個人翻譯的betterexplained上的部分文章

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如題網站是 https://betterexplained.com/
樓主是閑得沒事干的高三學生，在不久以前發現了這個奇妙的數學網站，感覺這個網站上的核心思想非常妙，所以想翻譯一下上面的部分文章。不過我個人英語和數學水平都非常有限，所以譯文連同原文一起放出來，要是有哪里翻譯得不對請務必指正！

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1l備用，

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首先是這篇，我非常喜歡里面的比喻:
Developing Your Intuition For Math
建立數學直觀

Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?
我們對一個想法最初的接觸塑造了我們的直覺。我們的直覺影響了我們對一個事物的喜愛程度。這是在指什么呢?

Suppose we want to define a “cat”:
假設我們想給"貓"下個定義:

Caveman definition: A furry animal with claws, teeth, a tail, 4 legs, that purrs when happy and hisses when angry…
原始人定義:有爪子，牙齒，一根尾巴和四條腿，高興時咕嚕咕嚕，生氣時滋滋滋的毛茸茸動物。
Evolutionary definition: Mammalian descendants of a certain species (F. catus), sharing certain characteristics…
進步的定義:某種特定哺乳動物的后代(學名:家貓)，有著共通的特征……
Modern definition: You call those definitions? Cats are animals sharing the following DNA: ACATACATACATACAT…
現代定義:你管這些叫定義?貓是著以下DNA序列的動物:ACATACATACATACAT……



The modern definition is precise, sure. But is it the best? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.
現代的定義當然很精確。但它是最好的嗎?你是這樣教小孩理解這個單詞的嗎?它真的讓我們理解了這種動物的本質嗎?現代定義很有用，但前提是理解了貓究竟是什么。所以我們不應該從這里開始。
Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.
很不幸地，數學的思維模式像DNA序列一樣。我們學的是現代的，嚴格精準的定義，而不是逐漸深入的理解。我們只剩下了神秘的公式(像DNA一樣）卻沒有對它的理解。

Let’s approach ideas from a different angle. I imagine a circle: the center is the idea you’re studying, and along the outside are the facts describing it. We start in one corner, with one fact or insight, and work our way around to develop our understanding. Cats have common physical traits leads to Cats have a common ancestor leads to A species can be identified by certain portions of DNA. Aha! I can see how the modern definition evolved from the caveman one.
讓我們從另一個角度看問題吧。想象一個圓:正中間是你正在學習的東西，周圍環繞著對它的描述。我們從某處事實或理解開始，找到方向來更新我們的認知。貓有相同的特征→貓有共同的祖先→特定的DNA片段可以用來識別一個物種。好！我已經看到了定義的演化——從原始人的到現代的。

But not all starting points are equal. The right perspective makes math click — and the mathematical “cavemen” who first found an idea often had an enlightening viewpoint. Let’s learn how to build our intuition.
但并非所有開始的地方都是平等的。好的角度可以使數學豁然開朗——而且數學式的原始人觀點通常都非常有啟發作用。讓我們開始吧。

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What is a Circle?
什么是圓?
Time for a math example: How do you define a circle?
數學例子來咯:你是怎么定義圓的?
Definitions of a circle

There are seemingly countless definitions. Here’s a few:
似乎有無窮多的定義。以下列出一些:

The most symmetric 2-d shape possible
高度對稱的平面圖形
The shape that gets the most area for the least perimeter (see the isoperimeter property)
最小周長能圍出最大面積的圖形(等周長圖形中)
All points in a plane the same distance from a given point (drawn with a compass, or a pencil on a string)
對給定的某點，所有到該點距離相等的點的集合(可以用圓規或者套著繩結的筆畫)
The points (x,y) in the equation x2 + y2 = r2 (analytic version of the geometric definition above)
滿足方程的點的集合(上面定義的解析幾何版)
The points in the equation r * cos(t), r * sin(t), for all t (really analytic version)
(參數方程)
The shape whose tangent line is always perpendicular to the position vector (physical interpretation)
切線總是垂直于矢徑的圖形(物理定義)
The list goes on, but here’s the key: the facts all describe the same idea! It’s like saying 1, one, uno, eins, “the solution to 2x + 3 = 5″ or “the number of noses on your face” — just different names for the idea of “unity”.
還有很多，但關鍵在此:所有的事都有一個共通的本質！和1，uno(意大利語)，eins(德語)
，方程2x＋3=5的解，你臉上鼻子的數量，都是一樣的——皆為同一事物的不同名稱。
But these initial descriptions are important — they shape our intuition. Because we see circles in the real world before the classroom, we understand their “roundness”. No matter what fancy equation we see (x2 + y2 = r2), we know deep inside that a circle is “round”. If we graphed that equation and it appeared square, or lopsided, we’d know there was a mistake.
但這些最初的描述很重要——它們塑造了我們的直覺。因為我們在上學前就見過真實的圓，我們就能理解"圓"。我們無論看見多奇特的方程，都能理解它的本質:圓是"○"。如果我們畫出這個方程的圖形，卻發現它是方的或者不對稱的，我們就知道它不是圓。

As children, we learn the “caveman” definition of a circle (a really round thing), which gives us a comfortable intuition. We can see that every point on our “round thing” is the same distance from the center. x2 + y2 = r2 is the analytic way of expressing that fact, using the Pythagorean theorem for distance. We started in one corner, with our intuition, and worked our way around to the formal definition.
從小以來，我們就學到了原始人定義的圓(非常圓的東西)，這給了我們非常美妙的直觀。我們能看到，我們"非常圓的東西"上的所有點到圓心的距離都相等。則是個用了勾股定理來轉述事實的解析幾何定義。我們從我們的直觀開始，走向了嚴格的定義。

Other ideas aren’t so lucky. Do we instinctively see the growth of e, or is it an abstract definition? Do we realize the rotation of i, or is it an artificial, useless idea?
但是其他的概念就沒有這么好了。我們能直接看到e是"增長"，又或者它只是個抽象的概念?我們理解了i是"旋轉"，又或者它只是個假的，沒用的概念?

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A Strategy For Developing Insight
建立理解的方法

I still have to remind myself about the deeper meaning of e and i — which seems as absurd as “remembering” that a circle is round or what a cat looks like! It should be the natural insight we start with.
我仍然會提醒我自己e和i的深層含義——但這有點像提醒自己圓是圓的或貓長什么樣一樣荒唐。這本該是我們開始的地方。

Missing the big picture drives me crazy: math is about ideas — formulas are just a way to express them. Once the central concept is clear, the equations snap into place. Here’s a strategy that has helped me:
我看不清整體就會瘋(這句不會，丟人.jpg)。數學理應是關于想法的——公式只是表現它們的一種途徑。只要核心概念清楚了，方程就成型了。以下是一種曾幫助過我的方法:

Step 1: Find the central theme of a math concept. This can be difficult, but try starting with its history. Where was the idea first used? What was the discoverer doing? This use may be different from our modern interpretation and application.
1.找到數學概念的本質。可能有點難，但也可以試試從歷史上的發展開始。該概念最初被用來做什么?提出者做了什么?注意，這用途可能跟我們現在的定義和應用不同。
Step 2: Explain a property/fact using the theme. Use the theme to make an analogy to the formal definition. If you’re lucky, you can translate the math equation (x2 + y2 = r2) into a plain-english statement (“All points the same distance from the center”).
2.試著以各種形式解釋本質。可以用類比。
Step 3: Explore related properties using the same theme. Once you have an analogy or interpretation that works, see if it applies to other properties. Sometimes it will, sometimes it won’t (and you’ll need a new insight), but you’d be surprised what you can discover.
3.用你發現的本質來探索相關性質。一旦你有一個有效的類比或解釋，看看它是否適用于其他性質。有時它會奏效，有時不會（你需要新的見解），但你會對你發現的東西驚訝。
Let’s try it out.
來試試。

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A Real Example: Understanding e
實例:理解e
Understanding the number e has been a major battle. e appears all of science, and has numerous definitions, yet rarely clicks in a natural way. Let’s build some insight around this idea. The following section will have several equations, which are simply ways to describe ideas. Even if the equation is gibberish, there’s a plain-english idea behind it.
理解數字e一直以來都是困難的。e出現在科學的各個分支中，有無數個定義，但卻很少以一種自然的方式出現。讓我們來建立對它的直觀。下面的部分有幾個等式，可以用來簡單描述一些概念。哪怕等式看上去像胡扯，我們仍然能發現它背后的簡明理解。

Here’s a few popular definitions of e:
一些e的定義(圖在后面):
The first step is to find a theme. Looking at e’s history, it seems it has something to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.
首先是找到實質。看看e的歷史，它總是和增長與利率有關。e是在做商業計算時被發現的，而不是抽象的數學概念所以利率，或稱增長，也許是它的核心。

Let’s look at the first definition, in the upper left. The key jump, for me, was to realize how much this looked like the formula for compound interest. In fact, it is the interest formula when you compound 100% interest for 1 unit of time, compounding as fast as possible.
首先看看它最初的定義。在這個圖的左上角。對我而言，關鍵是看出這個東西與復利計算公式有幾分相似。實際上，這就是你以100％的利率在單位時間內獲得的利息公式。

Definition 1: Define e as 100% compound growth at the smallest increment possible.
定義1:(我沒法描述這句，丟人.jpg→是山寨gpt給的翻譯)將e定義為在最小增量上以100%的復利增長。

Let’s look at the second definition: an infinite series of terms, getting smaller and smaller. What could this be?
看看第二個定義:無窮的多項式，而且越來越小。這又是個什么?

\displaystyle{e = {1 \over 0!} + {1 \over 1!} + {1 \over 2!} + {1 \over 3!} + \cdots}

After noodling this over using the theme of “interest” we see this definitions shows the components of compound interest. Now, insights don’t come instantly — this insight might strike after brainstorming “What could 1 + 1 + 1/2 + 1/6 + …” represent when talking about growth?”
用之前我們找到的"利率"的概念來艱難地理解這個，它顯示了這概念的組成。但是靈感還沒有來。也許在頭腦風暴后就會來了:"談論增長時，1 + 1 + 1/2 + 1/6 + …代表什么?"

Well, the first term (1 = 1/0!, remembering that 0! is 1) is your principal, the original amount. The next term (1 = 1/1!) is the “direct” interest you earned — 100% of 1. The next term (0.5 = 1/2!) is the amount of money your interest made (“2nd level interest”). The following term (.1666 = 1/3!) is your “3rd-level interest” — how much money your interest’s interest earned!
好吧，首項(0！=1)是你的本金。第二項是你"直接"的利息。100％的1。第三項是你的利息的利息。第四項是你利息的利息！

Money earns money, which earns money, which earns money, and so on — the sequence separates out these contributions.There’s much more to say, but that’s the “growth-focused” understanding of that idea.
錢生錢，錢生錢生錢——這個多項式展現了它們各自的貢獻。這里還能作文章，但是這個概念的核心理解就是"增長"。

Definition 2: Define e by the contributions each piece of interest makes
定義2:e是每一份利息做出的貢獻。
Neato.
好。

Now to the 3rd, and shortest definition. What does it mean? Instead of thinking “derivative” (which turns your brain into equation-crunching mode), think about what it means. The feeling of the equation. Make it your friend.
現在是第三個，最短的定義。它是什么?別再想導數了，你的大腦會進入方程計算模式。想想它的意義。方程的感覺。讓它成為你的朋友。

\displaystyle{\fracvq7ohtf{dx}Blah = Blah}

It’s the calculus way of saying “Your rate of growth is equal to your current amount”. Well, growing at your current amount would be a 100% interest rate, right? And by always growing it means you are always calculating interest — it’s another way of describing continuously compound interest!
它以微積分的形式講述了"你的增長速率等于你現在的量"。以你現在的量增長，當然就是100％的利率吧?而且它永遠在增長，你就得永遠計算利息——另一種描述復利的方式！

Definition 3: Define e as a function that always grows at 100% of your current value
定義3:定義e為一個永遠以你當前的量翻倍的方式增長的函數
Nice — e is the number where you’re always growing by exactly your current amount (100%), not 1% or 200%.
好——e是精確描述你以當前的量的100％增長的數字，不是1％也不是200％。

Time for the last definition — it’s a tricky one. Here’s my interpretation: Instead of describing how much you grew, why not say how long it took?
來到最后一個定義——這個很麻煩。這是我的理解:為什么不用增長的時間(次數）來替代增長量?

If you’re at 1 and growing at 100%, it takes 1 unit of time to get from 1 to 2. But once you’re at 2, and growing 100%, it means you’re growing at 2 units per unit time! So it only takes 1/2 unit of time to go from 2 to 3. Going from 3 to 4 only takes 1/3 unit of time, and so on.
你從1開始，增長率100％，需要一次增長才能到2。但一旦你增長到2，你就能每次增長2！所以只需要二分之一的次數增長到3。從3到4只需要三分之一的次數，以此類推。

The time needed to grow from 1 to A is the time from 1 to 2, 2 to 3, 3 to 4… and so on, until you get to A. The first definition defines the natural log (ln) as shorthand for this “time to grow” computation.
從1到A的次數是從1到2，2到3，3到4…直到增長到A。第一個定義就將自然對數㏑以縮寫的方式表示了增長次數。

ln(a) is simply the time to grow from 1 to a. We then say that “e” is the number that takes exactly 1 unit of time to grow to. Said another way, e is is the amount of growth after waiting exactly 1 unit of time!
lnA正是從1增長到A所需的次數。我們就可以說，e是一次增長會增長出的量。換句話說，e就是一次增長的數量的極限！

Definition 4: Define the time needed to grow continuously from 1 to a as ln(a). e is the amount of growth you have after 1 unit of time.
定義4:(和上面的表述一樣，偷個懶)
Whablamo! These are four different ways to describe the mysterious e. Once we have the core idea (“e is about 100% continuous growth”), the crazy equations snap into place — it’s possible to translate calculus into English. Math is about ideas!
哇，四種方法描述神秘的e！只要我們掌握了核心觀點，瘋狂的方程式就原型畢露了——可以將微積分翻譯成人話。數學是有關思想的！

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What’s the Moral?
意義何在?

In math class, we often start with the last, most complex idea. It’s no wonder we’re confused — we’re showing DNA and expecting students to see the cat.
數學課上，我們通常從最新的，最復雜的定義出發。我們不懂也就正常了——我們展示DNA，卻想讓學生們看見貓。

I’ve learned a few lessons from this approach, and it underlies how I understand and explain math:
我以這種方法學了一些，它構成了我對數學的理解和解釋:

Search for insights and apply them. That first intuitive insight can help everything else snap into place. Start with a definition that makes sense and “walk around the circle” to find others.
找到理解，應用它們。最初的理解可以使一切東西原型畢露。從能理解的定義開始，繞個圈子來找到其它定義。
Develop mental toughness. Banging your head against an idea is no fun. If it doesn’t click, come at it from different angles. There’s another book, another article, another person who explains it in a way that makes sense to you.
精神要堅強。嗯瞪著一個概念沒有意思。如果沒有靈感，換個角度就是了。總有人能把你教懂的。
It’s ok to be visual. We think of math as rigid and analytic — but visual interpretations are ok! Do what develops your understanding.
可視化很好。我們把數學視作固定的，嚴格的——但直觀圖形也可以！不擇手段地更新你的理解。
Imaginary numbers were puzzling until their geometric interpretation came to light, decades after their initial discovery. Looking at equations all day didn’t help mathematicians “get” what they were about.
虛數很怪，直到它的幾何意義被發現，而且是在它被發明的幾十年后。永遠瞪著方程式對數學家們領會它們的意義無益。
Math becomes difficult when we emphasize definitions over understanding. Remember that the modern definition is the most advanced step of thought, not necessarily the starting point. Don’t be afraid to approach a concept from a funny angle — figure out the plain-English sentence behind the equation. Happy math.
我們強調定義重于理解時，數學就會變得很難。雖然現代定義是最精確的，但它沒必要是開始的地方。不要抗拒從比較怪的角度來理解數學。找到方程式背后的直觀語言。享受數學吧。

—TBC—

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今天這篇比較短，但是我非常喜歡。這篇翻譯完過后就翻譯點實戰演練內容居多的文章_(:з」∠)_
——
Why Do We Learn Math?
我們為什么要學數學?

I cringe when hearing "Math teaches you to think".
"數學教會你思考"這樣的說法讓我尷尬。


It's a well-meaning but ineffective appeal that only satisfies existing fans (see: "Reading takes you anywhere!"). What activity, from crossword puzzles to memorizing song lyrics, doesn't help you think?
這種號召聽起來很好，但是沒什么用，它只會使已經喜歡數學的人滿意。(比如:閱讀可以帶你去你想去的任何地方！)難道縱橫字謎和記歌詞就不能教會你思考了嗎?

Math seems different, and here's why: it's a specific, powerful vocabulary for ideas.
數學看起來不一樣，原因在于:它使用了特定的，有力的詞匯來描述想法。

Imagine a cook who only knows the terms "yummy" and "yucky". He makes a bad meal. What's wrong? Hrm. There's no way to describe it! Too mild? Salty? Sweet? Sour? Cold? These specific critiques become hazy variations of the "yucky" bucket. He probably wouldn't think "Needs more umami".
設想一個廚師只知道"好吃"和"難吃"兩個詞。他做了頓難吃的飯。哪里有問題呢?額，沒法描述。味道淡了?咸了?甜了?酸了?涼了?這些具體的問題變成了模糊的"難吃"。也許廚師根本不會想到"再加點味精。"

Words are handholds that latch onto thoughts. You (yes, you!) think with extreme mathematical sophistication. Your common-sense understanding of quantity includes concepts refined over millennia: base-10 notation, zero, decimals, negatives.
詞語是用來抓住思想的把手。你(對，就是你！)用高度復雜的數學思想想象。過去幾千年以來，人們對數量的定義被一次又一次重新改寫:10進制，0，小數，負數。

What we call "Math" are just the ideas we haven't yet internalized.
被我們稱之為"數學"的東西其實是還沒有被我們內化的事物。

Let's explore our idea of quantity. It's a funny notion, and some languages only have words for one, two and many. They never thought to subdivide "many", and you never thought to refer to your East and West hands.
讓我們探索一下數量的概念。這個概念比較奇特，有些語言只有表示1、2和許多的詞匯。他們從來沒有想過細分"許多"這個概念。你也沒有想過稱你的右手和左手為"東手""西手"(這句實在不會)

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Here's how we've refined quantity over the years:
數量的概念是這樣被改寫的:

引用

We have "number words" for each type of quantity ("one, two, three... five hundred seventy nine")

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各種各樣的數(1，2，3…579)

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The "number words" can be written with symbols, not regular letters, like lines in the sand. The unary (tally) system has a line for each object.

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數詞可以用符號來表示，就像沙灘上劃出的線。"正"字每一筆畫都代表著一份。

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Shortcuts exist for large counts (Roman numerals: V = five, X = ten, C = hundred)

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大的數字有縮寫(羅馬數字5，10，100)

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We even have a shortcut to represent emptiness: 0

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甚至"什么都沒有"也有縮寫。

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The position of a symbol is a shortcut for other numbers. 123 means 100 + 20 + 3.

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符號的位置也可以表示縮寫。123=100+20+3

Numbers can have incredibly small, fractional differences: 1.1, 1.01, 1.001...

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數字可以非常小，只有極細微的區別:1.1，1.01，1.001…

引用

Numbers can be negative, less than nothing (Wha?). This represents "opposite" or "reverse", e.g., negative height is underground, negative savings is debt.

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數字甚至可以是負的，比"什么都沒有"還小。(啥?)它表示"相反的"或者"顛倒的"。比如，負的高度是在地下，而負的存款是負債。

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Numbers can be 2-dimensional (or more). This isn't yet commonplace, so it's called "Math" (scary M).

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數字可以是二維或以上的。但它還不夠常見，所以尚且被稱作"數學"(真可怕)

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Numbers can be undetectably small, yet still not zero. This is also called "Math".

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數字可以無窮小，但并不是零。這也是"數學"。

174781

Our concept of numbers shapes our world. Why do ancient years go from BC to AD? We needed separate labels for "before" and "after", which weren't on a single scale.我們對數字的理解塑造了我們的世界。為什么古代是從bc到ad?我們需要給公元前和公元后分別貼上標簽，因為它們并不是同一個時間尺度上的。

Why did the stock market set prices in increments of 1/8 until 2000 AD? We were based on centuries-old systems. Ask a modern trader if they'd rather go back.
為什么股市在公元2000年以前都是以增量的八分之一設置價格?因為它基于一個古老的系統。去問問現在的交易員愿不愿意回到過去吧。

Why is the decimal system useful for categorization? You can always find room for a decimal between two other ones, and progressively classify an item (1, 1.3, 1.38, 1.386).
為什么小數對分類有用呢?因為你總可以在兩個數字間找到更小的數字，并據此分類(1，1.3，1.38，1.386)

Why do we accept the idea of a vacuum, empty space? Because you understand the notion of zero. (Maybe true vacuums don't exist, but you get the theory.)
我們為什么能理解"什么都沒有"的概念?因為你理解了零。也許真空并不存在，但不妨礙你理解它。

Why is anti-matter or anti-gravity palatable? Because you accept that positives could have negatives that act in opposite ways.
為什么反物質和反重力可以被我們理解?因為你理解了負數就是正數相對的那一面。

How could the universe come from nothing? Well, how can 0 be split into 1 and -1?
為什么宇宙可以從什么都沒有發展到今天這樣?呃，我們能把0分成1和-1嗎?

Our math vocabulary shapes what we're capable of thinking about. Multiplication and division, which eluded geniuses a few thousand years ago, are now homework for grade schoolers. All because we have better ways to think about numbers.
我們的數學用詞決定了我們能理解什么。乘和除曾困擾了幾千年前的天才們，現在卻只是小學生的家庭作業。這全都是因為我們有了能更好理解數字的方法。

We have decent knowledge of one noun: quantity. Imagine improving our vocabulary for structure, shape, change, and chance. (Oh, I mean, the important-sounding Algebra, Geometry, Calculus and Statistics.)
我們很清楚一個詞:數量。再設想我們理解了結構，形狀，改變和機會的詞(用聽上去很正式的話來說就是代數，幾何，微積分和統計學。)

Caveman Chef Og doesn't think he needs more than yummy/yucky. But you know it'd blow his mind, and his cooking, to understand sweet/sour/salty/spicy/tangy.
原始人廚師當然不會覺得他需要比好吃和難吃更具體的詞。但是如果他理解了甜、酸、咸、辣、味道重的意思，他的思想和廚藝都會爆炸式進步。

We're still cavemen when thinking about new ideas, and that's why we study math.
在思考新的概念時，我們仍是原始人。這就是我們學數學的原因。

—TBC—