### 计算机代写|计算机图形学作业代写computer graphics代考|Imaginary Numbers

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• 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

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