Posted on Categories:Modern Algebra, 数学代写, 现代代数

# 数学代写|现代代数代考Modern Algebra代写|THE FIELD OF COMPLEX NUMBERS

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|现代代数代考Modern Algebra代写|THE FIELD OF COMPLEX NUMBERS

There is no real number $x$ such that $x^2=-1$, because the square of any nonzero element in an ordered integral domain must be positive (Lemma 28.1). The field of complex numbers, which contains the field of real numbers as a subfield, overcomes this deficiency. It does much more than that, in fact, as can be seen from the following theorem.

Theorem 32.1 (Fundamental Theorem of Algebra). Every polynomial equation
$$a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0=0$$
which is of degree at least 1 and whose coefficients $a_n, a_{n-1}, \ldots, a_1, a_0$ are complex numbers, has at least one solution in the field of complex numbers.

Notice the implications of this theorem. To have solutions for all equations $a x=b$ (coefficients integers), we extended the integers to the rational numbers. To have a solution for $x^2=2$, we went outside the rational numbers to the real numbers. To have a solution for $x^2=-1$, we are extending the real numbers to the complex numbers. The Fundamental Theorem of Algebra asserts that in looking for soiutions to polynomial equations there will be no need to extend further, because any such equation with complex numbers as coefficients will have a solution in the field of complex numbers. We shall not prove the Fundamental Theorem of Algebra, but we say more about polynomial equations in Chapters $\mathrm{X}$ and XI.

We now give a description of the complex numbers. The rational numbers were constructed using equivalence classes of ordered pairs of integers. The complex numbers will be constructed using ordered pairs of real numbers. Problem 32.18 suggests how they could be constructed using matrices, if one preferred. Suggestion: After studying the statement of Theorem 32.2, pass over the proof and read through Example 32.2; then return to the proof. This should make the operations in the theorem seem more natural.

## 数学代写|现代代数代考Modern Algebra代写|COMPLEX ROOTS OF UNITY

The other sections of this chapter have been concerned primarily with general properties and abstractions. This section has to do with computation. We look at some useful ways to represent complex numbers and show how they can be used to determine the complex roots of unity-the solutions of equations of the form $x^n=1$. These roots of unity are useful for examples; they also arise often enough in other areas of mathematics to make their inclusion here worthwhile.

Just as the points on a line can be used to represent real numbers geometrically, the points in a plane can be used to represent complex numbers geometrically. A rectangular coordinate system is chosen for the plane, and then each complex number $a+b i$ is represented by the point with coordinates $(a, b)$. Because
$$a+b i=c+d i \quad \text { iff } a=c \quad \text { and } \quad b=d$$

(for $a, b, c, d \in \mathbb{R}$ ), the correspondence
$$a+b i \leftrightarrow(a, b)$$
is one-to-one between complex numbers and points of the plane. Figure 33.1 shows some examples.

Addition of complex numbers corresponds to vector addition of points in the plane:
$$(a+b i)+(c+d i)=(a+c)+(b+d) i \leftrightarrow(a+c, b+d)=(a, b)+(c, d) .$$
To describe muitiplication of complex numbers geometrically, we turn to polar coordinates. Recall that the polar representation of a point with rectangular coordinates $(a, b)$ is $(r, \theta)$, where $r$ denotes the distance between the origin and the given point, and $\theta$ denotes the angle from the positive $x$-axis to the ray from the origin through the given point, with the positive direction taken counterclockwise (Figure 33.2).

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|THE FIELD OF COMPLEX NUMBERS

$$a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0=0$$

## 数学代写|现代代数代考Modern Algebra代写|COMPLEX ROOTS OF UNITY

$$a+b i=c+d i \quad \text { iff } a=c \quad \text { and } \quad b=d$$

(网址:$a, b, c, d \in \mathbb{R}$)，信件
$$a+b i \leftrightarrow(a, b)$$

$$(a+b i)+(c+d i)=(a+c)+(b+d) i \leftrightarrow(a+c, b+d)=(a, b)+(c, d) .$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。