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# 数学代写|复分析代写Complex analysis代考|The Puzzle

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## 数学代写|复分析代写Complex analysis代考|The Puzzle

We return to our historical puzzle. Why was the development of complex numbers so laboured and hesitant, whereas that of complex analysis was explosive? We suggest a possible answer (only personal opinion and thus open to dispute). It is somewhat different from the ‘foundations + breakthrough’ explanation offered earlier.

Looking at the early history of complex numbers, the overall impression is of countless generations of mathematicians beating out their brains against a brick wall in search of – what? A triviality. The definition of complex numbers as ordered pairs of points $(x, y)$, or as points in the plane, was obtained over and over and over again. It is even implicit in Bombelli’s work; it is there for all to see in Wallis’s; it crops up again by way of Wessel, Argand, and Gauss. Morris Kline remarks on page 629 of [11]:
That many men – Cotes, de Moivre, Euler, and Vandermonde – really thought of complex numbers as points in the plane follows from the fact that all, in attempting to solve $x^n-1=0$, thought of solutions … as the vertices of a regular polygon.
If the problem has such a simple solution, why was this not recognised sooner?
Perhaps the early mathematicians were not so much seeking a construction for complex numbers as a meaning, in the philosophical sense: ‘what are complex numbers?’ However, the development of complex analysis showed that the complex number concept was so useful that no mathematician in his right mind could possibly ignore it. The unspoken question became ‘what can we do with complex numbers?’, and once that had been given a satisfactory answer, the original philosophical question evaporated. There was no jubilation at Hamilton’s incisive answer to the 300 -year old foundational problem – it was ‘old hat’. Once mathematicians had woven the notion of complex numbers into a powerful coherent theory, the fears that they had concerning the existence of complex numbers became unimportant, because mathematicians lost interest in that issue.

## 数学代写|复分析代写Complex analysis代考|Is Mathematics Discovered or Invented?

Students trying to understand new concepts are in a similar position to the pioneers who first investigated them. At any stage in our education, we build not just on our current knowledge, but on a variety of beliefs and intuitions that are often vague, and may not be consciously recognised. As a trivial example, children familiar with counting numbers may find it hard to adapt their thinking to negative numbers, or rational numbers. When faced with questions like ‘what is 3 minus 7 ?’ or ‘what is 3 divided by 7 ‘, intuition based solely on whole numbers leads to the answer ‘can’t be done’. That makes it hard to understand -4 or $3 / 7$. In fact, these is not really trivial examples, because the world’s top mathematicians, centuries ago, were just as confused by the question ‘what is the square root of minus one?’ Even their terminology – ‘imaginary’ – reveals how puzzled they were. Intuitively they considered numbers to be ‘real’ – not in the sense we now use to distinguish real from complex, but as direct representations of real measurements. The new objects behaved like numbers in many ways, but they seemed not to correspond directly to reality.

In such circumstances, it can be tempting to discard existing intuition completely. But it is more sensible to adapt the intuition to fit the new circumstances. It is much easier to do arithmetic with negative numbers or fractions if you remember how to do it with whole numbers; it is much easier to do algebra with complex numbers if you bear in mind how to do it with real numbers. So the trick is to sort out which aspects of existing intuition remain valid, and which need to be refined into a broader kind of understanding.

One way to approach this issue is to take seriously a question that is often asked but seldom answered satisfactorily: is mathematics discovered or invented? One answer is to dismiss the question, and agree that neither word is entirely appropriate; moreover, they are not mutually exclusive. Most discoveries have elements of invention, most inventions have elements of discovery. Galileo would not have discovered the moons of Jupiter without the invention of the telescope. The telescope could not have been invented without discovering that sand could be melted to make glass.

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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。