计算机代写|计算机图形学代写computer graphics代考|COS426

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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|计算机图形学代写computer graphics代考|Negative Numbers

When negative numbers were first proposed, they were not accepted with open arms, as it was difficult to visualise $-5$ of something. For instance, if there are 5 donkeys in a field, and they are all stolen to make salami, the field is now empty, and there is nothing we can do in the arithmetic of donkeys to create a field of $-5$ donkeys. However, in applied mathematics, numbers have to represent all sorts of quantities such as temperature, displacement, angular rotation, speed, acceleration, etc., and we also need to incorporate ideas such as left and right, up and down, before and after, forwards and backwards, etc. Fortunately, negative numbers are perfect for representing all of the above quantities and ideas.

Consider the expression $4-x$, where $x \in \mathbb{N}^0$. When $x$ takes on certain values, we have
\begin{aligned} & 4-1=3 \ & 4-2=2 \ & 4-3=1 \ & 4-4=0 \end{aligned}
and unless we introduce negative numbers, we are unable to express the result of $4-5$. Consequently, negative numbers are visualised as shown in Fig. 2.1, where the number line shows negative numbers to the left of the natural numbers, which are positive, although the $+$ sign is omitted for clarity.

Moving from left to right, the number line provides a numerical continuum from large negative numbers, through zero, towards large positive numbers. In any calculations, we could agree that angles above the horizon are positive, and angles below the horizon, negative. Similarly, a movement forwards is positive, and a movement backwards is negative. So now we are able to write:
$$\begin{gathered} 4-5=-1 \ 4-6=-2 \ 4-7=-3 \ \text { etc., } \end{gathered}$$
without worrying about creating impossible conditions.

计算机代写|计算机图形学代写computer graphics代考|The Arithmetic of Positive and Negative Numbers

Once again, Brahmagupta compiled all the rules, Tables $2.1$ and $2.2$, supporting the addition, subtraction, multiplication and division of positive and negative numbers. The real fly in the ointment, being negative numbers, which cause problems for children, math teachers and occasional accidents for mathematicians. Perhaps, the one rule we all remember from our school days is that two negatives make a positive.
Another problem with negative numbers arises when we employ the square-root function. As the product of two positive or negative numbers results in a positive result, the square-root of a positive number gives rise to a positive and a negative answer. For example, $\sqrt{4}=\pm 2$. This means that the square-root function only applies to positive numbers. Nevertheless, it did not stop the invention of the imaginary object $i$, where $i^2=-1$. However, $i$ is not a number, but behaves like an operator, and is described later.

The commutative law in algebra states that when two elements are linked through some binary operation, the result is independent of the order of the elements.
The commutative law of addition is
\begin{aligned} a+b & =b+a \ \text { e.g. } 1+2 & =2+1 \end{aligned}
The commutative law of multiplication is
$$\begin{array}{r} a \times b=b \times a \ \text { e.g. } 1 \times 2=2 \times 1 . \end{array}$$
Note that subtraction is not commutative:
$$\begin{array}{r} a-b \neq b-a \ \text { e.g. } 1-2 \neq 2-1 . \end{array}$$

计算机代写|计算机图形学代写computer graphics代考|Negative Numbers

$$4-1=3 \quad 4-2=24-3=1 \quad 4-4=0$$

$$4-5=-14-6=-24-7=-3 \text { etc., }$$

计算机代写|计算机图形学代写computer graphics代考|The Arithmetic of Positive and Negative Numbers

Brahmagupta 再次编制了所有规则，表格 $2.1$ 和 $2.2$ ，支持正负数的加减乘除。真正美中不足的是负数， 这给孩子们、数学老师带来了麻烦，也给数学家们带来了偶尔的事故。也许，我们在学生时代都记得的一 条规则是，两个负面因素构成一个积极因素。

$$a+b=b+a \text { e.g. } 1+2=2+1$$

$$a \times b=b \times a \text { e.g. } 1 \times 2=2 \times 1 \text {. }$$

$$a-b \neq b-a \text { e.g. } 1-2 \neq 2-1 \text {. }$$

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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。