### 数学代写|抽象代数作业代写abstract algebra代考|MATH 355

statistics-lab™ 为您的留学生涯保驾护航 在代写抽象代数abstract algebra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。我们的专家在代写抽象代数abstract algebra代写方面经验极为丰富，各种代写抽象代数abstract algebra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

## 数学代写|抽象代数作业代写abstract algebra代考|Modular Arithmetic

Another application of the division algorithm that will be important to us is modular arithmetic. Modular arithmetic is an abstraction of a method of counting that you often use. For example, if it is now September, what month will it be 25 months from now? Of course, the answer is October, but the interesting fact is that you didn’t arrive at the answer by starting with September and counting off 25 months. Instead, without even thinking about it, you simply observed that $25=2 \cdot 12+1$, and you added 1 month to September. Similarly, if it is now Wednesday, you know that in 23 days it will be Friday. This time, you arrived at your answer by noting that $23=7 \cdot 3+2$, so you added 2 days to Wednesday instead of counting off 23 days. If your electricity is off for 26 hours, you must advance your clock 2 hours, since $26=2 \cdot 12+2$. Surprisingly, this simple idea has numerous important applications in mathematics and computer science. You will see a few of them in this section. We shall see many more in later chapters.

The following notation is convenient. When $a=q n+r$, where $q$ is the quotient and $r$ is the remainder upon dividing $a$ by $n$, we write $a \bmod n=r$. Thus,
\begin{aligned} 3 \bmod 2 &=1 \text { since } 3=1 \cdot 2+1, \ 6 \bmod 2 &=0 \text { since } 6=3 \cdot 2+0, \ 11 \bmod 3 &=2 \text { since } 11=3 \cdot 3+2, \ 62 \bmod 85 &=62 \text { since } 62=0 \cdot 85+62, \ -2 \bmod 15 &=13 \text { since }-2=(-1) 15+13 . \end{aligned}
In general, if $a$ and $b$ are integers and $n$ is a positive integer, then $a \bmod n=b \bmod n$ if and only if $n$ divides $a-b$ (Exercise $9) .$

In our applications, we will use addition and multiplication $\bmod n$. When you wish to compute $a b \bmod n$ or $(a+b) \bmod n$, and $a$ or $b$ is greater than $n$, it is easier to “mod first.” For example, to compute $(27 \cdot 36) \bmod 11$, we note that $27 \bmod 11=5$ and $36 \bmod 11=3$, so $(27 \cdot 36) \bmod 11=(5 \cdot 3) \bmod 11=4$. (See Exercise 11.)

Modular arithmetic is often used in assigning an extra digit to identification numbers for the purpose of detecting forgery or errors. We present two such applications.

## 数学代写|抽象代数作业代写abstract algebra代考|Complex Numbers

Recall that complex numbers $\mathrm{C}$ are expressions of the form $a+b \sqrt{-1}$, where $a$ and $b$ are real numbers. The number $\sqrt{-1}$ is defined to have the property $\sqrt{-1^{2}}=-1$. It is customary to use $i$ to denote $\sqrt{-1}$. Then, $i^{2}=-1$. Complex numbers written in the form $a+b i$ are said to be in standard form. In some instances it is convenient to write a complex number $a+b i$ in another form. To do this we represent $a+b i$ as the point $(a, b)$ in a plane coordinatized by a horizontal axis called the real axis and a vertical $i$ axis called the imaginary axis. The distance from the point $a+b i$ to the origin is $r=\sqrt{a^{2}+b^{2}}$ and is often denoted by $|a+b i|$ and called the norm of $a+b i$. If we draw the line segment from the origin to $a+b i$ and denote the angle formed by the line segment and the positive real axis by $\theta$, we can write $a+b i$ as $r(\cos \theta+i \sin \theta)$.

This form of $a+b i$ is called the polar form. An advantage of the polar form is demonstrated in parts 5 and 6 of Theorem 0.4.IEXAMPLE $11(-1+i)^{4}=\left(\sqrt{2}\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right)^{4}=$ $\sqrt{2^{4}}\left(\cos \frac{4 \cdot 3 \pi}{4}+i \sin \frac{4 \cdot 3 \pi}{4}\right)=4(\cos 3 \pi+i \sin 3 \pi)=-4 .$ The three cube roots of $i=\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}$ are $\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}=\frac{\sqrt{3}}{2}+\frac{1}{2} i$ $\cos \left(\frac{\pi}{6}+\frac{2 \pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{2 \pi}{3}\right)=-\frac{\sqrt{3}}{2}+\frac{1}{2} i$ $\cos \left(\frac{\pi}{6}+\frac{4 \pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{4 \pi}{3}\right)=-i$.

## 数学代写|抽象代数作业代写abstract algebra代考| Complex Numbers

$\sqrt{2^{4}}\left(\cos \frac{4 \cdot 3 \pi}{4}+i \sin \frac{4 \cdot 3 \pi}{4}\right)=4(\cos 3 \pi+i \sin 3 \pi)=-4$. 的三个立方根 $i=\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}$ 是 $\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}=\frac{\sqrt{3}}{2}+\frac{1}{2} i \cos \left(\frac{\pi}{6}+\frac{2 \pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{2 \pi}{3}\right)=-\frac{\sqrt{3}}{2}+\frac{1}{2} i$ $\cos \left(\frac{\pi}{6}+\frac{4 \pi}{3}\right)+i \sin \left(\frac{\pi}{6}+\frac{4 \pi}{3}\right)=-i$

## 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。