## 计算机代写|计算机图形学代写computer graphics代考|CS559

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|计算机图形学代写computer graphics代考|The Base of a Number System

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 hetween 0 and 9 , any decimal number hetween 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:
$$1_2, 10_2, 11_2, 100_2, 101_2, 110_2, 111_2, 1000_2, 1001_2, 1010_2, 1011_2, 1100_2, 1101_2 \text {. }$$
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|计算机图形学代写computer graphics代考|The Base of a Number System

$$a \times 1000+b \times 100+c \times 10+d \times 1$$

$$e \times 0.1+f \times 0.01+g \times 0.001+h \times 0.0001$$

$$\ldots a 10^3+b 10^2+c 10^1+d 10^0+e 10^{-1}+f 10^{-2}+g 10^{-3}+h 10^{-4} \ldots$$

$$\ldots a B^3+b B^2+c B^1+d B^0+e B^{-1}+f B^{-2}+g B^{-3}+h B^{-4} \ldots$$

## 计算机代写|计算机图形学代写computer graphics代考|Binary Numbers

$$\ldots a 2^3+b 2^2+c 2^1+d 2^0+e 2^{-1}+f 2^{-2}+g 2^{-3}+h 2^{-4} \ldots$$

$$1_2, 10_2, 11_2, 100_2, 101_2, 110_2, 111_2, 1000_2, 1001_2, 1010_2, 1011_2, 1100_2, 1101_2 .$$

$$\left(1 \times 2^4\right)+\left(1 \times 2^3\right)+\left(0 \times 2^2\right)+\left(1 \times 2^1\right)+\left(1 \times 2^0\right)+\left(1 \times 2^{-1}\right)+\left(1 \times 2^{-2}\right)(1 \times 16)$$

## 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|计算机图形学代写computer graphics代考|COS426

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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:
$$\begin{gathered} 4-5=-1 \ 4-6=-2 \ 4-7=-3 \ \text { etc., } \end{gathered}$$
without worrying about creating impossible conditions.

## 计算机代写|计算机图形学代写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.

The commutative law in algebra states that when two elements are linked through some binary operation, the result is independent of the order of the elements.
The commutative law of addition is
\begin{aligned} a+b & =b+a \ \text { e.g. } 1+2 & =2+1 \end{aligned}
The commutative law of multiplication is
$$\begin{array}{r} a \times b=b \times a \ \text { e.g. } 1 \times 2=2 \times 1 . \end{array}$$
Note that subtraction is not commutative:
$$\begin{array}{r} a-b \neq b-a \ \text { e.g. } 1-2 \neq 2-1 . \end{array}$$

## 计算机代写|计算机图形学代写computer graphics代考|Negative Numbers

$$4-1=3 \quad 4-2=24-3=1 \quad 4-4=0$$

$$4-5=-14-6=-24-7=-3 \text { etc., }$$

## 计算机代写|计算机图形学代写computer graphics代考|The Arithmetic of Positive and Negative Numbers

Brahmagupta 再次编制了所有规则，表格 $2.1$ 和 $2.2$ ，支持正负数的加减乘除。真正美中不足的是负数， 这给孩子们、数学老师带来了麻烦，也给数学家们带来了偶尔的事故。也许，我们在学生时代都记得的一 条规则是，两个负面因素构成一个积极因素。

$$a+b=b+a \text { e.g. } 1+2=2+1$$

$$a \times b=b \times a \text { e.g. } 1 \times 2=2 \times 1 \text {. }$$

$$a-b \neq b-a \text { e.g. } 1-2 \neq 2-1 \text {. }$$

## 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|计算机图形学代写computer graphics代考|CS4600

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|计算机图形学代写computer graphics代考|Symbols and Notation

One of the reasons why many people find mathematics inaccessible is due to its symbols and notation. Let’s look at symbols first. The English alphabet possesses a reasonable range of familiar character shapes:
$$\begin{gathered} \text { a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z } \ \text { A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z } \end{gathered}$$
which find their way into every branch of mathematics and physics, and permit us to write equations such as
$$E=m c^2$$
and
$$A=\pi r^2 .$$
It is important that when we see an equation, we are able to read it as part of the text. In the case of $E=m c^2$, this is read as ‘ $E$ equals $m, c$ squared’, where $E$ stands for energy, $m$ for mass, $c$ the speed of light, which is multiplied by itself. In the case of $A=\pi r^2$, this is read as ‘ $A$ equals pi, $r$ squared’, where $A$ stands for area, $\pi$ the ratio of a circle’s circumference to its diameter, and $r$ the circle’s radius. Greek symbols, which happen to look nice and impressive, have also found their way into many equations, and often disrupt the flow of reading, simply because we don’t know their English names. For example, the English theoretical physicist Paul Dirac (1902-1984), derived an equation for a moving electron using the symbols $\alpha_i$ and $\beta$, which are $4 \times 4$ matrices, where
$$\alpha_i \beta+\beta \alpha_i=0$$
and is read as
‘the sum of the products alpha- $i$ beta, and beta alpha- $i$, equals zero.’
Although we do not come across moving electrons in this book, we do have to be familiar with the following Greek symbols:
$$\begin{array}{llll} \alpha & \text { alpha } & \nu & \mathrm{nu} \ \beta & \text { hera } & \xi & \mathrm{xi} \end{array}$$

## 计算机代写|计算机图形学代写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:
$$n \in \mathbb{N}$$
which reads ‘ $n$ is a member of $(\in)$ the set $\mathbb{N}$ ‘, i.e. $n$ is a whole number. Similarly:
$$x \in \mathbb{R}$$
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 1st 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.

## 计算机代写|计算机图形学代写computer graphics代考|Symbols and Notation

a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,

$$E=m c^2$$

$$A=\pi r^2 .$$

$$\alpha_i \beta+\beta \alpha_i=0$$

“乘积 alpha- 的总和” $i$ beta 和 beta alpha- $i$ ，等于零。

$\alpha$ alpha $\nu$ nu $\beta$ hera $\xi$ xi

## 计算机代写|计算机图形学代写computer graphics代考|Sets of Numbers

$$n \in \mathbb{N}$$

$$x \in \mathbb{R}$$

$$P=2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, \ldots \quad \mathbb{N}=1,2,3,4,5,6,7,8,9,10,11$$

$$2,1,3,2,5,3,7,4,11,5,13,6,17,7,19,8,23,9, \ldots$$

## 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|计算机图形学代写computer graphics代考|CS559

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|计算机图形学代写computer graphics代考|Associative Law

The associative law in algebra states that when three or more elements are linked together through a binary operation, the result is independent of how each pair of elements is grouped.
The associative law of addition is
\begin{aligned} a+(b+c) &=(a+b)+c \ \text { e.g. } 1+(2+3) &=(1+2)+3 . \end{aligned}
The associative law of multiplication is
$$a \times(b \times c)=(a \times b) \times c$$ $$\text { e.g. } 1 \times(2 \times 3)=(1 \times 2) \times 3 \text {. }$$
However, note that subtraction is not associative:
$$\begin{array}{r} a-(b-c) \neq(a-b)-c \ \text { e.g. } 1-(2-3) \neq(1-2)-3, \end{array}$$
which may seem surprising, but at the same time confirms the need for clear axioms.

## 计算机代写|计算机图形学代写computer graphics代考|The Base of a Number System

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:
$$\ldots a 10^3+b 10^2+c 10^1+d 10^0+e 10^{-1}+f 10^{-2}+g 10^{-3}+h 10^{-4} \ldots \ldots$$
Remember that any number raised to the power 0 equals 1 . By adding extra terms both left annd right, any number cān bé accoommodatêd.

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
$$\ldots a B^3+b B^2+c B^1+d B^0+c B^{-1}+f B^{-2}+g B^{-3}+h B^{-4} \ldots$$
where the values of $a$ to $h$ range between 0 and $B-1$.

## 计算机代写|计算机图形学代写computer graphics代考|Associative Law

$$a+(b+c)=(a+b)+c \text { e.g. } 1+(2+3) \quad=(1+2)+3 .$$

$$\begin{gathered} a \times(b \times c)=(a \times b) \times c \ \text { e.g. } 1 \times(2 \times 3)=(1 \times 2) \times 3 . \end{gathered}$$

$$a-(b-c) \neq(a-b)-c \text { e.g. } 1-(2-3) \neq(1-2)-3,$$

## 计算机代写|计算机图形学代写computer graphics代考|The Base of a Number System

$$a \times 1000+b \times 100+c \times 10+d \times 1$$

$$e \times 0.1+f \times 0.01+g \times 0.001+h \times 0.0001$$

$$\ldots a 10^3+b 10^2+c 10^1+d 10^0+e 10^{-1}+f 10^{-2}+g 10^{-3}+h 10^{-4} \ldots \ldots$$

$$\ldots a B^3+b B^2+c B^1+d B^0+c B^{-1}+f B^{-2}+g B^{-3}+h B^{-4} \ldots$$

## 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|计算机图形学代写computer graphics代考|COS426

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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 \ \text { etc., } \end{aligned}
without worrying about creating impossible conditions.

## 计算机代写|计算机图形学代写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, hut hehaves like an operator, and is described later.

The commutative law in algebra states that when two elements are linked through some binary operation, the result is independent of the order of the elements.
The commutative law of addition is
$$\begin{array}{r} a+b=b+a \ \text { e.g. } 1+2=2+1 . \end{array}$$
The commutative law of multiplication is
\begin{aligned} a \times b &=b \times a \ \text { e.g. } 1 \times 2 &=2 \times 1 . \end{aligned}
Note that subtraction is not commutative:
$$\begin{array}{r} a-b \neq b-a \ \text { e.g. } 1-2 \neq 2-1 . \end{array}$$

## 计算机代写|计算机图形学代写computer graphics代考|Negative Numbers

$$4-1=3 \quad 4-2=24-3=1 \quad 4-4=0$$

$$4-5=-14-6 \quad=-24-7=-3 \text { etc., }$$

## 计算机代写|计算机图形学代写computer graphics代考|The Arithmetic of Positive and Negative Numbers

Brahmagupta 再次编制了所有规则，表格 $2.1$ 和 $2.2$ ，支持正负数的加减乘除。真正美中不足的是负数，这给 孩子们、数学老师带来了麻烦，也给数学家们带来了偶尔的事故。也许，我们在学生时代都记得的一条规则 是，两个负面因素构成一个积极因素。

$$a+b=b+a \text { e.g. } 1+2=2+1 .$$

$$a \times b=b \times a \text { e.g. } 1 \times 2=2 \times 1 .$$

$$a-b \neq b-a \text { e.g. } 1-2 \neq 2-1 \text {. }$$

## 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|计算机图形学代写computer graphics代考|CS4600

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|计算机图形学代写computer graphics代考|Symbols and Notation

One of the reasons why many people find mathematics inaccessible is due to its symbols and notation. Let’s look at symbols first. The English alphabet possesses a reasonable range of familiar character shapes:
$$\begin{gathered} \text { a,b,c,d,ef,f,g,h,i,i,j,k,l,m,n,o,p,q,r,r,s,t,u,v,w, } x, y, z \ \text { A,B,B,C,D,E,F,G, H,I, J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z } \end{gathered}$$
which find their way into every branch of mathematics and physics, and permit us to write equations such as
$$E=m c^2$$
and
$$A=\pi r^2 .$$
It is important that when we see an equation, we are able to read it as part of the text. In the case of $E=m c^2$, this is read as ‘ $E$ equals $m, c$ squared’, where $E$ stands for energy, $m$ for mass, $c$ the speed of light, which is multiplied by itself. In the case of $A=\pi r^2$, this is read as ‘ $A$ equals pi, $r$ squared’, where $A$ stands for area, $\pi$ the ratio of a circle’s circumference to its diameter, and $r$ the circle’s radius. Greek symbols, which happen to look nice and impressive, have also found their way into many equations, and often disrupt the flow of reading, simply because we don’t know their English names. For example, the English theoretical physicist Paul Dirac (1902-1984), derived an equation for a moving electron using the symbols $\alpha_i$ and $\beta$, which are $4 \times 4$ matrices, where
$$\alpha_i \beta+\beta \alpha_i=0$$
and is read as
‘the sum of the products alpha-i beta, and beta alpha-i, equals zero.’
Although we do not come across moving electrons in this book, we do have to be familiar with the following Greek symbols:
$\begin{array}{llll}\alpha & \text { alpha } & \nu & \mathrm{nu} \ \beta & \text { beta } & \xi & \mathrm{xi}\end{array}$

## 计算机代写|计算机图形学代写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:
$$n \in \mathbb{N}$$
which reads ‘ $n$ is a member of $(E)$ the set $\mathbb{N}$ ‘, i.e. $n$ is a whole number. Similarly:
$$x \in \mathbb{R}$$
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 1st prime, and 3 is the 2nd 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’.

## 计算机代写|计算机图形学代写computer graphics代考|Symbols and Notation

a,b,c,d,ef,f,g,h,i,i,j,k,l,m,n, o,p,q,r,r ,s,t,u,v,w , x, y, z A,B,B,C,D,E,F, G, H,I, J,K,L, M, N, O,P,

$$E=m c^2$$

$$A=\pi r^2 .$$

$$\alpha_i \beta+\beta \alpha_i=0$$

“alpha-i beta 和 beta alpha-i 的乘积之和等于零。”

$\alpha$ alpha $\nu$ nu $\beta$ beta $\xi$ xi

## 计算机代写|计算机图形学代写computer graphics代考|Sets of Numbers

$$n \in \mathbb{N}$$

$$x \in \mathbb{R}$$

$$P=2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, \ldots \quad \mathbb{N}=1,2,3,4,5,6,7,8,9,10,11,12,13,$$

$$2,1,3,2,5,3,7,4,11,5,13,6,17,7,19,8,23,9, \ldots$$

## 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|计算机图形学代写computer graphics代考|CS480

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|计算机图形学代写computer graphics代考|The Base of a Number System

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|计算机图形学代写computer graphics代考|The Base of a Number System

$$a \times 1000+b \times 100+c \times 10+d \times 1$$

$$e \times 0.1+f \times 0.01+g \times 0.001+h \times 0.0001$$

$$\ldots a 10^3+b 10^2+c 10^1+d 10^0+e 10^{-1}+f 10^{-2}+g 10^{-3}+h 10^{-4} \ldots$$

$$\ldots a B^3+b B^2+c B^1+d B^0+e B^{-1}+f B^{-2}+g B^{-3}+h B^{-4} \ldots$$

## 计算机代写|计算机图形学代写computer graphics代考|Binary Numbers

$$\ldots a 2^3+b 2^2+c 2^1+d 2^0+e 2^{-1}+f 2^{-2}+g 2^{-3}+h 2^{-4} \ldots$$

$$1_2, 10_2, 11_2, 100_2, 101_2, 110_2, 111_2, 1000_2, 1001_2, 1010_2, 1011_2, 1100_2, 1101_2 .$$

$$\left(1 \times 2^4\right)+\left(1 \times 2^3\right)+\left(0 \times 2^2\right)+\left(1 \times 2^1\right)+\left(1 \times 2^0\right)+\left(1 \times 2^{-1}\right)+\left(1 \times 2^{-2}\right)(1 \times 16)+(1 \times 8)$$

## 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|计算机图形学代写computer graphics代考|CSCl420

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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:
$$\begin{gathered} 4-5=-1 \ 4-6=-2 \ 4-7=-3 \ \text { etc., } \end{gathered}$$
without worrying about creating impossible conditions.

## 计算机代写|计算机图形学代写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.

The commutative law in algebra states that when two elements are linked through some binary operation, the result is independent of the order of the elements.
The commutative law of addition is
$$\begin{array}{r} a+b=b+a \ \text { e.g. } 1+2=2+1 . \end{array}$$
The commutative law of multiplication is
$$\begin{array}{r} a \times b=b \times a \ \text { e.g. } 1 \times 2=2 \times 1 . \end{array}$$
Note that subtraction is not commutative:
$$\begin{array}{r} a-b \neq b-a \ \text { e.g. } 1-2 \neq 2-1 \end{array}$$

## 计算机代写|计算机图形学代写computer graphics代考|Negative Numbers

$$4-1=3 \quad 4-2=24-3=1 \quad 4-4=0$$

$$4-5=-14-6=-24-7=-3 \text { etc., }$$

## 计算机代写|计算机图形学代写computer graphics代考|The Arithmetic of Positive and Negative Numbers

Brahmagupta 再次编制了所有规则，表格 $2.1$ 和 $2.2$ ，支持正负数的加减乘除。真正美中不足的是负数，这给孩 子们、数学老师带来了麻烦，也给数学家们带来了偶尔的事故。也许，我们在学生时代都记得的一条规则是，两 个负面因素构成一个积极因素。

$$a+b=b+a \text { e.g. } 1+2=2+1 .$$

$$a \times b=b \times a \text { e.g. } 1 \times 2=2 \times 1 \text {. }$$

$$a-b \neq b-a \text { e.g. } 1-2 \neq 2-1$$

## 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|计算机图形学代写computer graphics代考|CS559

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|计算机图形学代写computer graphics代考|Symbols and Notation

One of the reasons why many people find mathematics inaccessible is due to its symbols and notation. Let’s look at symbols first. The English alphabet possesses a reasonable range of familiar character shapes:
$$\begin{gathered} \text { a,b,cc, d,e,f,g,h,i,j,k,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z } \ \text { A,B,C,D,E,F,G,H,I,J,K,L, M, N,O,P,Q,R,S,T,U,V,W,X,Y,Z } \end{gathered}$$
which find their way into every branch of mathematics and physics, and permit us to write equations such as
$$E=m c^2$$
and
$$A=\pi r^2 .$$
It is important that when we see an equation, we are able to read it as part of the text. In the case of $E=m c^2$, this is read as ‘ $E$ equals $m, c$ squared’, where $E$ stands for energy, $m$ for mass, $c$ the speed of light, which is multiplied by itself. In the case of $A=\pi r^2$, this is read as ‘ $A$ equals pi, $r$ squared’, where $A$ stands for area, $\pi$ the ratio of a circle’s circumference to its diameter, and $r$ the circle’s radius. Greek symbols, which happen to look nice and impressive, have also found their way into many equations, and often disrupt the flow of reading, simply because we don’t know their English names. For example, the English theoretical physicist Paul Dirac (1902-1984), derived an equation for a moving electron using the symbols $\alpha_i$ and $\beta$, which are $4 \times 4$ matrices, where
$$\alpha_i \beta+\beta \alpha_i=0$$
and is read as
‘the sum of the products alpha-i beta, and beta alpha- $i$, equals zero.’
Although we do not come across moving electrons in this book, we do have to be familiar with the following Greek symbols:
$\alpha$ alpha $\quad \nu \quad$ nu
$\beta$ beta $\quad \xi \quad$ xi

## 计算机代写|计算机图形学代写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:
$$n \in \mathbb{N}$$
which reads ‘ $n$ is a member of $(\epsilon)$ the set $\mathbb{N}$ ‘, i.e. $n$ is a whole number. Similarly:
$$x \in \mathbb{R}$$
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 1st 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.

## 计算机代写|计算机图形学代写computer graphics代考|Symbols and Notation

a,b,cc, d,e,f,g,h,i,j,k,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z A,B,C,D, E,F,G,H,I,J,K,L, M, N, O,P,Q,R

$$E=m c^2$$

$$A=\pi r^2 .$$

$$\alpha_i \beta+\beta \alpha_i=0$$

“产品 alpha-i beta 和 beta alpha- 的总和” $i$ ，等于零。

$\alpha \alpha \quad \nu \quad$ 不是
$\beta$ 贝塔 $\xi \mathrm{xi}$

## 计算机代写|计算机图形学代写computer graphics代考|Sets of Numbers

$$n \in \mathbb{N}$$

$$x \in \mathbb{R}$$

$$P=2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, \ldots \quad \mathbb{N}=1,2,3,4,5,6,7,8,9,10,11,12,$$

$$2,1,3,2,5,3,7,4,11,5,13,6,17,7,19,8,23,9, \ldots$$

## 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。

## CS代写|计算机图形学作业代写computer graphics代考|CS480

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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

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

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 I Iindu-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:
$$\ldots a 10^3+b 10^2+c 10^1+d 10^0+e 10^{-1}+f 10^{-2}+g 10^{-3}+h 10^{-4} \ldots \ldots$$
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 $B-1$.

## CS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 $14_8$ in decimal? Well, it stands for:
$$1 \times 8^1+4 \times 8^0=12$$
Thus $356.4_8$ 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!

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

$$a \times 1000+b \times 100+c \times 10+d \times 1$$

$$e \times 0.1+f \times 0.01+g \times 0.001+h \times 0.0001$$

$$\ldots a 10^3+b 10^2+c 10^1+d 10^0+e 10^{-1}+f 10^{-2}+g 10^{-3}+h 10^{-4} \ldots \ldots$$

$$\ldots a B^3+b B^2+c B^1+d B^0+e B^{-1}+f B^{-2}+g B^{-3}+h B^{-4} \ldots$$

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

$$\ldots a 8^3+b 8^2+c 8^1+d 8^0+e 8^{-1}+f 8^{-2}+g 8^{-3}+h 8^{-4} \ldots$$

$$1_8, 2_8, 3_8, 4_8, 5_8, 68,78,10_8, 11_8, 12_8, 13_8, 14_8, 15_8, 16_8, 17_8, 20_8, 21_8$$

$$1 \times 8^1+4 \times 8^0=12$$

$$\left(3 \times 8^2\right)+\left(5 \times 8^1\right)+\left(6 \times 8^0\right)+\left(4 \times 8^{-1}\right)(3 \times 64)+(5 \times 8)+(6 \times 1)+(4 \times 0.125)(192+40$$

## 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。