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# 数学代写|数学分析作业代写Mathematical Analysis代考|The Real and Complex Number Fields

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Real and Complex Number Fields

An organized study of mathematics must be rooted in a proper understanding of number systems. Authors of textbooks such as this one are often divided between two extremes: either they provide an extensive development of number systems from scratch or they ignore the entire matter and consider knowledge of the real numbers to be a prerequisite. This presentation is a compromise between the two extremes. It is assumed that the reader has a thorough knowledge of integers and the rational number field, including such topics as divisibility, prime factorizations, the infinitude of the set of prime numbers, and the construction of $\mathbb{Q}$ in the usual manner as the quotient field of $\mathbb{Z}$. We basically accept the completeness of real numbers as an axiom, then prove the Cauchy criterion and the Bolzano-Weierstrass property, which we decided to develop at length since it is a cornerstone theorem. The section concludes with the definition of complex numbers and a study of their basic properties, including completeness. Although the section is not totally self-contained, there is value in its inclusion because it illustrates a number of important proof techniques and provides a succinct summary of the properties of real and complex number fields.

Definition. Let $F$ be a nonempty set endowed with two binary operations, + (addition) and $\times$ (multiplication). The triple $(F,+, \times)$ is said to be a field if the following conditions are satisfied for all $a, b, c \in F$ :
(a) $a+b=b+a$
(b) $a+(b+c)=(a+b)+c$.
(c) There is an element $0 \in F$ such that $a+0=a$.
(d) For every $a \in F$, there is an element $-a \in F$ such that $a+(-a)=0$.
(e) $a \times b=b \times a$.
(f) $a \times(b \times c)=(a \times b) \times c$.
(g) There is an element $1 \in F$ such that $a \times 1=a$.
(h) For every $a \neq 0$, there is an element $a^{-1}$ such that $a \times a^{-1}=1$.
(i) $a \times(b+c)=a \times b+a \times c$.
We often omit the symbol for multiplication and write $a b$ or $a . b$ for $a \times b$. The element 0 is called the additive identity, and 1 is called the multiplicative identity of the filed. A field must clearly contain at least two elements.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Complex Numbers

Definition. A complex number $z$ is an ordered pair $(x, y)$ of real numbers. We use the symbol $\mathbb{C}$ for the set of complex numbers.

The definition so far makes $\mathbb{C}$ nothing more than the Euclidean plane $\mathbb{R}^2$. This is why the set of complex numbers is also called the complex plane. What sets $\mathbb{C}$ apart from $\mathbb{R}^2$ is the following pair of binary operations.

Definition. For complex numbers $z=(x, y)$ and $w=(a, b)$, we define the sum $z+w=(x+a, y+b)$ and the product $z w=(a x-b y, a y+b x)$. The real field $\mathbb{R}$ is embedded into the complex plane in a natural way: we identify a real number $x$ with the complex number $(x, 0)$. Under the operations of complex addition and multiplication, the subset $\tilde{\mathbb{R}}={(x, 0) \in \mathbb{C}: x \in \mathbb{R}}$ is closed in the sense that if $z$ and $w$ are in $\tilde{\mathbb{R}}$, then $z+w$ and $z w$ are in $\tilde{\mathbb{R}}$. Indeed $z+w=(x+a, 0)$ and $z w=(a x, 0)$. From now on, we make no distinction between $\mathbb{R}$ and $\tilde{\mathbb{R}}$ and simply write $x$ for $(x, 0)$. With this understanding, we see that if $x \in \mathbb{R}$, then $x w=(x, 0)(a, b)=(x a, x b)$. It is also straightforward to verify that the elements $0=(0,0)$ and $1=(1,0)$ satisfy $z+0=z$ and $z .1=z$ for all $z \in \mathbb{C}$. Thus 0 and 1 are the identity elements for complex addition and multiplication, respectively.
Definition. The complex number $i=(0,1)$ is called the imaginary number. Now $i^2=(0,1) .(0,1)=(-1,0)=-1$. We therefore think of $i$ as the square root of -1.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Real and Complex Number Fields

(a) $a+b=b+a$
(b) $a+(b+c)=(a+b)+c$。
(c)有一个要素$0 \in F$使$a+0=a$。
(d)对于每一个$a \in F$，都有一个元素$-a \in F$，使得$a+(-a)=0$。
(e) $a \times b=b \times a$。
(f) $a \times(b \times c)=(a \times b) \times c$。
(g)有一个要素$1 \in F$使得$a \times 1=a$。
(h)对于每个$a \neq 0$，有一个元素$a^{-1}$使得$a \times a^{-1}=1$。
(i) $a \times(b+c)=a \times b+a \times c$。

## MATLAB代写

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