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Most languages are written in a left-to-right (LTR) direction, but some languages, such as Arabic, Hebrew, and Urdu, are written in a right-to-left (RTL) direction. For example, Arabic is currently used by 600 million people worldwide. Consequently, if a software interface supports RTL languages like Arabic, certain UI adjustments are necessary.

RTL adaptation involves many details, and this article attempts to summarize some key points.

## UI and Functional Interfaces

Common software interfaces need to mirror the layout of functional areas. For example:

- If navigation expands from left to right, in RTL pages, it should expand from right to left.
- If there is a sidebar or side menu, it should be placed on the right.
- The close button of a window should also be placed on the right.

For instance, the following images show a settings page in both LTR and RTL layouts:

![](https://image-assets.dreams.plus/images20250117122440.png)

![](https://image-assets.dreams.plus/images20250117122547.png)

Similarly, the order of icons in bottom menus or horizontal menus should also be reversed, from right to left.

If the website has a breadcrumb component, the order should also be flipped. For example, `Home > Category > Subcategory` becomes `Subcategory < Category < Home`.

For timeline components, the sequence of events should be arranged from right to left. This is especially important for **line charts and other statistical charts with time axes, where the order should also be from right to left.**

For progress bars, the direction of progress should also be adjusted accordingly.

![](https://image-assets.dreams.plus/images20250117122828.png)

- For toggle switch components, remember that the "on" state indicator should be on the left, and the "off" state should be on the right, as shown below:

![](https://image-assets.dreams.plus/images20250117121142.png)

## Text and Symbols

Arabic text is right-aligned by default.

Although Arabic is written in an RTL direction, some content remains LTR in mixed contexts. For example, the commonly used Indo-Arabic numerals are still written from left to right. For instance, the number "one thousand two hundred thirty-four" is written as $1234$, not $4321$. In Arabic text mixed with numbers, the Arabic text is RTL, but the numbers are LTR. Similarly, if Arabic text includes English words, the English words remain LTR.

```plaintext
هذا نص مختلط، هذا هو الرقم 1234، هذه هي الكلمة الإنجليزية English
```

On the other hand, mathematical formulas written in Arabic are special and fall into two categories: one uses Arabic letters as symbols, and the other uses Latin or Greek letters, as we do.

These two cases require different handling.

**First, if the mathematical formula uses Arabic letters as variables or unknowns, the formula must be mirrored.**

Not only should the derivation direction be rewritten from right to left, but the equation should also be reversed. For example, "1 + 1 = 2" should be written as $2 = 1 + 1$. The following image shows a formula where the calculation proceeds from right to left:

![](https://image-assets.dreams.plus/images20250117104949.png)

Symbol directions should also be mirrored. For example, in the following screenshot from an Arabic advanced mathematics course, note the direction of the integral symbol:

![](https://image-assets.dreams.plus/images20250117105825.png)

The same applies to mathematical roots:

![](https://image-assets.dreams.plus/images20250117105939.png)

**However, if the mathematical formula uses Latin or Greek letters as symbols, following international conventions, it does not need to be mirrored.** For example, in the following formula, which uses $x, e, i$ as variables, the writing direction does not need to be mirrored:

![](https://image-assets.dreams.plus/images20250117125222.png)

## Icons

For icons, there is one important consideration: although Arabic is written in an RTL direction, **Arabic people are still right-handed and prefer using their right hand for tasks.** Therefore, some icons should not be simply mirrored.

For example, the magnifying glass icon: most people are accustomed to holding the magnifying glass with their right hand, so the handle should remain on the right side and not be mirrored.

![](https://image-assets.dreams.plus/images20250117114401.png)

Similarly, for the notebook icon, Arabic people still write with their right hand, so the pen should remain on the right side and not be mirrored.

![](https://image-assets.dreams.plus/images20250117114659.png)

Generally, most realistic icons are designed based on the right-handed habits of humans, so in RTL layouts, there is usually no need to worry about flipping icons.

**However, if the icon involves text structure, it still needs to be flipped.** For example, in the following images, icons representing a "directory tree" and "text" need to be adjusted.

![](https://image-assets.dreams.plus/images20250117123437.png)

## Motion and Behavior

In RTL design, pay attention to the direction of arrows representing motion. For example, in pagination components or slide buttons, the "previous" and "next" buttons have arrows whose **directions are not reversed, but their logic is reversed.**

In our standard, the left arrow `⬅` represents "previous," and the right arrow `➡` represents "next."

However, in Arabic regions, **the left arrow `⬅` represents "next," and the right arrow `➡` represents "previous."** For example, in [this Arabic news website](https://www.alarabiya.net/editor-choice/archive), the pagination component at the bottom of the news list looks like this. When the current page is the first page, clicking the left arrow button on the left side takes you to the next page.

![](https://image-assets.dreams.plus/images20250117112944.png)

This reversed motion behavior can also be applied to other components. For example, in a carousel component, for RTL layouts, swiping to the right means "previous," and swiping to the left means "next." Similarly, the "forward" and "backward" buttons in navigation should also be adjusted.

Another point to note is in dialog boxes: **the cancel button (Cancel) is usually on the left, and the confirm button (Confirm) is on the right.**

For example, in the following dialog box, the cancel button is on the left, and the confirm button is on the right.

![](https://image-assets.dreams.plus/images20250117115156.png)

Similarly, in UI interaction design, frequently used buttons or buttons you want users to click should be placed on the **right**, while less frequently used buttons or those you don't want users to click should be placed on the **left.**

In animation design, corresponding adjustments should also be made. "Enter" animations should be designed to move "from right to left," and "exit" animations should move "from left to right."

On mobile devices, special attention should be paid to swipe and drag directions. Adjustments should also be made in drag menus.

![](https://image-assets.dreams.plus/images20250117122034.png)

## Frontend Design Implementation

If your frontend page supports i18n and you want to support RTL languages like Arabic or Hebrew, the points mentioned above should be considered. In frontend design, here are some insights on how to support RTL.

Generally, browsers default to LTR layout. If you want to switch to RTL, simply add the `dir` attribute to the `<html>` or `<body>` DOM element to set the entire page to RTL.

```html
<html dir="rtl">
  <head>
  </head>
  <body>
  </body>
</html>
```

This will change the text flow direction. For example, for this blog page:

![](https://image-assets.dreams.plus/images20250117124134.png)

In CSS, there are two methods to adapt to RTL and LTR.

The first is the manual method, where you can set rules for RTL and LTR elements separately using selectors. For example, the following rules set padding for left and right sides differently.

```css
/* Styles for LTR */
[dir="ltr"] .container {
  padding-left: 20px;
  padding-right: 10px;
}

/* Styles for RTL */
[dir="rtl"] .container {
  padding-left: 10px;
  padding-right: 20px;
}
```

The second method is to use CSS logical properties. For example, the `margin-inline-start` rule automatically detects the current page direction and adjusts accordingly.

```css
.container {
  text-align: start; /* Equivalent to left in LTR and right in RTL */
  margin-inline-start: 20px; /* Equivalent to left margin in LTR and right margin in RTL */
  margin-inline-end: 10px; /* Equivalent to right margin in LTR and left margin in RTL */
}
```

In existing CSS frameworks like Tailwind CSS, there is out-of-the-box RTL support. You can use the `rtl:` or `ltr:` prefixes to automatically apply classes based on the current layout.

Additionally, **avoid using classes like `ml-*, mr-*, left-*, right-*` or rules like `margin-left`.** These are based on physical directions rather than logical directions, which is detrimental to automatic RTL adaptation.

Instead, use classes like `ms-*, me-*, start-*, end-*` or rules like `margin-start` and `margin-end`. CSS will automatically adjust the spacing based on the current flow direction to match RTL and LTR.

[Here is the documentation for this feature.](https://tailwindcss.com/docs/hover-focus-and-other-states#rtl-support)

As mentioned earlier, components like "forward" and "backward" buttons or toggle switches have reversed logic. Therefore, in React, you can set a global state to indicate whether the current layout is RTL or LTR, and modify button behavior accordingly. Additionally, set corresponding RTL and LTR properties for each i18n language, and adjust the layout state automatically when switching languages.


Most languages ​​are written in LTR (left to right), but some are written in RTL (right to left), such as Arabic, Hebrew, Urdu, etc. The UI interface using these languages ​​also needs some detailed adjustments.

International UI Design Details: RTL Layout.


在 2024 年上半年，我开始了为期一年的背单词计划，每天坚持记忆、复习 200-400 个单词，坚持了将近一年，大约累计背了 16000 个单词，把 IELTS，TOFEL，专业四级和大半个 SAT 词书都背完了。一路走过来再回顾这些历程，虽然过程极度痛苦，日日夜夜，但好歹也算走过来了，想想也挺不容易的哈。

![](https://image-assets.dreams.plus/images20250116194715.png)

在背单词的过程中，我也累积了一些经验。我打算把这些经验总结一下，然后再分成几个篇幅来依次说说。本次想要分享的是：挑选词典。在 2024 年上半年，我买入了四本词典，都是牛津学习词典系列。

![牛津短语动词词典，和新旧两版同义词词典](https://image-assets.dreams.plus/imagesee97854c285b3202284930edd6fef07.webp)

![牛津习语词典](https://image-assets.dreams.plus/images20250116170944.webp)

我选的这些词典，它与像牛津词典、朗文词典等知名的综合大词典不同，它属于“专题词典”，针对某一目的或者主题来收录词条，使用目的性较强，专门为学习者设计，使用“专题词典”来对英语进行有目的的、针对性的学习，效率更高。这也是我目前正在学习的方法。

英语的学习难点在于词汇、词法。据统计，一个成年美国人的基础词汇量是 20000-30000 左右，也是一个美国高中生的词汇水平。也就是说，如果你想毫无障碍的阅读大多数美国语料，那么你的词汇量必须达到两万以上，才能勉强不卡壳。所以学习英语的过程也是在不断征服单词的过程。

## 同义词词典

首先是介绍同义词词典。

对任何有过背单词经历的同学来讲，英语的同义词现象非常常见，每次背单词的时候经常会受到同义词的困扰，每一个“含义”几乎都有 3-4 个同义词！

因为英语在历史的发展进程中吸收了大量的外来语，所以现在几乎是世界上词汇量最大的语言，而其中的同义词、近义词就占据了大半。确实，相比其他语言如日语、法语、西班牙语等等，**英语的同义词现象时最严重的。**

举个例子，“拒绝”的表达，就有 Refuse、Reject、Deny、Rebuff、Dismiss、Turn down、Oppose、Withhold、Repudiate、Resist 等等，它们都有“拒绝”之意。如果你单独把这些词一个一个去记，那么很难把它们“内化”成自己的一部分。在从外界接收、背诵了一大堆单词之后，久而久之，大脑还是处于无组织、混乱的状态。

假如你在英语写作和英语对话中，想要表达“拒绝”时，你能在大脑中一瞬间枚举出以上所有的表达吗？而且上面的同义词，在不同语境下都会有一些很细微的差别，你能思索一番后，举一反三，选取出最合适的表达词汇吗？例如：refuse 和 decline 都表示“拒绝”，但 decline 更礼貌，而 refuse 更直接。

还有，雅思考试、托福考试对单词的考题，大都是以“同义词替换”来考察的。需要识别同义词和近义词来回答阅读和词汇题。

所以归纳总结就非常有必要。这样才能把消极词汇转化为主动词汇。在背单词时，把不同的单词按照同一个或者相似的 idea 进行总结就尤为重要。归纳同义词和近义词不仅是词汇学习的一种技巧，更是一种提升语言能力的策略。它可以让学习者从“记单词”转向“用单词”，从而在实际交流中更自信、精准地表达自己。

还有，从同义词入手，可以显著提高背单词的效率，对提升词汇量都有着不小的帮助。

![](https://image-assets.dreams.plus/images20250116173639.webp)

这也是同义词词典存在的意义。我推荐牛津英语同义词词典，它按照首字母编序，为每个单词都枚举出了对应的同义词和近义词。

![](https://image-assets.dreams.plus/images20250116175401.webp)

如上图，举个例子，tedious 在表达“乏味、无聊”时的同义词，就有 boring,dreary,dull,monotonous,tire 等。它把对应的意思的相关近义词、同义词全部枚举出来，对归纳总结和提升词汇量都有不小的帮助。

它由于是 2001 年的早期初版，在淘宝上可能买不到新书了，只有二手书，价格低廉，大约 15 元以内就可以包邮到家。当然，不想花钱也可以下载电子书。同义词词典的初版在 Zlibrary 上可以找到电子版：[牛津同义词词典 - Zlibrary 链接](https://z-library.sk/book/22266968/4f2cc7/%E7%89%9B%E6%B4%A5%E8%8B%B1%E8%AF%AD%E5%90%8C%E4%B9%89%E8%AF%8D%E8%AF%8D%E5%85%B8.html)。

但是这本书还有一个新版，是 2014 年出版的，它还对近义词加上了辨析内容。

![](https://image-assets.dreams.plus/images20250116181118.webp)

它与初版相比，更注重于辨析，所以它只收录了常见同义词，删减了很多，而且价格也比初版更贵（160 元）

比如对“拒绝”的同义词，它的收录内容预览如下。

![](https://image-assets.dreams.plus/images20250116175951.webp)

**但是，我还是建议选初版的同义词词典。** 一来，我个人推荐在大量阅读的过程中学习同义词辨析，而不是词典中。二来，新版的同义词词典只收录了一些初级的、常用的词汇，对于进阶一点的词汇都删减了。**而且价格不菲，性价比不高。** 新版的词典我在网上没有找到电子版本。

所以还是推荐使用初版的，虽然初版词典没有辨析内容，但是它相较于新版的，词量更加全面。

当然，为了学习方便，我把这本词典做成了词表，[发布在了 Github 上](https://github.com/PrinOrange/my-english-word-books/tree/main/dictionaries/%E7%89%9B%E6%B4%A5%E8%8B%B1%E8%AF%AD%E5%90%8C%E4%B9%89%E8%AF%8D%E8%AF%8D%E5%85%B8/plain)，读者可以下载，导入不背单词、墨墨单词等单词 App 上。

同义词词典适合作为在学习过程中的总结阶段中使用，总结同义词可以帮助建立单词之间的联想，形成语义网络，助于更全面地理解一个词的含义和使用场景。长期积累有助于内化语言逻辑，逐步形成地道的英语语感。

## 短语动词词典

在英语学习阶段，只背“单词”本身是不够的，因为在英语中还有大量的“短语动词”，也就是 verb + adv./prep. 的形式，这类词的含义经常让人捉摸不透。

比如 take up 这个词，就是 verb. + prep. 的组合。

按照我们朴素的理解，“take”是“带走”的意思，“up”是“上面”的意思。乍一看，我肯会以为它是“把某件物体带到上面”的意思。

但实际上并不是，它是“占据”的意思。而“up”在这里不是介词而是副词，表示“彻底”的意思。和“把某件物体带到上面”没有任何关系。也就是说，我们无法单单从“take”或者“up”上推测这个短语动词的含义。类似的还有“take off”，是“起飞”的意思。

再换一个，比如 take over, make over, pull over，你能按照朴素的理解来推测它的含义吗？

但它们的含义是“接管”、“转让”和“停车”。

这就是短语动词的恶心之处，一个含义加上另一个含义，就生成了一个完全毫无关联的、新的含义。

短语动词很多是常见的动词加上一个常见的介词或副词，给人一种“很熟悉”的感觉，但实际上完全是另一个意思。而偏偏在英语的文本中（尤其是在互联网上），短语动词的使用频率又往往会占到四成水平。短语动词广泛用于英语的口语和非正式交流中，几乎每一个母语者都会经常使用。尤其是在英语口语中，母语者往往倾向于使用短语动词而不是正式词汇，因为它们更简单、自然。

对于非母语者来说，不积累足够多的短语动词，就会造成在阅读上的南辕北辙。**对于短语动词，其实没有什么好的办法，只能死记硬背，把它们“当成一个独立的新单词”来背。这也是一个难点所在**。

也不要指望自己像做数学题那样“推测”含义，因为短语动词的组合极为灵活，在英语的发展过程中大都是“约定成俗”的用语，**甚至没有什么逻辑，并不是简单的词义相加就能推测出来的。**

所以，我推荐“牛津短语动词词典”

![](https://image-assets.dreams.plus/images20250116184439.webp)

这本词典收录了大量的短语动词，并且总结出了一些常用义项。比如 run into 这个词，词典就给出了三种解释：遭遇、遇到、纠缠三重含义。

![](https://image-assets.dreams.plus/images20250116184809.webp)

它也是以词为单位的，把每个动词都列举出所有对应的短语动词，并依次编排，比如 rush 这个词

![](https://image-assets.dreams.plus/images20250116185052.webp)

短语动词大都是动介组合或者动副组合，其中的副词和介词又被称为小品词，像 turn out,make out，等，out 就是小品词，在短语动词中起到辅助意义的作用。

这本词典在附录中也总结了很多小品词，例举出了小品词的不同含义，总结的很全面，比如 over。

![](https://image-assets.dreams.plus/images20250116191451.webp)

这本书总结了 6000+ 左右的短语动词，已经满足了九成以上的场景覆盖需求，对于学习者来说已经完全足够了。这本书的价格也很亲民，40 元左右，影印书更便宜，9.9 元包邮。当然，它在 Zlibrary 上也有电子版：[牛津短语动词词典 - ZLibrary 链接](https://z-library.sk/book/18213158/3388a4/%E7%89%9B%E6%B4%A5%E7%9F%AD%E8%AF%AD%E5%8A%A8%E8%AF%8D%E8%AF%8D%E5%85%B8-%E8%8B%B1%E6%B1%89%E5%8F%8C%E8%A7%A3%E7%89%88%E7%AC%AC%E4%BA%8C%E7%89%88-oxford-phrasal-verbs-dictionarychineseenglishenglishchinese2nd-edition.html)

## 习语词典

在进阶阶段，接触到英语电影、小说的时候，里面会经常有一些修辞性的、本土化的表达，称之为习语“idiom”，类似于中文里面的“成语”。同样也不能只从字面上理解。

比如“得陇望蜀”就是个修辞性词语，来形容一方野心勃勃、贪得无厌。但是在上个世纪中，中方对苏联的外交翻译中，因为外交工作人员的失误，直接翻译成了字面意义上的“占领了甘肃和四川”，导致双方出现战略误解。

英语习语跟成语一样，它也跟英语文化紧密结合。如果想要深入接触英语文化，尤其是阅读英语本土文学作品，理解它们的表达，那么去学习一些习语则是必要的。许多习语反映了英语国家的文化、历史、传统和价值观。例如："Spill the beans"（泄露秘密）来源于古希腊的投票习俗。通过学习习语，可以更好地了解英语文化和母语者的思维方式。

对英语学习者来说是非常重要的，因为习语不仅是语言的重要组成部分，还能够丰富表达方式，帮助更好地理解英语文化和母语者的思维方式。

另一方面，英语习语有比正是表达更为灵活微妙的表达能力，更有感情色彩，这是常规表达中难以传递的。例如："It's raining cats and dogs."（下着倾盆大雨）、"Hit the nail on the head."（一针见血），"a piece of cake"（小菜一碟）。

学习习语时，你会接触到许多表达方式，增强对英语的敏感度和语感。例如："Break the ice" 不仅可以用作“打破僵局”，还能帮助你理解类似的表达如 "icebreaker"（破冰者/活动）。

除此之外，使用习语可以让你的英语表达更加生动自然，贴近母语者的语言风格。例如：与其说 "I'm very tired," 不如说 "I'm worn out."（我累坏了）。熟练使用习语能让你的语言表达从“标准化”提升到“地道化”。

这本书收录的也相对较为全面，选取一页来看

![](https://image-assets.dreams.plus/images20250116191057.webp)

它收录了大约一万句常用习语，不仅可以用来查阅，它也给出了很多用法例句，也非常适合在茶余饭后，没事的时候，来消遣时随意翻上一两页（不用刻意去背），坚持一年，对自己的表达能力的提升都有极大的帮助。

不过这本书在网上我也没找到电子版，但是价格也比较便宜，30-40 左右，也可以去拼夕夕拿 9.9 包邮的影印书，非常实惠。

以上推荐的三本词典大约如此。这些主题词典都会对英语进阶有很大的帮助。当然，更重要的一条：

**注重积累**。


根据我的一段学习英语的历程，本期推荐三本好用针对非母语者的英语学习词典：牛津短语动词词典、牛津习语词典和同义词学习词典。适合进阶阶段使用。

英语学习词典推荐


The time I start writing blog is when I was enrolled in University after the College Entrance Matriculation in 2020. And I had my some thoughts over technology and programming, which I would like to put them down as fruition for studying progress.

Now the 2025 is coming, also it's my fifth year since I starting my blog.

During my teenhood when I'm a senior high school student, my parent restricted my time spending on computer so that I have less chances with my computer. At that time, I didn’t have the ability to build my own blog website. So I back then prefer using QZone, or writing my thoughts on specified notebook.

And for a long time I accrued more valuable notes about programming and thinking, they entails with C++, Java and some software theories until 2021. But unfortunately, my QZone was banned for some censorship and most of my blog was deleted, years of hard work went down the drain, its a great pity, and that's when I made a decision of building my own blog site. I can not tolerate the deteriorating cyber environment and living in censorship with trepidation.

So I learned React.js, next.js and frontend UI design, finalize this blog website as my own my blog, I have my own free writing space now.

Throughout the years topics for my blog have always been "Thoughts on Programming and Life" but in fact the point is, my blog touches more programming than life experience.

That's because, the contents on life experience would be better on social media platforms of microblog, like twitter, weibo and QZone. I expect more fallbacks from my cyber friends when I publish my content to get more emotional values, because most of my friends are online in social media platforms rather than distributed blog site. I would hard to earn this effect if I publish on blog site.

Secondly, the most of contents over life experience are fragmentation that I would write them down anytime and anywhere. For example, when you cooked culinary delicacies, got out of kitchen and want to post them in public on mood. It's convenience for everyone to open App in several seconds and type them in mobile, and publish, all in one go. Rather than write them in blog site, I must pay more focus on the text formatting after I power up computer and launch blog platform. And naturally the long blog text is not suitable for fragment writing for no convenience.

Thus I write my life experience on social media(more on [my twitter](https://x.com/codemica)) and technologies on this blog site. But the cons are also remarkable, long time fragmentation writing conduces less deep thinkings and expression, I also noticed that my expression becoming more superficial than years before.

The next steps I will take efforts to do is pick my ability of deep writing up again, especially the content over life experience, which I would put more thoughts in.


This is my fifth year since I started writing blog. After these years I noticed that I have lost more about writing.

The Fifth Anniversary for My Blog


This issue can be seen as a classic case of historical inertia combined with outdated reforms in the education system. This encapsulates the core of the problem.

Frankly, I find it difficult to tolerate the myriad justifications people offer to defend this situation. It is merely a residual consequence of the glacial pace of educational reform, yet some have framed it as "developing logical thinking" or claimed that "university isn't a training camp." Such arguments are utterly untenable.

In reality, even if computer science majors studied more mathematics instead of physics, it would be far more beneficial.

Computer science fundamentally diverges from traditional engineering disciplines such as civil engineering, mechanical engineering, electronic engineering, and chemical engineering. These traditional fields are rooted in physics, whereas computer science is deeply grounded in mathematics.

Traditional engineering disciplines primarily engage with the physical world. Their foundations lie in natural sciences, particularly physics, and they boast histories spanning decades, if not centuries. The primary focus of these disciplines is the transformation of the physical realm.

Consider this: as a mechanical engineering major, it is inconceivable to bypass the study of mechanics—how else would you design gears? If you are in precision instrumentation, knowledge of optics and electronics is indispensable for designing sensors. For chemical engineering students, quantum mechanics and thermodynamics are fundamental to understanding molecular interactions. Similarly, hydraulic engineering requires mastery of fluid mechanics.

Each traditional engineering discipline is built upon natural sciences like physics, dedicated to leveraging these principles to reshape the material world.

In stark contrast, computer science follows an entirely distinct trajectory. It is a "formal science," entirely man-made, with most of its concepts abstracted directly from mathematics.

Modern computer science theory originated with Alan Turing, whose work on computability theory employed pure mathematical constructs to define the concept of a "computer." These theoretical foundations were later implemented to create various types of computing systems.

Computer science, at its core, studies theoretical frameworks. While engineers apply these frameworks to design chips and systems, the discipline itself focuses on abstract structures and models rather than the tangible, physical world.

Formal sciences—such as mathematics, logic, and computer science—emphasize abstract, logical constructs instead of physical phenomena. They rely on deductive reasoning and logical proofs rather than empirical observation, which aligns computer science closely with mathematics.

For instance, many foundational concepts in computer science, such as Turing machines, operating system models, and the Von Neumann architecture, are mathematical abstractions. These constructs do not exist in nature; they are products of human ingenuity.

Even programming concepts such as functions, pointers, parameters, and operators—or areas like complexity analysis—are rooted in mathematical models. Algorithm design exemplifies this reliance on mathematical abstraction.

So, what justification exists for requiring computer science majors to study physics? Will they be calculating the forces involved in moving servers? Or perhaps determining the melting point of silicon to fabricate CPUs?

Let us be candid: computer science is fundamentally a repository of mathematical abstractions. This alignment with mathematics is entirely unsurprising, given the discipline's origins.

And for those who argue, "We study electronics to design circuits for CPUs," it is crucial to clarify: digital circuits and logic gates are mathematical abstractions as well. The physical implementation of chips falls under the domain of electronic engineering, not computer science. The role of computer science is to provide the models, theories, and blueprints, while other fields handle physical realization.

Furthermore, most computer science graduates today pursue careers in software development. It is unrealistic to assume they are designing CPUs.

In China, computer science was only formalized as a university discipline in the 1980s, following the reinstatement of the national college entrance exams. However, the theoretical foundations of computer science had already been established during World War II. By the time China integrated it into its educational system, we were already lagging behind Western nations by no less than half a century.

During that era, computers were a rarity in China, and expertise in teaching the subject was even scarcer. Lacking domestic models to emulate, we resorted to copying. Since physics was a staple of other STEM fields, it was incorporated into computer science curricula as well.

And yet, we remain entrenched in these antiquated practices. Despite nearly four decades, our approach to computer science education continues to stagnate. For example, certain universities still teach C++ using VC6, a tool that is unequivocally obsolete by contemporary standards.

In conclusion, the shortcomings of computer science education in China can be attributed to historical inertia and outdated reforms. Unlike traditional engineering disciplines rooted in physics, computer science is grounded in mathematics and focuses on abstract models. The inclusion of subjects like physics in the curriculum reflects a failure to adapt to the unique nature and practical demands of this field.


China's computer science education is outdated, often including irrelevant subjects like physics, due to historical and systemic inertia.

Should Computer Science Majors Learn College Physics


In the United States, the date format is uniquely distinctive and stands out globally. Unlike the European "small to large" DMY format or the East Asian "large to small" YMD format, the U.S. uses MDY. Moreover, unlike other Europeans who prefer pure numeric date formats, Americans use a semi-numeric, semi-English format: English month + numeric date + numeric year. For example, today is November 27, 2024. While a British person might write it as 27/11/2024, an American would write it as November 27, 2024!

Below is a map of date formats around the world:

![](https://image-assets.dreams.plus/images20241128101343.webp)

In the map, blue areas represent DMY, yellow areas represent YMD, red areas use a mix of DMY and MDY, green areas use a mix of DMY and YMD, and pink areas exclusively use MDY. The only country that uses this date format is the United States.

A British author once complained American written form, and wrote this on his book.
> “Most British people write the day first, followed by the month. Businesspeople and older women still insist on putting the month first.” \
> —— _Have His Carcase_, by Dorothy Leigh Sayers

I am genuinely puzzled by the logic behind the American date format. Many former British colonies or Commonwealth countries have adopted the British DMY format. But why didn't the United States, once a British colony heavily influenced by British culture, inherit the DMY format?

In fact, the American insistence on their date format has led to some amusing anecdotes. For instance, a British woman and an American man planned a date and wrote "10/12/1997" in their letters. However, they never met. Why? Because the American man understood it as October 12, while the British woman understood it as December 10!

If the world only had YMD and DMY formats, even when Asians and Westerners read each other's texts, there would be no ambiguity.

For example, dates like "2024-11-11" and "11-11-2024" can be easily understood by looking at the position of the year, allowing one to quickly determine the date format and identify the corresponding month and day. The world would be a much better place.

However, the problem lies in the strong influence of the United States in computer science and technology, which has forced their culture, including the anti-human MDY date format, onto the world, including Europe.

If you encounter an English text with a date like "11/11/2024" and cannot determine whether it was written by an American or a British person, how would you interpret it?

As a result, many English-speaking countries, including several European nations, have had to compromise: when dates are ambiguous, the month should be written in abbreviated form. For example, the British would write the above date as 11 November, 2024, or 11 Nov. 2024.

Thus, the world is now caught in a three-way battle between MDY, YMD, and DMY.

The origins of the YMD and DMY formats reflect the long-term practical needs of Eastern and Western civilizations. The reasons for these differences are varied and inconclusive, but there are two plausible explanations from cultural needs and cognitive linguistics.

The first explanation is based on cultural needs.

For a long time, the Chinese did not adopt the Western Gregorian calendar but used the era name system. Era names held significant political importance in ancient China, reflecting the will of the state and the emperor. After Emperor Wu of Han, official documents often placed the era name at the beginning to show its importance. China, being an agricultural society, developed the lunar calendar, where agricultural activities were often organized by months for planting, cultivating, and harvesting. Hence, the Chinese prioritized the year and month. Other East Asian countries like Korea, Japan, and Mongolia were also influenced by the Chinese lunar calendar.

Europeans, on the other hand, had a different approach. During the Middle Ages, Europe was deeply religious, and Europeans frequently participated in religious activities such as Sundays and sermon days. Church activities emphasized specific dates of holidays and religious events, with months being secondary information and years changing less frequently, thus placed last. For example: "Die 31 mensis Decembris anno Domini 2024" (December 31, 2024). This tradition of prioritizing the "day" has continued in many European countries, influencing their date formats.

The second explanation comes from cognitive linguistics.

From a linguistic perspective, the Western DMY format is used because European languages generally follow a logical sequence from specific to abstract, with the date as a modifier expressed first, close to the core words of month and year. In many Indo-European languages, modifiers (such as specific dates or temporal features) usually precede the main information. Western address formats also share this characteristic, starting with the house number and street, followed by the city, state, or province. This format aligns with the European linguistic habit of "modifiers being close to the modified object" and fits their cultural emphasis on logic and linear time flow. Chinese, using a large-to-small logic (year → month → day), emphasizes the progressive relationship of time, aligning with the Chinese narrative structure of "background to specifics."

In my opinion, the best date format should be one that is friendly to both humans and computers, which is the YMD format popular in East Asian countries. YMD is more intuitive for humans and most convenient for organizing documents.

While the differences between YMD and DMY can be explained from various angles, what is the logic behind the American MDY format? What is the rationale for using MDY?

After extensive online research, I have not found a concrete reason.

I strongly suspect that some American just came up with this date format on a whim.

Since most languages (like Chinese and English) are read from left to right, imagine you need to sort a pile of documents by date. When you see a date like "20241111," you can easily scan the numbers 2-0-2-4-1-1-1-1 in sequence, quickly identifying the year and month prefix, and categorize it into a "broad range" for easier sorting. However, if the date format is the British DMY, like "12072024," you need to interrupt your reading flow, start scanning from the end (4-2-0-2), then correct the order to "2-0-2-4" to identify the year, and then scan "7-0," correcting it to "0-7" to identify July. Thus, when scanning a DMY date, extracting the year and month requires changing direction four times. While manageable in the short term, it becomes a significant mental burden with large workloads, greatly increasing the chance of errors due to carelessness.

The YMD format is also very computer-friendly. YMD dates can be directly sorted in computers, as the string format "2024-07-12" is sequentially encoded, making dictionary sorting straightforward.

Imagine you have two files:

```plaintext
2024-11-12-document1.docx
2024-12-07-document2.docx
```

With the date format prefix, the former file name will always have a higher dictionary order than the latter, allowing file explorers to sort by date directly using the file names.

Numeric dates like "20240712" also make it easy to compare two dates using simple computer comparison operators like `>` and `<`, without needing additional processing.

These advantages are not present in the DMY format.

American stubbornness is not limited to date formats; their measurement standards are also a mess. For example, Fahrenheit and imperial units (miles, inches, feet, gallons, ounces, etc.). The imperial system originated during the Roman Empire in Europe and has long been a confusing mess with no fixed standards, changing over different historical periods. It wasn't until the Middle Ages that Britain standardized these units.

However, when Britain established the imperial system, there were no strict, standard measurements. For example, an inch was defined as the length of three barleycorns laid end to end, and a foot was the distance from King Edward's nose to his outstretched hand. The problem is, using barleycorns to define an inch means a thousand barleycorns could have a thousand different lengths, and using body parts for length definitions means a thousand people could have a thousand different lengths. It remains a confusing mess.

The long-standing confusion in measurements was only resolved with the development of modern French natural science, leading to the strict definition of the metric system. The metric system uses reproducible physical experiments for definitions, making it much more rigorous than the imperial system.

Thus, today, the entire world, except the United States, uses the metric system. But the U.S. remains an outlier, stubbornly clinging to the imperial tradition under the guise of "cultural heritage" and "preserving freedom and diversity." The American refusal to adopt the metric system is as irrational as the Lilliputians in _Gulliver's Travels_ arguing over which end of an egg to crack. It's truly bizarre.


In the United States, the date format is unique in the world. It is different from the European "small to large" DMY and the East Asian countries' "large to small" YMD. It is actually MDY.

Complain on American Date Format

Let us implement rendering markdown in React and we'll embark on fundamental conceptions of Markdown compiling procession with usages and examples for unified plugins.

How to Render Markdown Document in React


British economist John Maynard Keynes and Austrian economist Friedrich Hayek are two of the most renowned figures in the history of 20th-century economic thought. Their ideas not only had a profound impact on the past economic and social landscape, but they will continue to influence current and future economic and social development until a more convincing economic ideology takes hold. The debate between Keynes and Hayek is known as “the most classic and famous duel in economic history.” American journalist Nicholas Wapshott, who was a senior editor at *The Times* of London and *The New York Sun*, systematically examined this debate in his book *Keynes Hayek: The Clash that Defined Modern Economics*. As we revisit this classic academic feast, we also reflect on the serious economic question: “To what extent should the government intervene in the market?”

Today, people are already quite familiar with the starkly opposing economic views of Keynes and Hayek. Keynes believed that the root cause of an economic crisis is the lack of confidence among capitalists in investment prospects, leading to reduced investment, and thus advocated for effective government intervention and control over social economic activities. Hayek, on the other hand, believed that the root cause of economic crises was over-investment by capitalists and argued that government intervention would only damage the mechanisms of market development and functioning. Therefore, he firmly defended the idea of a free market order. Keynes argued that the government could expand public spending through fiscal deficits and national debt, increase state investment and consumption, and indirectly advocated for the "nationalization of currency." Hayek, by contrast, proposed that currency issuance should be handed over to private banks, which would provide sound and stable money based on their own interests, advocating for the "denationalization of currency."

Keynes supported inflationary policies to increase corporate profits, stimulate investment, expand employment, and achieve stable economic development. Hayek argued that inflation would widen the income gap, cause asset prices to soar, and eventually lead to greater unemployment and economic vicious cycles. He believed that business cycles would naturally return to full employment. Keynes argued for increased investment in public works, improving social welfare, and providing relief spending for those living below the poverty line. Hayek, however, believed that public goods do not directly satisfy any specific demand but provide certain conditions under which individuals and smaller groups can enjoy better opportunities to meet their respective needs.

The debate between Keynes and Hayek is characterized by a vast ideological chasm, and throughout their lives, they could never reconcile with each other’s viewpoints. Their debates were later carried on by their passionate disciples, such as John Kenneth Galbraith and Milton Friedman. In terms of personality, the two were also very different. Keynes was highly articulate and charismatic, often painting an optimistic vision of the world, which was embraced by an entire generation of politicians and economists on both sides of the Atlantic. In contrast, Hayek was meticulous and persistent, swimming against the tide, but found support among market advocates and libertarians. To this day, the two distinct schools of thought in economics continue their “battle.”

This economic battle has left government decision-makers oscillating between the two scholars' ideas, influencing the ups and downs of the world economy, affecting the lives and livelihoods of billions. The global economic crisis of 1929–1933 led to the widespread adoption of Keynesianism, represented by the "New Deal" of Franklin D. Roosevelt, and it seemed that Keynes had triumphed over Hayek. However, the oil crisis of the 1970s plunged the global economy into the dilemma of "stagflation," and the neoliberalism represented by the "Washington Consensus" became the dominant policy-making ideology, giving Hayek a comeback. By the end of the 20th century, the failure of neoliberal economic policies in Latin America, Russia, Southeast Asia, and Eastern Europe, coupled with the impact of the 2008 global financial crisis, suggested that Keynesianism might once again be on the rise.

In terms of policy practice, the struggle between Keynesianism and neoliberalism may continue, and it is still too early to conclude who will ultimately prevail. Interestingly, the Nobel Prize in Economics, which is seen as a "weather vane" for academic trends, has also been swayed by economic crises and shifts in government decision-making. Since the Nobel Prize in Economics was first awarded in 1968, Keynes, who had passed away by then, was naturally ineligible. Hayek, however, won the prize in 1974. During the 1970s and 1980s, when neoliberalism was on the rise, figures like Milton Friedman, Robert Lucas, Finn Kydland, and Edward Prescott also won the prize. Later, however, new Keynesians like Joseph Stiglitz and Peter Diamond, who carried on Keynes' legacy, were awarded the prize as well. In 2012, with the resurgence of neoliberalism, Alvin Roth and Lloyd Shapley took home the Nobel Prize in Economics.

In fact, despite their dramatic disagreements, Keynes and Hayek had similar aims. They both studied the same subject: the cycles of boom and bust in commercial activity that emerged from the ruins of World War I. Both sought to establish a theoretical framework that combined real and monetary phenomena to explain the dynamics of the economy, especially the business cycle. Their economic theories both originated from Knut Wicksell's theory of monetary equilibrium and the concept of "natural interest rates." Furthermore, they shared a deep commitment to the basic principles of a market economy governed by freedom, democracy, and the rule of law. If we look purely at logical reasoning, both of their theoretical conclusions and policy recommendations could explain the causes of economic recessions and offer solutions. The historical choices made by policymakers ultimately explain the root cause of their disagreements—it was a matter of experience and practical judgment.

It’s worth mentioning that although their debate was often heated, with sharp criticisms exchanged, the mutual respect they maintained beyond academia, and their personal friendship, is even more commendable. Although Hayek had differing views on credit from Keynes, he praised Keynes' *A Treatise on Money* as "an outstanding description and classification of different forms of money." Similarly, Keynes was deeply moved by Hayek's great work *The Road to Serfdom*. Keynes not only provided housing for Hayek when he first arrived in London, but in their debate letters, they openly discussed their friendship. Hayek once said, “Although I still disagreed with Keynes' views and had fierce debates with him, we maintained the best of personal friendships. And as a man, I held him in the highest esteem in many ways.”

Their ideas not only had a profound impact on the past economic and social landscape, but they will continue to influence current and future economic and social development until a more convincing economic ideology takes hold.

The Historical Debate Between Hayek and Keynes


## The Largest Ear of Wheat

It is said that Socrates once led his students for a walk in a wheat field. His students asked him: how can one find the most suitable life partner?

Socrates looked at the wheat field and gave them a task: "Go into the wheat field and find the largest ear of wheat, but as you walk, you can only move forward and cannot turn back."

The students began walking while choosing, and the field was full of large ears of wheat. But which one was the largest? They walked with their heads down, checking one ear of wheat after another. They shook their heads at this one; they shook their heads at that one. They always believed that the largest ear was still ahead. Although they picked a few, they were not satisfied and threw them away. They thought there were plenty of chances and no need to decide too early.

The students continued walking with their heads down, carefully selecting as they went, for quite a long time. Suddenly, they heard Socrates' deep, resounding voice: "You have reached the end." At that moment, the students, empty-handed, realized what had happened and felt disheartened.

Socrates said to them: "There is certainly a largest ear of wheat in this field, but you may not encounter it; and even if you do, you may not make the correct judgment. Therefore, the largest ear is the one you just picked."

This story tells us that life is like walking in a wheat field, also searching for the largest ear of wheat. Some people, upon seeing a full and ripe ear, seize the opportunity and pick it without hesitation; others keep looking around and miss their chances again and again. Of course, the goal should be the largest ear, but holding the one in hand is the most tangible reality.

**So, the question arises: Is there a method that can help us find the largest ear of wheat as efficiently as possible?**

## Treasure Box Game

Let’s consider another example. Imagine there is a gambling game with the following rules: There are $m$ treasure boxes, each containing a random prize amount between $0$ and $x$ dollars, uniformly distributed. **However, you don’t know what the maximum amount $x$ in the boxes is.** Once you open a treasure box and see the prize, you have two choices: either take the prize and stop, or discard the box and open another one. However, once discarded, a box cannot be chosen again.

Now the question arises: how should you strategize to get the most money possible? Naturally, the more money, the better.

What would happen if we didn’t use any strategy and randomly picked a treasure box?

Let’s assume the amounts in the treasure boxes range from $1000, 999, 998, ..., 0$, with the top prize being $1000$. What is the probability of drawing the top prize?

Since each specific amount (including the top prize of $1000) has an equal chance of appearing, and the prize is uniformly distributed, the probability of drawing the top prize can be calculated as follows:

$$
P_{Jackpot} = \frac{1}{(\text{MaxMoney} - \text{MinMoney} + 1)}
$$

In this case, the maximum prize is $1000$ and the minimum is $0$, so:

$$
P_{Jackpot} = \frac{1}{(1000 - 0 + 1)} = \frac{1}{1001}
$$

Therefore, the probability of randomly picking a treasure box containing the top prize of $1000$ is $0.0999\% ≈ 0.1\%$. The odds are very low indeed.

Since chasing the jackpot is unrealistic, let’s lower our expectations a bit. What if we aim for a prize of at least $800$? What is the probability of winning that?

We can use the formula for uniform distribution. For a prize uniformly distributed between $0$ and $1000$, the probability density function is constant. The formula for probability is:

$$
P(a \leq X \leq b) = \frac{b - a}{\text{range}}
$$

In this case, the range is from $0$ to $1000$, so the total range is $1000$. We want to calculate the probability for the interval from $800$ to $1000$, so:

$$
P(800 \leq X \leq 1000) = \frac{1000 - 800}{1000} = \frac{200}{1000} = 0.2
$$

Thus, the probability of randomly selecting a treasure box with a prize between $800$ and $1000$ is $20\%$.

While the probability has increased significantly, $20\%$ is still relatively low.

So, if you randomly pick a treasure box and decide to keep it, the chances of getting a high prize are still slim, which clearly isn’t the optimal strategy.

## Optimization

**In most cases, we don’t know the maximum prize inside the treasure boxes.** Just like in the story of Socrates' students picking the largest ear of wheat, they didn’t know how big the largest ear was.

Let’s change our approach. Even though we don’t know the maximum prize, we do know that if we sample a portion of the treasure boxes, the probability of finding the jackpot in the sample is equal to its probability in the entire set. By sampling some treasure boxes, we can estimate the value of the jackpot based on the maximum prize in the sample.

Let’s denote the jackpot in the entire set as $M$. Since the jackpot is uniformly distributed among the treasure boxes, we can take a sample from the boxes and record the maximum prize in that sample, denoted as $M'$. Clearly, this sample maximum $M'$ will be close to the overall jackpot $M$.

Sampling a portion of the boxes to estimate the jackpot $M$ is feasible, and the more samples we take, the closer $M'$ will be to $M$.

However, the sample size cannot be too small, or the difference between $M'$ and $M$ will be large, making the estimate ineffective. On the other hand, the sample size cannot be too large either, because there will be fewer treasure boxes left to choose from, reducing the chance of finding the jackpot. Consider an extreme example: if we sample $999$ treasure boxes out of $1000$, $M'$ will likely be almost equal to $M$, but with only one treasure box remaining, the chance of hitting the jackpot becomes negligible.

Thus, it is important to balance the sample size with the total number of treasure boxes. Is there an optimal sample size where $M'$ is close to $M$ and the probability of winning a high prize is maximized afterward?

Next, we will use mathematical methods to derive this “balance point.” Suppose the total number of treasure boxes is $m$, and the sample size is $r$. We will find the expression for the optimal $r$.

## Mathematical Derivation

Assume we have a total of $N$ treasure boxes, and we select the first $r$ boxes as a sample. In this scenario, we aim to find the optimal sample size $r$ that maximizes the probability of selecting the highest prize after discarding the sample.

### Probability Derivation

For a fixed $r$, suppose the highest prize appears in the $k$-th treasure box. To win the highest prize after discarding the sample, it is necessary that the second-highest prize appears in the first $k-1$ boxes, and specifically, this second-highest prize must also appear within the sample (the first $r$ boxes). The probability of this happening is $\frac{r}{k-1}$.

This can be visualized with a diagram, where the red arrow indicates the positions where the second-highest prize occurs:

![](https://image-assets.dreams.plus/20240922115506.webp)

Thus, we derive the following equation:

$$
P = \frac{1}{N} \sum_{k=r+1}^{N} \frac{r}{k-1} = \frac{r}{N} \sum_{k=r+1}^{N} \frac{1}{k-1}
$$

In this equation, $\frac{1}{N}$ represents the probability of the highest prize appearing in the $k$-th treasure box, and $\frac{r}{k-1}$ represents the probability that the second-highest prize is within the sample of the first $r$ boxes. $P$ is the probability of winning the highest prize after discarding the sample.

### Summation Approximation

The above equation contains a harmonic series, which does not have a simple closed form but can be approximated using the following formula:

$$
\sum_{k=1}^{N} \frac{1}{k} = \ln(N) + \gamma
$$

where $\ln(N)$ is the natural logarithm and $\gamma$ is Euler's constant. Here, we want to approximate the summation from $r+1$ to $N$. To do this, we rewrite the sum as:

$$
\sum_{k=r+1}^{N} \frac{1}{k-1} = \sum_{k=1}^{N-(r+1)+1} \frac{1}{k} = \sum_{k=1}^{N-r} \frac{1}{k}
$$

Thus, the sum starts from $k=1$ and follows the same harmonic series approximation:

$$
\sum_{k=r+1}^{N} \frac{1}{k-1} = \left( \ln(N) + \gamma \right) - \left( \ln(r) + \gamma \right) = \ln(N) - \ln(r)
$$

Finally, using logarithmic properties, we simplify this to:

$$
\ln(N) - \ln(r) = \ln\left(\frac{N}{r}\right)
$$

### Substituting into the Original Equation

Now, we substitute the simplified sum into the original probability equation:

$$
P = \frac{r}{N} \ln\left(\frac{N}{r}\right)
$$

Next, we introduce a new variable $x = \frac{r}{N}$ to represent the proportion of the total boxes chosen as the sample, yielding:

$$
P = x \ln\left(\frac{1}{x}\right)
$$

### Maximizing the Probability

To maximize $P$, we take the derivative of $P = x \ln\left(\frac{1}{x}\right)$ with respect to $x$:

$$
\frac{dP}{dx} = \frac{d}{dx} \left(x \ln\left(\frac{1}{x}\right)\right)
$$

Using the product rule:

$$
\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}
$$

where $u = x$ and $v = \ln\left(\frac{1}{x}\right)$, we calculate the derivatives:

$$
\frac{du}{dx} = 1
$$

and

$$
\frac{dv}{dx} = \frac{d}{dx} \ln\left(\frac{1}{x}\right) = \frac{d}{dx} \left(-\ln(x)\right) = -\frac{1}{x}
$$

Substituting these into the product rule:

$$
\frac{dP}{dx} = x \left(-\frac{1}{x}\right) + \ln\left(\frac{1}{x}\right) \cdot 1 = -1 + \ln\left(\frac{1}{x}\right)
$$

To find the maximum probability $P$, we set the derivative equal to zero:

$$
-1 + \ln\left(\frac{1}{x}\right) = 0
$$

Thus, we have:

$$
\ln\left(\frac{1}{x}\right) = 1
$$

Exponentiating both sides, we get:

$$
\frac{1}{x} = e
$$

Therefore, $x = \frac{1}{e}$, which means the probability is maximized when $\frac{r}{N} = \frac{1}{e}$.

### Maximum Probability

At this optimal point, the maximum probability is:

$$
P = P_{max} = \frac{1}{e} \approx 0.368
$$

This shows that the optimal strategy is to sample approximately $\frac{1}{e}$ (about 36.8%) of the total treasure boxes and then select the next box that is greater than all the ones in the sample. This strategy maximizes the chance of selecting the highest prize.

## Program Simulation

We can use TypeScript to simulate the process described above.

To verify the probability of winning the highest prize using this strategy, we can calculate the ratio of successful trials where the highest prize is obtained to the total number of trials. Below is a TypeScript program that simulates this process:

```typescript
function simulateGamblingGame(m: number, x: number, trials: number): number {
  let successCount = 0;

  for (let trial = 0; trial < trials; trial++) {
    const boxes = Array.from({ length: m }, () => Math.random() * x);
    const firstPart = boxes.slice(0, Math.floor(m * 0.37));

    // Remember the highest prize in the first 37% of boxes
    const p = Math.max(...firstPart);

    // Find the maximum prize in all boxes
    const maxPrize = Math.max(...boxes);

    // Check from the 37% onward if a prize higher than p is found
    for (const prize of boxes.slice(Math.floor(m * 0.37))) {
      if (prize > p) {
        if (prize === maxPrize) {
          successCount++;
        }
        break; // Stop once a higher prize is found
      }
    }
  }

  return successCount / trials; // Return the success probability
}

// Set parameters
const m = 100; // Total number of boxes
const x = 1000; // Maximum prize in each box
const trials = 100000; // Number of trials

const successProbability = simulateGamblingGame(m, x, trials);
console.log(`Probability of winning the highest prize: ${successProbability}`);
```

### Explanation

1. **Parameters**:
   - `m`: Total number of treasure boxes.
   - `x`: Maximum prize in each box.
   - `trials`: Number of simulation trials. More trials increase accuracy.
2. **Simulation Process**:

   - Randomly generate the prize for each box.
   - Calculate the maximum prize in the first 37% of the boxes (`p`).
   - Find the highest prize (`maxPrize`) in all boxes.
   - Starting from the 38% box, check if there is a prize higher than `p` and if it equals `maxPrize`.

3. **Result**:
   - Return the probability of winning the highest prize after many trials.

Running this simulation produces a result like this:

```plaintext
Probability of winning the highest prize: 0.367999
```

As shown, the probability of winning the highest prize increases to around 37%, which is a significant improvement from 0.1%.

---

To further verify, let's simulate the probability of winning the top $t\%$ prizes by dividing the prize ranges into $10\%$ increments (e.g., top 10%, top 20%, etc.).

### Modified TypeScript Code for Different Prize Ranges

```typescript
function simulateGamblingGameWithRanges(
  m: number,
  x: number,
  trials: number
): number[] {
  const successCounts = Array(10).fill(0);
  // Array to store the number of successes in each range

  for (let trial = 0; trial < trials; trial++) {
    const boxes = Array.from({ length: m }, () => Math.random() * x);

    const k = Math.floor(m * 0.37);
    const firstPart = boxes.slice(0, k);

    const p = Math.max(...firstPart);

    let chosenPrize = 0;

    // Find the first box greater than p after the first 37% boxes
    for (const prize of boxes.slice(k)) {
      if (prize > p) {
        chosenPrize = prize;
        break;
      }
    }

    // If no box greater than p is found, select the last box
    if (chosenPrize === 0) {
      chosenPrize = boxes[boxes.length - 1];
    }

    const sortedBoxes = [...boxes].sort((a, b) => b - a);

    // Determine the prize range that the chosen prize falls into
    for (let i = 1; i <= 9; i++) {
      const thresholdPrize = sortedBoxes[Math.floor((m * i * 10) / 100)];
      if (chosenPrize >= thresholdPrize) {
        successCounts[i - 1]++;
      }
    }

    // The top 100% prize is always chosen, so increment this count
    successCounts[9]++;
  }

  // Convert counts to probabilities
  const successProbabilities = successCounts.map((count) => count / trials);

  return successProbabilities;
}

// Set parameters
const m = 100; // Number of boxes
const x = 1000; // Maximum prize
const trials = 100000; // Number of trials

const successProbabilities = simulateGamblingGameWithRanges(m, x, trials);

// Output the probabilities for each prize range
for (let i = 1; i <= 10; i++) {
  console.log(
    `Probability of winning a prize in the top ${i * 10}%: ${(
      successProbabilities[i - 1] * 100
    ).toFixed(2)}%`
  );
}
```

### Results from the Program

```plaintext
Probability of winning a prize in the top 10%: 66.96%
Probability of winning a prize in the top 20%: 70.67%
Probability of winning a prize in the top 30%: 74.42%
Probability of winning a prize in the top 40%: 78.06%
Probability of winning a prize in the top 50%: 81.70%
Probability of winning a prize in the top 60%: 85.45%
Probability of winning a prize in the top 70%: 89.14%
Probability of winning a prize in the top 80%: 92.84%
Probability of winning a prize in the top 90%: 96.60%
Probability of winning a prize in the top 100%: 100.00%
```

We see that, using this strategy, we maximize the chances of getting a high-value prize.

## Conclusion

The simulation verifies that selecting 37% of the boxes first and then choosing the next box with a prize greater than the maximum from the first 37% is the optimal strategy. This maximizes the probability of winning the highest prize, known as the **37% rule**.

This strategy can also be applied in real-life scenarios, such as making optimal choices in uncertain situations.


This article proposes the classic 37% rule in game theory and its mathematical derivation process.

Socrates: The Story of Searching for the Largest Ear of Wheat


## Wallis's Formula

Let's start with a definite integral:

$$
\int_{0}^{\frac{\pi}{2}} \sin^n x \, dx
$$

We can derive the following result:

$$
\int_{0}^{\frac{\pi}{2}} \sin^n x \, dx = -\int_{0}^{\frac{\pi}{2}} \sin^{n-1} x \, d(\cos x)
$$

Using integration by parts, where $\int u \, dv = uv - \int v \, du$, we get:

$$
-\int_{0}^{\frac{\pi}{2}} \sin^{n-1} x \, d(\cos x) \\ = -\left( \sin^{n-1} x \cos x \right) \Big|_{0}^{\frac{\pi}{2}} + \int_{0}^{\frac{\pi}{2}} \cos x \, d\left(\sin^{n-1} x \right)
$$

$$
= \int_{0}^{\frac{\pi}{2}} \cos^2 x \cdot (n-1) \sin^{n-2} x \, dx \\ = (n-1) \int_{0}^{\frac{\pi}{2}} \left( \sin^{n-2} x - \sin^n x \right) \, dx
$$

Thus:

$$
\int_{0}^{\frac{\pi}{2}} \sin^n x \, dx = (n-1) \int_{0}^{\frac{\pi}{2}} \left( \sin^{n-2} x - \sin^n x \right) \, dx
$$

We get:

$$
n \int_{0}^{\frac{\pi}{2}} \sin^n x \, dx = (n-1) \int_{0}^{\frac{\pi}{2}} \sin^{n-2} x \, dx \\ \implies \int_{0}^{\frac{\pi}{2}} \sin^n x \, dx = \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} \sin^{n-2} x \, dx
$$

Therefore:

**For even $n$, i.e., $n = 2k$:**

$$
\int_{0}^{\frac{\pi}{2}} \sin^{2k} x \, dx = \frac{2k-1}{2k} \int_{0}^{\frac{\pi}{2}} \sin^{2k-2} x \, dx \\ = \frac{2k-1}{2k} \cdot \frac{2k-3}{2k-2} \cdots \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \sin^0 x \, dx \\= \frac{2k-1}{2k} \cdot \frac{2k-3}{2k-2} \cdots \frac{1}{2} \cdot \frac{\pi}{2}
$$

**For odd $n$, i.e., $n = 2k + 1$:**

$$
\int_{0}^{\frac{\pi}{2}} \sin^{2k+1} x \, dx \\ = \frac{2k}{2k+1} \int_{0}^{\frac{\pi}{2}} \sin^{2k-1} x \, dx \\ = \frac{2k}{2k+1} \cdot \frac{2k-2}{2k-1} \cdots \frac{2}{3} \int_{0}^{\frac{\pi}{2}} \sin^1 x \, dx \\ = \frac{2k}{2k+1} \cdot \frac{2k-2}{2k-1} \cdots \frac{2}{3}
$$

Here, we need the double factorial:

$$
m!! = m (m-2) (m-4) \cdots
$$

Thus:

$$
\int_{0}^{\frac{\pi}{2}} \sin^{2k} x \, dx = \frac{(2k-1)!!}{(2k)!!} \cdot \frac{\pi}{2}
$$

$$
\int_{0}^{\frac{\pi}{2}} \sin^{2k+1} x \, dx = \frac{(2k)!!}{(2k+1)!!}
$$

Since $\sin^{2k+1} x < \sin^{2k} x < \sin^{2k-1} x$ for $x \in (0, \frac{\pi}{2})$, we have:

$$
\int_{0}^{\frac{\pi}{2}} \sin^{2k+1} x \, dx < \int_{0}^{\frac{\pi}{2}} \sin^{2k} x \, dx < \int_{0}^{\frac{\pi}{2}} \sin^{2k-1} x \, dx
$$

Substituting the above formulas:

$$
\frac{(2k)!!}{(2k+1)!!} < \frac{(2k-1)!!}{(2k)!!} \cdot \frac{\pi}{2} < \frac{(2k-2)!!}{(2k-1)!!}
$$

We can examine the difference between the bounds for $\int_{0}^{\frac{\pi}{2}} \sin^{2k} x \, dx$:

$$
\lim_{k \rightarrow \infty} \left( \frac{(2k-2)!!}{(2k-1)!!} - \frac{(2k)!!}{(2k+1)!!} \right) \\ = \lim_{k \rightarrow \infty} \left( \frac{(2k-2)!!}{(2k-1)!!} - \frac{2k}{2k+1} \cdot \frac{(2k-2)!!}{(2k-1)!!} \right) \\ = \lim_{k \rightarrow \infty} \frac{1}{2k+1} \cdot \frac{(2k-2)!!}{(2k-1)!!} = 0
$$

Note: Here $\frac{2k-2}{2k-1} < 1$

For the inequality:

$$
\frac{(2k)!!}{(2k+1)!!} < \frac{(2k-1)!!}{(2k)!!} \cdot \frac{\pi}{2} < \frac{(2k-2)!!}{(2k-1)!!}
$$

Multiply all terms by $\frac{(2k)!!}{(2k-1)!!}$:

$$
\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} < \frac{\pi}{2} < \frac{(2k-2)!! (2k)!!}{(2k-1)!!^2}
$$

We can further examine the difference between the bounds for $\frac{\pi}{2}$:

$$
\lim_{k \rightarrow \infty} \left( \frac{(2k-2)!! (2k)!!}{(2k-1)!!^2} - \frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} \right) \\= \lim_{k \rightarrow \infty} \left( \frac{1}{2k} - \frac{1}{2k+1} \right) \cdot \frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} \\= \lim_{k \rightarrow \infty} \frac{1}{2k} \cdot \frac{(2k)!!^2}{(2k+1)!!^2} = 0
$$

Thus, we obtain **Wallis's Formula**:

$$
\lim_{k \rightarrow \infty} \frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \frac{\pi}{2}
$$

**This formula introduces Wallis's result in preparation for deriving Stirling's formula.**

## Stirling's Formula

Stirling's formula reveals a relationship between $n!$, $n^n$, and $e^n$.

> As $n$ approaches infinity, $n!$ is approximately $\sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$.

First, introduce a limit:

$$
\lim_{n \rightarrow \infty} \left( \frac{n!}{n^n} \right)^{\frac{1}{n}} = \frac{1}{e}
$$

Proof:

$$
\lim_{n \rightarrow \infty} \left( \frac{n!}{n^n} \right)^{\frac{1}{n}} = \lim_{n \rightarrow \infty} e^{\frac{1}{n} \ln \left( \frac{n!}{n^n} \right)}
$$

which is equivalent to proving:

$$
\lim_{n \rightarrow \infty} \frac{1}{n} \ln \left( \frac{n!}{n^n} \right) = -1
$$

We have:

$$
\lim_{n \rightarrow \infty} \frac{1}{n} \ln \left( \frac{n!}{n^n} \right) \\ = \lim_{n \rightarrow \infty} \frac{1}{n} \left( \ln \frac{1}{n} + \ln \frac{2}{n} + \cdots + \ln \frac{n}{n} \right) \\ = \int_{0}^{1} \ln x \, dx = (x \ln x - x) \Big|_{0}^{1} = -1
$$

QED.

Thus,

$$
\frac{n!}{n^n} \sim \frac{1}{e^n}
$$

We would like to verify:

$$
\lim_{n \rightarrow \infty} \frac{n!}{n^n} e^n = 1
$$

Let’s test this.

**Using the ratio test:**

$$
a_n = \frac{n!}{n^n} e^n, \quad a_{n+1} = \frac{(n+1)!}{(n+1)^{n+1}} e^{n+1}
$$

$$
\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \frac{e}{\left(1 + \frac{1}{n}\right)^n} > 1
$$

Although the limit of the expression is $e$, it is always approaching $e$ from below, so it is still greater than 1, indicating that $a_n$ is divergent.

Mathematicians have attempted various approaches and found:

$$
a_n = \frac{n!}{n^{n + \frac{1}{2}}} e^n
$$

To prove that $a_n$ converges, use the **ratio test** again:

$$
a_n = \frac{n!}{n^{n + \frac{1}{2}}} e^n, \quad a_{n+1} = \frac{(n+1)!}{(n+1)^{n + \frac{3}{2}}} e^{n+1}
$$

$$
\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \frac{e}{\left(1 + \frac{1}{n}\right)^{n + \frac{1}{2}}}
$$

Compare this with:

$$
\left(1 + \frac{1}{n}\right)^{n + \frac{1}{2}} = e^{(n + \frac{1}{2}) \ln \left(1 + \frac{1}{n}\right)}
$$

and $e$.

Thus, compare:

$$
(n + \frac{1}{2}) \ln \left(1 + \frac{1}{n}\right)
$$

with 1.

Specifically, compare $\ln \frac{n+1}{n}$ with $\frac{1}{n + \frac{1}{2}}$:

Since:

$$
\frac{1}{n + 1} < \ln \left(1 + \frac{1}{n}\right) < \frac{1}{n}
$$

**Note:** The right side follows from the inequality $\ln(1 + x) < x$ (for $x \neq 0$).

**The left side follows from:** $1 - \frac{n}{n+1} < \ln \left(\frac{n+1}{n}\right)$, where $\frac{n}{n+1}$ is considered as a whole. This is high school knowledge.

Because $\frac{1}{x}$ is a concave function:

$$
\int_{n}^{n+1} \frac{1}{x} \, dx = \ln \frac{n+1}{n} > \frac{1}{n + \frac{1}{2}}
$$

Therefore:

$$
\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \frac{e}{\left(1 + \frac{1}{n}\right)^{n + \frac{1}{2}}} < 1
$$

Thus, $a_n$ is monotonically decreasing and $a_n > 0$, so it must converge, i.e., it has a limit.

Let:

$$
\lim_{n \rightarrow \infty} \frac{n!}{n^{n + \frac{1}{2}}} e^n = a
$$

Then, we need to find this limit.

From the earlier Wallis formula:

$$
\lim_{k \rightarrow \infty} \frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \frac{\pi}{2}
$$

$$
(2n)!! = 2n (2n-2) \cdots 2 = 2^n n!
$$

$$
(2n-1)!! = \frac{(2n)!}{(2n)!!} = \frac{(2n)!}{2^n n!}
$$

$$
(2n+1)!! = \frac{(2n+2)!}{(2n+2)!!} = \frac{(2n+2)!}{2^{n+1} (n+1)!}
$$

Thus, Wallis' formula becomes:

$$
\lim_{n \rightarrow \infty} \frac{16^n (n!)^4 (2n+2)}{(2n)!(2n+2)!} = \frac{\pi}{2}
$$

Using:

$$
n! \sim a e^{-n} n^{n + \frac{1}{2}}
$$

Substitute into:

$$
\lim_{n \rightarrow \infty} \frac{a^2 e^2}{4 \left(\frac{n+1}{n}\right)^{2n + \frac{3}{2}}} = \frac{\pi}{2}
$$

Solve for:

$$
a = \sqrt{2 \pi}
$$

Thus, as $n$ approaches infinity:

$$
n! \sim \sqrt{2 \pi} e^{-n} n^{n + \frac{1}{2}}
$$

or equivalently:

$$
n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n
$$


Stirling's formula finds a relationship of n!,nⁿ,eⁿ while n trends to infinity.

The Stirling Formula


## Content

The Box-Muller transform is a method used to generate normally distributed random variables from uniformly distributed random variables.

Specifically, it involves selecting two random variables, $U_1$ and $U_2$, which are uniformly distributed over the interval $[0,1]$. The random variables $X$ and $Y$ are constructed as follows:

$$
X = \cos(2\pi U_1) \sqrt{-2\ln U_2}
$$

$$
Y = \sin(2\pi U_1) \sqrt{-2\ln U_2}
$$

Then, $X$ and $Y$ follow a normal distribution with a mean of 0 and a variance of 1.

## Derivation

Assume that $X$ and $Y$ are independent, normally distributed random variables with a mean of 0 and a variance of 1. Let $p(X)$ and $p(Y)$ be their probability density functions. Thus, we have:

$$
p(X) = \frac{1}{\sqrt{2\pi}} e^{-\frac{X^2}{2}}, \\ p(Y) = \frac{1}{\sqrt{2\pi}} e^{-\frac{Y^2}{2}}
$$

Since $X$ and $Y$ are independent, their joint probability density function is:

$$
p(X,Y) = \frac{1}{2\pi} e^{-\frac{X^2 + Y^2}{2}}
$$

We now perform a coordinate transformation by defining:

$$
X = R \cos(\theta), \\ Y = R \sin(\theta)
$$

Thus, the joint distribution in polar coordinates becomes:

$$
\int_{0}^{\infty} \int_{0}^{2\pi} \frac{1}{2\pi} e^{-\frac{R^2}{2}} R \, d\theta \, dR = 1
$$

From this, we can derive the distribution functions for $R$ and $\theta$, denoted as $P_R$ and $P_\theta$:

$$
P_R(R \leq r) = \\ \int_{0}^{r} \int_{0}^{2\pi} \frac{1}{2\pi} e^{-\frac{R^2}{2}} R \, d\theta \, dR = 1 - e^{-\frac{r^2}{2}}
$$

$$
P_\theta(\theta \leq \phi) = \\ \int_{0}^{\phi} \int_{0}^{\infty} \frac{1}{2\pi} e^{-\frac{R^2}{2}} R \, dR \, d\theta = \frac{\phi}{2\pi}
$$

It is evident that $\theta$ follows a uniform distribution over $[0, 2\pi]$. Let

$$
F_R = 1 - e^{-\frac{r^2}{2}}
$$

The inverse function is given by:

$$
R = F_R^{-1}(z) \\ = \sqrt{-2 \ln(1 - z)}
$$

When $z$ is uniformly distributed over $[0,1]$, the distribution function of $R$ is $F_R(r)$. Therefore, we can select two random variables $U_1$ and $U_2$, uniformly distributed over $[0,1]$, such that:

$$
\theta = 2\pi U_1, \\ 1 - z = U_2, \\ R = \sqrt{-2 \ln U_2}
$$

Substituting these into the earlier expressions, we obtain:

$$
X = R \cos(\theta), \\ Y = R \sin(\theta)
$$

Thus, the original expressions for $X$ and $Y$ are recovered, where they follow a normal distribution with mean 0 and variance 1.

## Verification

To generate a random variable $X \sim N(0,1)$ following a standard normal distribution, we note that the inverse of the error function $\mathrm{erf}^{-1}$ is not elementary. Using a series expansion to approximate this inverse can introduce significant computational error.

Elementary functions, on the other hand, have well-established high-precision algorithms. Thus, we aim to construct a normally distributed random variable from a uniformly distributed one using elementary functions.

Let $U, V \in (0,1)$ be uniformly distributed random variables. Consider the following transformations:

$$
X = \sqrt{-2\ln U} \cos(2\pi V), \\ Y = \sqrt{-2\ln U} \sin(2\pi V)
$$

Next, we verify that $X$ and $Y$ are independent and follow a standard normal distribution.

Define:

$$
R = \sqrt{X^2 + Y^2}, \\ \Theta = \arctan\left(\frac{Y}{X}\right)
$$

Note that $\Theta = 2\pi V$ and $R = \sqrt{-2\ln U}$. We now show that $R$ and $\Theta$, under the polar coordinate transformation, are independent and that they yield standard normal random variables.

Consider the probability density function of one variable when the other

is fixed. The probability density function of $\Theta$ is:

$$
P_\Theta(\Theta) = \frac{1}{2\pi}
$$

which is straightforward. For $R$, the cumulative distribution function is:

$$
P(R \leq R_0) = \\ P\left(U \leq \exp\left(\frac{-R_0^2}{2}\right)\right)
$$

Thus, the probability density function of $R$ is:

$$
P_R(R) = R e^{-\frac{R^2}{2}}
$$

Hence, the joint probability density of $R$ and $\Theta$ is:

$$
P(R, \Theta) = \frac{1}{2\pi} R e^{-\frac{R^2}{2}}
$$

Since the polar coordinates yield:

$$
R \, dR \, d\Theta = dX \, dY
$$

we obtain the joint probability density of $X$ and $Y$:

$$
P(X,Y) = \frac{1}{2\pi} e^{-\frac{X^2 + Y^2}{2}} \\ = \left(\frac{1}{\sqrt{2\pi}} e^{-\frac{X^2}{2}}\right) \left(\frac{1}{\sqrt{2\pi}} e^{-\frac{Y^2}{2}}\right)
$$

This is precisely the probability density function of two independent, normally distributed random variables $X$ and $Y$, each following a standard normal distribution.

## Code Implementation

```typescript
function randInterval(): number {
  return Math.random();
}

function boxMuller(mu: number, sigma: number): [number, number] {
  const u = randInterval();
  const v = randInterval();
  const x = Math.cos(2 * Math.PI * u) * Math.sqrt(-2 * Math.log(v));
  const y = Math.sin(2 * Math.PI * u) * Math.sqrt(-2 * Math.log(v));
  return [x * sigma + mu, y * sigma + mu];
}

// Usage example
const [value1, value2] = boxMuller(0, 1);
console.log(value1, value2);
// Outputs two normally distributed random numbers
```

To verify that the `boxMuller` function generates numbers that follow a normal distribution, one can generate a large set of random numbers and plot a histogram to compare with a standard normal distribution. You can use libraries such as `plotly` to visualize the histogram and evaluate the output.

Here is a test program that generates a dataset using the `boxMuller` function and performs basic statistical tests, along with plotting the histogram. Assuming you are running in a Node.js environment, you can use `plotly-nodejs` or directly use `plotly.js` in the browser to generate the graph.

First, install the required dependencies:

```bash
npm i plotly.js-dist
```

Then, execute the following code:

```typescript
import * as fs from "fs";
import * as plotly from "plotly.js-dist";

// Generate random numbers following a normal distribution using Box-Muller
function randInterval(): number {
  return Math.random();
}

function boxMuller(mu: number, sigma: number): [number, number] {
  const u = randInterval();
  const v = randInterval();
  const x = Math.cos(2 * Math.PI * u) * Math.sqrt(-2 * Math.log(v));
  const y = Math.sin(2 * Math.PI * u) * Math.sqrt(-2 * Math.log(v));
  return [x * sigma + mu, y * sigma + mu];
}

// Generate n normally distributed random numbers
function generateNormalDistribution(
  mu: number,
  sigma: number,
  n: number
): number[] {
  const values: number[] = [];
  for (let i = 0; i < n / 2; i++) {
    const [value1, value2] = boxMuller(mu, sigma);
    values.push(value1, value2);
  }
  return values;
}

// Generate the data
const mu = 0;
const sigma = 1;
const sampleSize = 10000; // Sample size
const values = generateNormalDistribution(mu, sigma, sampleSize);

// Calculate the mean and variance of the generated random numbers
const mean = values.reduce((acc, val) => acc + val, 0) / values.length;
const variance =
  values.reduce((acc, val) => acc + (val - mean) ** 2, 0) / values.length;

console.log(`Mean: ${mean}`);
console.log(`Variance: ${variance}`);

// Plot the histogram
const trace = {
  x: values,
  type: "histogram",
  xbins: {
    size: 0.1, // Bin size for the histogram
  },
  marker: {
    color: "blue",
  },
  opacity: 0.7,
  name: "Box-Muller Distribution",
};

const data = [trace];

const layout = {
  title: "Generated Normal Distribution",
  xaxis: { title: "Value" },
  yaxis: { title: "Frequency" },
  bargap: 0.05,
};

const graphOptions = {
  filename: "box-muller-distribution",
  fileopt: "overwrite",
};
fs.writeFileSync("box-muller.html", plotly.plot(data, layout, graphOptions));

console.log("Box-Muller distribution data saved as box-muller.html.");
```

This program uses the `boxMuller` function to generate 10,000 random numbers that follow a normal distribution. It calculates the mean and variance of the generated dataset. The random numbers are visualized using a histogram, allowing for a direct comparison with a standard normal distribution. The histogram is saved as a file named `box-muller.html`, which can be opened in a browser for viewing.


The Box-Muller transformation is a method to construct a random variable that follows a Gaussian distribution by using a random variable that follows a uniform distribution.

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