t \leq 1$ is interpreted as: $t$ is greater than or equal to 0 , and is less than or equal to 1 . Basically, this means $t$ varies between 0 and 1 .

计算机代写|计算机图形学作业代写computer graphics代考|Laws of Indices

23×22=25=32 2422=22=4 (22)3=26=64

计算机代写|计算机图形学作业代写computer graphics代考|Logarithms

102=100 然后 日志⁡100=2 103=1000 然后 日志⁡1000=3

ln⁡(20×50)=ln⁡20+ln⁡50≈3+3.9=6.9 和6.9≈1000

计算机代写|计算机图形学作业代写computer graphics代考|Further Notation

<少于>比…更棒
≤小于或等于
≥大于或等于
≈大约等于
≡相当于

• 不等于
|X|的绝对值X.
例如，0≤吨≤1被解释为：吨大于或等于 0 ，并且小于或等于 1 。基本上，这意味着吨在 0 和 1 之间变化。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考|Algebra

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Introduction

Some people, including me, find learning a foreign language a real challenge; one of the reasons being the inconsistent rules associated with its syntax. For example, why is a table feminine in French, ‘la table’, and a bed masculine, ‘le lit’? They both have four legs! The rules governing natural language are continuously being changed by each generation, whereas mathematics appears to be logical and consistent. The reason for this consistency is due to the rules associated with numbers and the way they are combined together and manipulated at an abstract level. Such rules, or axioms, generally make our life easy, however, as we saw with the invention of negative numbers, extra rules have to be introduced, such as ‘two negatives make a positive’, which is easily remembered. However, as we explore mathematics, we discover all sorts of inconsistencies, such as there is no real value associated with the square-root of a negative number. It’s forbidden to divide a number by zero. Zero divided by zero gives inconsistent results. Nevertheless, such conditions are easy to recognise and avoided. At least in mathematics, we don’t have to worry about masculine and feminine numbers!

As a student, I discovered Principia Mathematica [1], a three-volume work written by the British philosopher, logician, mathematician and historian Bertrand Russell (1872-1970), and the British mathematician and philosopher Alfred North Whitehead (1861-1947), in which the authors attempt to deduce all of mathematics using the axiomatic system developed by the Italian mathematician Giuseppe Peano (18581932). The first volume established type theory, the second was devoted to numbers, and the third to higher mathematics. The authors did intend a fourth volume on geometry, but it was too much effort to complete. It made extremely intense reading. In fact, I never managed to get pass the first page! It took the authors almost 100 pages of deep logical analysis in the second volume to prove that $1+1=2$ !

计算机代写|计算机图形学作业代写computer graphics代考|Background

Modern algebraic notation has evolved over thousands of years where different civilisations developed ways of annotating mathematical and logical problems. The word ‘algebra’ comes from the Arabic ‘al-jabr w’al-muqabal’ meaning ‘restoration and reduction’. In retrospect, it does seem strange that centuries passed before the ‘equals’ sign (=) was invented, and concepts such as ‘zero’ (CE 876) were introduced, especially as they now seem so important. But we are not at the end of this evolution, because new forms of annotation and manipulation will continue to emerge as new mathematical objects are invented.

One fundamental concept of algebra is the idea of giving a name to an unknown quantity. For example, $m$ is often used to represent the slope of a $2 \mathrm{D}$ line, and $c$ is the line’s $y$-coordinate where it intersects the $y$-axis. René Descartes formalised the idea of using letters from the beginning of the alphabet $(a, b, c, \ldots$ ) to represent arbitrary quantities, and letters at the end of the alphabet $(p, q, r, s, t, \ldots, x, y, z)$ to represent quantities such as pressure $(p)$, time $(t)$ and coordinates $(x, y, z)$.

With the aid of the basic arithmetic operators: $+,-, \times, /$ we can develop expressions that describe the behaviour of a physical process or a logical computation. For example, the expression $a x+b y-d$ equals zero for a straight line. The variables $x$ and $y$ are the coordinates of any point on the line and the values of $a, b$ and $d$ determine the position and orientation of the line. The $=$ sign permits the line equation to be expressed as a self-evident statement:
$$0=a x+b y-d$$
Such a statement implies that the expressions on the left- and right-hand sides of the = sign are ‘equal’ or ‘balanced’, and in order to maintain equality or balance,

whatever is done to one side, must also be done to the other. For example, adding $d$ to both sides, the straight-line equation becomes
$$d=a x+b y .$$
Similarly, we could double or treble both expressions, divide them by 4 , or add 6 , without disturbing the underlying relationship. When we are first taught algebra, we are often given the task of rearranging a statement to make different variables the subject. For example, $(3.1)$ can be rearranged such that $x$ is the subject:
\begin{aligned} y &=\frac{x+4}{2-\frac{1}{z}} \ y\left(2-\frac{1}{z}\right) &=x+4 \ x &=y\left(2-\frac{1}{z}\right)-4 . \end{aligned}
Making $z$ the subject requires more effort:
\begin{aligned} y &=\frac{x+4}{2-\frac{1}{z}} \ y\left(2-\frac{1}{z}\right) &=x+4 \ 2 y-\frac{y}{z} &=x+4 \ 2 y-x-4 &=\frac{y}{z} \ z &=\frac{y}{2 y-x-4} \end{aligned}
Parentheses are used to isolate part of an expression in order to select a subexpression that is manipulated in a particular way. For example, the parentheses in $c(a+b)+d$ ensure that the variables $a$ and $b$ are added together before being multiplied by $c$, and finally added to $d$.

计算机代写|计算机图形学作业代写computer graphics代考|Solving the Roots of a Quadratic Equation

Problem solving is greatly simplified if one has solved it before, and having a good memory is always an advantage. In mathematics, we keep coming across problems that have been encountered before, apart from different numbers. For example,

$(a+b)(a-b)$ always equals $a^{2}-b^{2}$, therefore factorising the following is a trivial exercise:
\begin{aligned} a^{2}-16 &=(a+4)(a-4) \ x^{2}-49 &=(x+7)(x-7) \ x^{2}-2 &=(x+\sqrt{2})(x-\sqrt{2}) . \end{aligned}
A perfect square has the form:
$$a^{2}+2 a b+b^{2}=(a+b)^{2}$$
Consequently, factorising the following is also a trivial exercise:
\begin{aligned} a^{2}+4 a b+4 b^{2} &=(a+2 b)^{2} \ x^{2}+14 x+49 &=(x+7)^{2} \ x^{2}-20 x+100 &=(x-10)^{2} \end{aligned}
Now let’s solve the roots of the quadratic equation $a x^{2}+b x+c=0$, i.e. those values of $x$ that make the equation equal zero. As the equation involves an $x^{2}$ term, we will exploit any opportunity to factorise it. We begin with the quadratic where $a \neq 0$ :
$$a x^{2}+b x+c=0 .$$
Step 1: Subtract $c$ from both sides to begin the process of creating a perfect square:
$$a x^{2}+b x=-c$$
Step 2: Divide both sides by $a$ to create an $x^{2}$ term:
$$x^{2}+\frac{b}{a} x=-\frac{c}{a} .$$
Step 3: Add $b^{2} / 4 a^{2}$ to both sides to create a perfect square on the left side:
$$x^{2}+\frac{b}{a} x+\frac{b^{2}}{4 a^{2}}=\frac{b^{2}}{4 a^{2}}-\frac{c}{a}$$
Step 4: Factorise the left side:
$$\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}}{4 a^{2}}-\frac{c}{a}$$

0=一种X+b是−d

d=一种X+b是.

计算机代写|计算机图形学作业代写computer graphics代考|Solving the Roots of a Quadratic Equation

(一种+b)(一种−b)总是等于一种2−b2，因此分解以下是一个简单的练习：

X2+b一种X=−C一种.

X2+b一种X+b24一种2=b24一种2−C一种

(X+b2一种)2=b24一种2−C一种

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考|Imaginary Numbers

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Imaginary Numbers

Imaginary numbers were invented to resolve problems where an equation such as $x^{2}+16=0$, has no real solution (roots). The simple idea of declaring the existence of a quantity $i$, such that $i^{2}=-1$, permits the solution to be expressed as
$$x=\pm 4 i$$
For example, if $x=4 i$ we have
\begin{aligned} x^{2}+16 &=16 i^{2}+16 \ &=-16+16 \ &=0 \end{aligned}
and if $x=-4 i$ we have
\begin{aligned} x^{2}+16 &=16 i^{2}+16 \ &=-16+16 \ &=0 \end{aligned}
But what is $i$ ? In 1637 , the French mathematician René Descartes (1596-1650), published La Géométrie, in which he stated that numbers incorporating $\sqrt{-1}$ were ‘imaginary’, and for centuries this label has stuck. Unfortunately, it was a derogatory remark, as there is nothing ‘imaginary’ about $i$-it simply is an object that when introduced into various algebraic expressions, reveals some amazing underlying patterns. $i$ is not a number in the accepted sense, it is a mathematical object or construct that squares to $-1$. In some respects it is like time, which probably does not really exist, but is useful in describing the universe. However, $i$ does lose its mystery when interpreted as a rotational operator, which we investigate below.
As $i^{2}=-1$ then it must be possible to raise $i$ to other powers. For example,
$$i^{4}=i^{2} i^{2}=1$$
and
$$i^{5}=i i^{4}=i$$
Table $2.6$ shows the sequence up to $i^{6}$.

计算机代写|计算机图形学作业代写computer graphics代考|Complex Numbers

A complex number has a real and imaginary part: $z=a+i b$, and represented by the set $\mathbb{C}$ :
$$z=a+b i \quad z \in \mathbb{C}, \quad a, b \in \mathbb{R}, \quad i^{2}=-1$$
Some examples are
\begin{aligned} &z=1+i \ &z=3-2 i \ &z=-23+\sqrt{23} i \end{aligned}
Complex numbers obey all the normal laws of algebra. For example, if we multiply $(a+b i)$ by $(c+d i)$ we have
$$(a+b i)(c+d i)=a c+a d i+b c i+b d i^{2}$$
Collecting up like terms and substituting $-1$ for $i^{2}$ we get
$$(a+b i)(c+d i)=a c+(a d+b c) i-b d$$
which simplifies to

$$(a+b i)(c+d i)=a c-b d+(a d+b c) i$$
which is another complex number.
Something interesting happens when we multiply a complex number by its complex conjugate, which is the same complex number but with the sign of the imaginary part reversed:
$$(a+b i)(a-b i)=a^{2}-a b i+b a i-b^{2} i^{2} .$$
Collecting up like terms and simplifying we obtain
$$(a+b i)(a-b i)=a^{2}+b^{2}$$
which is a real number, as the imaginary part has been cancelled out by the action of the complex conjugate.

计算机代写|计算机图形学作业代写computer graphics代考|Infinity

The term infinity is used to describe the size of unbounded systems. For example, there is no end to prime numbers: i.e. they are infinite; so, too, are the sets of other numbers. Consequently, no matter how we try, it is impossible to visualise the size of infinity. Nevertheless, this did not stop Georg Cantor from showing that one infinite set could be infinitely larger than another.

Cantor distinguished between those infinite number sets that could be ‘counted’, and those that could not. For Cantor, counting meant the one-to-one correspondence of a natural number with the members of another infinite set. If there is a clear correspondence, without leaving any gaps, then the two sets shared a common infinite size, called its cardinality using the first letter of the Hebrew alphabet aleph: $\aleph$. The cardinality of the natural numbers $\mathbb{N}$ is $\aleph_{0}$, called aleph-zero.

Cantor discovered a way of representing the rational numbers as a grid, which is traversed diagonally, back and forth, as shown in Fig. 2.5. Some ratios appear several times, such as $\frac{2}{2}, \frac{3}{3}$ etc., which are not counted. Nevertheless, the one-toone correspondence with the natural numbers means that the cardinality of rational numbers is also $\aleph_{0}$.

A real surprise was that there are infinitely more transcendental numbers than natural numbers. Furthermore, there are an infinite number of cardinalities rising to $\aleph_{\aleph}$. Cantor had been alone working in this esoteric area, and as he published his results, he shook the very foundations of mathematics, which is why he was treated so badly by his fellow mathematicians.

计算机代写|计算机图形学作业代写computer graphics代考|Imaginary Numbers

X=±4一世

X2+16=16一世2+16 =−16+16 =0

X2+16=16一世2+16 =−16+16 =0

计算机代写|计算机图形学作业代写computer graphics代考|Complex Numbers

(一种+b一世)(C+d一世)=一种C+一种d一世+bC一世+bd一世2

(一种+b一世)(C+d一世)=一种C+(一种d+bC)一世−bd

(一种+b一世)(一种−b一世)=一种2−一种b一世+b一种一世−b2一世2.

(一种+b一世)(一种−b一世)=一种2+b2

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

The hexadecimal number system has $B=16$, and $a$ to $h$ can be 0 to 15 , which presents a slight problem, as we don’t have 15 different numerical characters. Consequently, we use 0 to 9 , and the letters $A, B, C, D, E, F$ to represent $10,11,12,13,14,15$ respectively:

$$\ldots a 16^{3}+b 16^{2}+c 16^{1}+d 16^{0}+e 16^{-1}+f 16^{-2}+g 16^{-3}+h 16^{-4} \ldots$$
and the first 17 hexadecimal numbers are:
$$1_{16}, 2_{16}, 3_{16}, 4_{16}, 5_{16}, 6_{16}, 7_{16}, 8_{16}, 9_{16}, A_{16}, B_{16}, C_{16}, D_{16}, E_{16}, F_{16}, 10_{16}, 11_{16}$$
Thus $1 E .8_{16}$ is converted to decimal as follows:
$$\begin{gathered} (1 \times 16)+(E \times 1)+\left(8 \times 16^{-1}\right) \ (16+14)+(8 / 16) \ 30.5 . \end{gathered}$$
Although it is not obvious, binary, octal and hexadecimal numbers are closely related, which is why they are part of a programmer’s toolkit. Even though computers work with binary, it’s the last thing a programmer wants to use. So to simplify the manmachine interface, binary is converted into octal or hexadecimal. To illustrate this, let’s convert the 16-bit binary code 1101011000110001 into octal.
Using the following general binary integer
$$a 2^{8}+b 2^{7}+c 2^{6}+d 2^{5}+e 2^{4}+f 2^{3}+g 2^{2}+h 2^{1}+i 2^{0}$$
we group the terms into threes, starting from the right, because $2^{3}=8$ :
$$\left(a 2^{8}+b 2^{7}+c 2^{6}\right)+\left(d 2^{5}+e 2^{4}+f 2^{3}\right)+\left(g 2^{2}+h 2^{1}+i 2^{0}\right)$$
Simplifying:
$$\begin{gathered} 2^{6}\left(a 2^{2}+b 2^{1}+c 2^{0}\right)+2^{3}\left(d 2^{2}+e 2^{1}+f 2^{0}\right)+2^{0}\left(g 2^{2}+h 2^{1}+i 2^{0}\right) \ 8^{2}\left(a 2^{2}+b 2^{1}+c 2^{1}\right)+8^{1}\left(d 2^{2}+e 2^{1}+f 2^{0}\right)+8^{0}\left(g 2^{2}+h 2^{1}+i 2^{0}\right) \ 8^{2} R+8^{1} S+8^{0} T \end{gathered}$$
where
\begin{aligned} &R=a 2^{2}+b 2^{1}+c \ &S=d 2^{2}+e 2^{1}+f \ &T=g 2^{2}+h 2^{1}+i \end{aligned}
and the values of $R, S, T$ vary between 0 and 7 . Therefore, given 1101011000 110001 , we divide the binary code into groups of three, starting at the right, and adding two leading zeros.

计算机代写|计算机图形学作业代写computer graphics代考|Subtracting Binary Numbers

Two’s complement is a technique for converting a binary number into a form such that when it is added to another binary number, it results in a subtraction. There are two stages to the conversion: inversion, followed by the addition of 1 . For example, 24 in binary is 0000000000110000 , and is inverted by switching every 1 to 0 , and vice versa: 1111111111100111 . Next, we add 1: 1111111111101000, which now represents $-24$. If this is added to binary $36: 0000000000100100$, we have
\begin{tabular}{l}
$0000000000100100=+36$ \
$1111111111101000=-24$ \
\hline $0000000000001100=+12$ \
\hline
\end{tabular}
Note that the last high-order addition creates a carry of 1, which is ignored. Here is another example, $100-30$ :
\begin{tabular}{rll}
& 0000000000011110 & $=+30$ \
inversion & 111111111100001 & \
add 1 & 0000000000000001 & \
\hline & 1111111111100010 & $=-30$ \
add 100 & 0000000001100100 & $=+100$ \
\hline & 0000000001000110 & $=+70$ \
\hline
\end{tabular}

计算机代写|计算机图形学作业代写computer graphics代考|Algebraic and Transcendental Numbers

Polynomial equations with rational coefficients have the form:
$$f(x)=a x^{n}+b x^{n-1}+c x^{n-2} \ldots+C$$
such as
$$y=3 x^{2}+2 x-1$$
and their roots belong to the set of algebraic numbers $A$. A consequence of this definition implies that all rational numbers are algebraic, since if
$$x=\frac{p}{q}$$
then
$$q x-p=0$$
which is a polynomial. Numbers that are not roots to polynomial equations are transcendental numbers and include most irrational numbers, but not $\sqrt{2}$, since if
$$x=\sqrt{2}$$
then

$$x^{2}-2=0$$
which is a polynomial.

计算机图形学代写

116,216,316,416,516,616,716,816,916,一种16,乙16,C16,D16,和16,F16,1016,1116

(1×16)+(和×1)+(8×16−1) (16+14)+(8/16) 30.5.

(一种28+b27+C26)+(d25+和24+F23)+(G22+H21+一世20)

26(一种22+b21+C20)+23(d22+和21+F20)+20(G22+H21+一世20) 82(一种22+b21+C21)+81(d22+和21+F20)+80(G22+H21+一世20) 82R+81小号+80吨

R=一种22+b21+C 小号=d22+和21+F 吨=G22+H21+一世

计算机代写|计算机图形学作业代写computer graphics代考|Subtracting Binary Numbers

\begin{表格}{l} $0000000000100100=+36$ \ $1111111111101000=-24$ \ \hline $0000000000001100=+12$ \ \hline \end{表格}\begin{表格}{l} $0000000000100100=+36$ \ $1111111111101000=-24$ \ \hline $0000000000001100=+12$ \ \hline \end{表格}

\begin{tabular}{rll} & 0000000000011110 & $=+30$ \ 反转 & 111111111100001 & \ 添加 1 & 0000000000000001 & \ \hline & 111111111100010 & $=-30$ \ 添加 100 & 00000=0+1001 $\hline & 0000000001000110 &$=+70$\ \hline \end{表格}\begin{tabular}{rll} & 0000000000011110 &$=+30$\ 反转 & 111111111100001 & \ 添加 1 & 0000000000000001 & \ \hline & 111111111100010 &$=-30$\ 添加 100 & 00000=0+1001$ \hline & 0000000001000110 & $=+70$ \ \hline \end{表格}

计算机代写|计算机图形学作业代写computer graphics代考|Algebraic and Transcendental Numbers

F(X)=一种Xn+bXn−1+CXn−2…+C

X=pq

qX−p=0

X=2

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考| The Base of a Number System

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Background

Over recent millennia, mankind has invented and discarded many systems for representing number. People have counted on their fingers and toes, used pictures (hieroglyphics), cut marks on clay tablets (cuneiform symbols), employed Greek symbols (Ionic system) and struggled with, and abandoned Roman numerals (I, V, X, L, C, D, M, etc.), until we reach today’s decimal place system, which has Hindu-Arabic and Chinese origins. And since the invention of computers we have witnessed the emergence of binary, octal and hexadecimal number systems, where 2,8 and 16 respectively, replace the 10 in our decimal system.

The decimal number 23 stands for ‘two tens and three units’, and in English is written ‘twenty-three’, in French ‘vingt-trois’ (twenty-three), and in German ‘dreiundzwanzig’ (three and twenty). Let’s investigate the algebra behind the decimal system and see how it can be used to represent numbers to any base. The expression:
$$a \times 1000+b \times 100+c \times 10+d \times 1$$
where $a, b, c, d$ take on any value between 0 and 9 , describes any whole number between 0 and 9999 . By including
$$e \times 0.1+f \times 0.01+g \times 0.001+h \times 0.0001$$
where $e, f, g, h$ take on any value between 0 and 9 , any decimal number between 0 and $9999.9999$ can be represented.
Indices bring the notation alive and reveal the true underlying pattern:
$$\ldots a 10^{3}+b 10^{2}+c 10^{1}+d 10^{0}+e 10^{-1}+f 10^{-2}+g 10^{-3}+h 10^{-4} \ldots$$
Remember that any number raised to the power 0 equals 1 . By adding extra terms both left and right, any number can be accommodated.

In this example, 10 is the base, which means that the values of $a$ to $h$ range between 0 and 9,1 less than the base. Therefore, by substituting $B$ for the base we have
$$\ldots a B^{3}+b B^{2}+c B^{1}+d B^{0}+e B^{-1}+f B^{-2}+g B^{-3}+h B^{-4} \ldots$$
where the values of $a$ to $h$ range between 0 and $B-1$.

计算机代写|计算机图形学作业代写computer graphics代考|Octal Numbers

The octal number system has $B=8$, and $a$ to $h$ range between 0 and 7 :
$$\ldots a 8^{3}+b 8^{2}+c 8^{1}+d 8^{0}+e 8^{-1}+f 8^{-2}+g 8^{-3}+h 8^{-4} \ldots$$
and the first 17 octal numbers are:
$$1_{8}, 2_{8}, 3_{8}, 4_{8}, 5_{8}, 6_{8}, 7_{8}, 10_{8}, 11_{8}, 12_{8}, 13_{8}, 14_{8}, 15_{8}, 16_{8}, 17_{8}, 20_{8}, 21_{8}$$
The subscript 8 reminds us that although we may continue to use the words ‘twentyone’, it is an octal number, and not a decimal. But what is $14_{\mathrm{g}}$ in decimal? Well, it stands for:
$$1 \times 8^{1}+4 \times 8^{0}=12$$
Thus $356.4_{g}$ is converted to decimal as follows:

$$\begin{gathered} \left(3 \times 8^{2}\right)+\left(5 \times 8^{1}\right)+\left(6 \times 8^{0}\right)+\left(4 \times 8^{-1}\right) \ (3 \times 64)+(5 \times 8)+(6 \times 1)+(4 \times 0.125) \ (192+40+6)+(0.5) \ 238.5 . \end{gathered}$$
Counting in octal appears difficult, simply because we have never been exposed to it, like the decimal system. If we had evolved with 8 fingers, instead of 10 , we would be counting in octal!

计算机代写|计算机图形学作业代写computer graphics代考|Binary Numbers

The binary number system has $B=2$, and $a$ to $h$ are 0 or 1 :
$$\ldots a 2^{3}+b 2^{2}+c 2^{1}+d 2^{0}+e 2^{-1}+f 2^{-2}+g 2^{-3}+h 2^{-4} \ldots$$
and the first 13 binary numbers are:
$$1_{2}, 10_{2}, 11_{2}, 100_{2}, 101_{2}, 110_{2}, 111_{2}, 1000_{2}, 1001_{2}, 1010_{2}, 1011_{2}, 1100_{2}, 1101_{2}$$
Thus $11011.11_{2}$ is converted to decimal as follows:
$$\begin{gathered} \left(1 \times 2^{4}\right)+\left(1 \times 2^{3}\right)+\left(0 \times 2^{2}\right)+\left(1 \times 2^{1}\right)+\left(1 \times 2^{0}\right)+\left(1 \times 2^{-1}\right)+\left(1 \times 2^{-2}\right) \ (1 \times 16)+(1 \times 8)+(0 \times 4)+(1 \times 2)+(1 \times 0.5)+(1 \times 0.25) \ (16+8+2)+(0.5+0.25) \ 26.75 . \end{gathered}$$
The reason why computers work with binary numbers-rather than decimal-is due to the difficulty of designing electrical circuits that can store decimal numbers in a stable fashion. A switch, where the open state represents 0 , and the closed state represents 1 , is the simplest electrical component to emulate. No matter how often it is used, or how old it becomes, it will always behave like a switch. The main advantage of electrical circuits is that they can be switched on and off trillions of times a second, and the only disadvantage is that the encoded binary numbers and characters contain a large number of bits, and humans are not familiar with binary.

计算机代写|计算机图形学作业代写computer graphics代考|Background

…一种103+b102+C101+d100+和10−1+F10−2+G10−3+H10−4…

…一种乙3+b乙2+C乙1+d乙0+和乙−1+F乙−2+G乙−3+H乙−4…

计算机代写|计算机图形学作业代写computer graphics代考|Octal Numbers

…一种83+b82+C81+d80+和8−1+F8−2+G8−3+H8−4…

18,28,38,48,58,68,78,108,118,128,138,148,158,168,178,208,218

1×81+4×80=12

计算机代写|计算机图形学作业代写computer graphics代考|Binary Numbers

…一种23+b22+C21+d20+和2−1+F2−2+G2−3+H2−4…

12,102,112,1002,1012,1102,1112,10002,10012,10102,10112,11002,11012

(1×24)+(1×23)+(0×22)+(1×21)+(1×20)+(1×2−1)+(1×2−2) (1×16)+(1×8)+(0×4)+(1×2)+(1×0.5)+(1×0.25) (16+8+2)+(0.5+0.25) 26.75.

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考| Negative Numbers

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Zero

The concept of zero has a well-documented history, which shows that it has been used by different cultures over a period of two-thousand years or more. It was the Indian mathematician and astronomer Brahmagupta $(598-\mathrm{c} .-670)$, who argued that zero was just as valid as any natural number, with the definition: the result of subtracting any number from itself. However, even today, there is no universal agreement as to whether zero belongs to the set $\mathbb{N}$, consequently, the set $\mathbb{N}^{0}$ stands for the set of natural numbers including zero.

In today’s positional decimal system, which is a place value system, the digit 0 is a placeholder. For example, 203 stands for: two hundreds, no tens and three units. Although $0 \in \mathbb{N}^{0}$, it does have special properties that distinguish it from other members of the set, and Brahmagupta also gave rules showing this interaction.
If $x \in \mathbb{N}^{0}$, then the following rules apply:
The expression $0 / 0$ is called an indeterminate form, as it is possible to show that under different conditions, especially limiting conditions, it can equal anything. So for the moment, we will avoid using it until we cover calculus.

计算机代写|计算机图形学作业代写computer graphics代考|Negative Numbers

When negative numbers were first proposed, they were not accepted with open arms, as it was difficult to visualise $-5$ of something. For instance, if there are 5 donkeys in a field, and they are all stolen to make salami, the field is now empty, and there is nothing we can do in the arithmetic of donkeys to create a field of $-5$ donkeys. However, in applied mathematics, numbers have to represent all sorts of quantities such as temperature, displacement, angular rotation, speed, acceleration, etc., and we also need to incorporate ideas such as left and right, up and down, before and after, forwards and backwards, etc. Fortunately, negative numbers are perfect for representing all of the above quantities and ideas.

Consider the expression $4-x$, where $x \in \mathbb{N}^{0}$. When $x$ takes on certain values, we have
\begin{aligned} &4-1=3 \ &4-2=2 \ &4-3=1 \ &4-4=0 \end{aligned}
and unless we introduce negative numbers, we are unable to express the result of $4-5$. Consequently, negative numbers are visualised as shown in Fig. $2.1$, where the number line shows negative numbers to the left of the natural numbers, which are positive, although the $+$ sign is omitted for clarity.

Moving from left to right, the number line provides a numerical continuum from large negative numbers, through zero, towards large positive numbers. In any

calculations, we could agree that angles above the horizon are positive, and angles below the horizon, negative. Similarly, a movement forwards is positive, and a movement backwards is negative. So now we are able to write:
\begin{aligned} &4-5=-1 \ &4-6=-2 \ &4-7=-3 \end{aligned}
etc.,
without worrying about creating impossible conditions.

计算机代写|计算机图形学作业代写computer graphics代考|The Arithmetic of Positive and Negative Numbers

Once again, Brahmagupta compiled all the rules, Tables $2.1$ and 2.2, supporting the addition, subtraction, multiplication and division of positive and negative numbers. The real fly in the ointment, being negative numbers, which cause problems for children, math teachers and occasional accidents for mathematicians. Perhaps, the one rule we all remember from our school days is that two negatives make a positive.
Another problem with negative numbers arises when we employ the square-root function. As the product of two positive or negative numbers results in a positive result, the square-root of a positive number gives rise to a positive and a negative answer. For example, $\sqrt{4}=\pm 2$. This means that the square-root function only applies to positive numbers. Nevertheless, it did not stop the invention of the imaginary object $i$, where $i^{2}=-1$. However, $i$ is not a number, but behaves like an operator, and is described later.

计算机代写|计算机图形学作业代写computer graphics代考|Negative Numbers

4−1=3 4−2=2 4−3=1 4−4=0

4−5=−1 4−6=−2 4−7=−3

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

计算机代写|计算机图形学作业代写computer graphics代考|Numbers

statistics-lab™ 为您的留学生涯保驾护航 在代写计算机图形学computer graphics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算机图形学computer graphics代写方面经验极为丰富，各种代写计算机图形学computer graphics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

计算机代写|计算机图形学作业代写computer graphics代考|Background

Over the centuries mathematicians have realised that in order to progress, they must give precise definitions to their discoveries, ideas and concepts, so that they can be built upon and referenced by new mathematical inventions. In the event of any new discovery, these rrrdefinitions have to be occasionally changed or extended. For example, once upon a time integers, rational and irrational numbers, satisfied all the needs of mathematicians, until imaginary quantities were invented. Today, complex numbers have helped shape the current number system hierarchy. Consequently, there must be clear definitions for numbers, and the operators that act upon them. Therefore, we need to identify the types of numbers that exist, what they are used for, and any problems that arise when they are stored in a computer.

计算机代写|计算机图形学作业代写computer graphics代考|Counting

Our brain’s visual cortex possesses some incredible image processing features. For example, children know instinctively when they are given less sweets than another child, and adults know instinctively when they are short-changed by a Parisian taxi driver, or driven around the Arc de Triumph several times, on the way to the airport! Intuitively, we can assess how many donkeys are in a field without counting them,

and generally, we seem to know within a second or two, whether there are just a few, dozens, or hundreds of something. But when accuracy is required, one can’t beat counting. But what is counting?

Well normally, we are taught to count by our parents by memorising first, the counting words ‘one, two, three, four, five, six, seven, eight, nine, ten, ..’ and second, associating them with our fingers, so that when asked to count the number of donkeys in a picture book, each donkey is associated with a counting word. When each donkey has been identified, the number of donkeys equals the last word mentioned. However, this still assumes that we know the meaning of ‘one, two, three, four,..’ etc. Memorising these counting words is only part of the problem-getting them in the correct sequence is the real challenge. The incorrect sequence ‘one, two, five, three, nine, four, ..’ etc., introduces an element of randomness into any calculation, but practice makes perfect, and it’s useful to master the correct sequence before going to university!

计算机代写|计算机图形学作业代写computer graphics代考|Sets of Numbers

A set is a collection of arbitrary objects called its elements or members. For example, each system of number belongs to a set with given a name, such as $\mathbb{N}$ for the natural numbers, $\mathbb{R}$ for real numbers, and $\mathbb{Q}$ for rational numbers. When we want to indicate that something is whole, real or rational, etc., we use the notation:
$$n \in \mathbb{N}$$
$$x \in \mathbb{R}$$
stands for ‘ $x$ is a real number.’
A well-ordered set possesses a unique order, such as the natural numbers $\mathbb{N}$. Therefore, if $P$ is the well-ordered set of prime numbers and $\mathbb{N}$ is the well-ordered set of natural numbers, we can write:
\begin{aligned} &P={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, \ldots} \ &\mathbb{N}={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, \ldots} \end{aligned}
By pairing the prime numbers in $P$ with the numbers in $\mathbb{N}$, we have:
$${{2,1},{3,2},{5,3},{7,4},{11,5},{13,6},{17,7},{19,8},{23,9}, \ldots}$$
and we can reason that 2 is the lst prime, and 3 is the 2 nd prime, etc. However, we still have to declare what we mean by $1,2,3,4,5, \ldots$ etc., and without getting too philosophical, I like the idea of defining them as follows. The word ‘one’, represented

by 1, stands for ‘oneness’ of anything: one finger, one house, one tree, one donkey, etc. The word ‘two’, represented by 2 , is ‘one more than one’. The word ‘three’, represented by 3 , is ‘one more than two’, and so on.

We are now in a position to associate some mathematical notation with our numbers by introducing the $+$ and $=$ signs. We know that $+$ means add, but it also can stand for ‘more’. We also know that = means equal, and it can also stand for ‘is the same as’. Thus the statement:
$$2=1+1$$
is read as ‘two is the same as one more than one.’
We can also write:
$$3=1+2$$
which is read as ‘three is the same as one more than two.’ But as we already have a definition for 2 , we can write
\begin{aligned} 3 &=1+2 \ &=1+1+1 \end{aligned}
Developing this idea, and including some extra combinations, we have:
\begin{aligned} &2=1+1 \ &3=1+2 \ &4=1+3=2+2 \ &5=1+4=2+3 \ &6=1+5=2+4=3+3 \ &7=1+6=2+5=3+4 \end{aligned}
etc.
and can be continued without limit. These numbers, $1,2,3,4,5,6$, etc., are called natural numbers, and are the set $\mathbb{N}$.

计算机代写|计算机图形学作业代写computer graphics代考|Sets of Numbers

n∈ñ
X∈R

2,1,3,2,5,3,7,4,11,5,13,6,17,7,19,8,23,9,…

2=1+1

3=1+2

3=1+2 =1+1+1

2=1+1 3=1+2 4=1+3=2+2 5=1+4=2+3 6=1+5=2+4=3+3 7=1+6=2+5=3+4

，并且可以无限制地继续。这些数字，1,2,3,4,5,6等，称为自然数，是集合ñ.

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。