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。