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• e to the pi i equals negative 1

e^(PI i) = -1

• is one of the most famous equations in math, but it's also one of the most

是數學裏最有名的等式中的一個 ，然後它也是一個最

• confusing.

搞不清的。

• Those watching this video likely fall into one of three categories

那些觀看本視頻也許可以落在下面三個範疇中的一個：

• 1) you know what each term means, but the statement as a whole seems nonsensical,

1）你知道每個術語的意思，但作為一個整體的表達式似乎不可理解，

• 2) you were lucky enough to see what this means and some long formulae explaining why it

2）在一堂微積分課中，你是很幸運的來知道看到這是什麼意思

• works in a calculus class,

和一些很長的公式解釋著為什麽是這樣，

• but it still feels like black magic, or 3) it's not entirely clear with the

但它仍然感覺像黑魔法，或者 3）它不是與完全清楚

• terms themselves are.

這些項的本身是什麽。

• Those in this last category might be in the best position to understand the explanation

這些最後的範疇中可能是處於最好的地位來理解

我打算給出的解釋

• since it doesn't require any calculus or advanced math, but will instead require an

因為它不需要任何微積分或高等數學，而是將代之以

• open-minded to reframing how we think about numbers.

開放態度來重新考慮我們如何看待數字。

• Once we do this, it will become clear what the question means,

一旦我們做到這一點，它會變得清晰這問題意味著什麼，

• why it's true and most importantly why it makes intuitive sense.

為什麼它是對的，最重要的是為什麽它 在直覺上有道理。

• First let's get one thing straight,

首先讓我們把一件事澄清一下，

• what we we write as e to the x is not repeated multiplication

我們我們寫為E的X次方不是重複的乘法

• that would only make sense when x is a number that we can count 1, 2, 3 and so on,

這只會在x是一個數字，我們可以數1，2，3等，才可理解的那樣，

• and even then you'd have to define the number the number e first. To understand what this

即使如此，你不得不先來定義數字e

• function actually does,

來理解這個函數實際所做的，

• we first need to learn how to think about numbers as actions.

我們首先需要學會怎樣來把一些數字為考慮成一些行動。

• We are first taught to think about numbers as counting things, and addition and

最初教我們把數字想成數東西，而加法

• multiplication are thought of with respect counting.

和乘法都被當作數數目的。

• However is more thinking becomes tricky when we talk about fractional amounts,

不過我們在談到分數的時候，就越想越難了，

• very tricky when we talk about irrational amounts, and downright nonsensical

非常棘手，在我們談到無理數的時候，更難了，

• when we introduce things like the square root of -1.

而在我們引入像-1的平方根的時候，那就簡直無法理解了。

• Instead we should think of each number as simultaneously being three things

相反，我們應該把各個數字同時看作三件事

• a point on an infinitely extending line, an action which slides that line along itself,

上無限延伸線上的一個點，一個沿著那根綫自身的滑動，

• in which case we call it an "adder", and an action which stretches the line

在這種情況下，我們把它稱為“加法器”，它一直伸展這根線的

• in which case we call it a "multiplier". When you think about numbers as adders,

在這種情況下，我們把它叫做一個“乘法器”。當你想到數字作為加法器時，

• you could imagine adding it with all numbers as points on the line

你也許可以想像把所有的數字作爲點子立刻加到綫上。

但是不是這樣，

• so that we can reframe how you think about it.

為了我們就可以對它怎麽來思考，忘掉你已經知 道的加法。

• Thing of adders purely as sliding the line with the following rule:

把加法想成純粹是按以下規則在綫上的滑動：

• You slide until the point corresponding to zero ends up where the point corresponding

你滑著直到相對於零點的那個點

• with the adder itself started

對上自己開始的加法。

• When you successively applied two adders, the effect will be the same as just applying some other adder.

在你連續做了兩個加法，其效果將和做另一個加法一樣。

• This is how we define their sum. Likewise,

這是我們怎樣定義它們的和。同樣，

• forget the you already know anything about multiplication, and think of a

忘記你已經知道了的關於乘法並把

• multiplier purely as a way to stretch the line.

乘上的數純粹想成作為一種拉伸這根綫的方法。

• Now the rule is to fix zero in place, and bring the point corresponding with

此時的規則是把零點固定，並把相對應

• one, to where the point corresponding with the multiplier itself started off,

的乘數自己開始的點，

• keeping everything evenly spaced as you do so. Just as with adders

在你這樣做的時候，保持距離均等。就像你在做加法的時候一樣。

• we can now redefine multiplication as the successive application

我們現在可以重新定義乘法為連續使用

• of two different actions. The life's ambition of e to the x

兩種不同的行動。 E的x次方就是

• is to transform adders into multipliers, and to do so is naturally as possible.

是加法的數轉化為乘數，並盡可能自然地來做到。

• For instance, if you take two adders, successfully apply them,

舉例來說，如果你拿兩個加數，成功地把它們加起來，

• then pump the resulting sum through the function, it is the same as first putting

然後通過函數產生出和，它就和把各個

• each adder through the function separately,

加數分別通過這個函數是一樣的。

• then successively applying the two multipliers you get. More succinctly,

然而你把兩個乘數連乘。更簡潔了，

• e to the x plus y equals e to the x time e to the y.

e的（x加y）次方等於e的x次方乘以 e的y次方。

• If e to the x was thought it as repeated multiplication, this property

如果d的x次方被考慮從是重複著的乘法，

• would be a consequence.

這個性質會有一個後果。

• but really it goes the other way around. You should think this property is defining

但實際上它是反過來的。你應該把這個性質是定義著

• e to the x, and the fact that the special case and counting numbers has

e的x次方，并且事實上是特殊的情況下

• anything to do with repeated multiplication

計數的數字才和重復的乘法有關這這種

• is a consequence the property.

性質的一個結果。

• Multiple functions satisfy this property,

很多函數符合這個性質，

• but when you try to define one explicitly, one stands out as being the most natural,

但是如果你想明確地來定義一個， 一個作為最自然特別明顯的，

• and we express it with this infinite sum. By the way,

而我們我們以這一無限的和來表達它。順便提一下，

• the number e is just to find to be the value of this function at one.

數字e只不過是一個可以用這個函數來找到的一個數值。

• The numbers is not nearly as special as the function as a whole, and the convention to

和函數作爲一個整躰而言這些數字也沒有什麽特別的，

• write this function as e to the x

而通常把這個函數寫成e的x次方

• is a vestige of its relationship with repeated multiplication.

是和重複乘法的有關係的一種遺跡。

• The other less natural function satisfying this property

其他滿足這個屬性而不太自然的函數

• are the exponentials with different bases. Now the expression "e to the pi i"

是不同不同基數的指數（冪）。現在，表達式“e的π i次方”

• at least seems to have some meaning,

至少看來有一些意義，

但在你要想理解它的時候，你不應該把它想成這個無限項之和。

• You only need to think about turning adders into multipliers. You see,

你只需要來想把加數變成乘數。你知道，

• we can also play this game a sliding and stretching in the 2d plane,

在2-維平面我們也可以玩這個滑動和 拉伸的遊戲，

• and this is what complex numbers are. Each numbers simultaneously a point on

而這就是複數。每個數字同時是平面上

• the plane

的一個點。

• an adder, which slides the plane so that the point for 0

一個加數，它在該平面上滑動，因此零點

• lands on the point for the number, and multiplayer which fixes zero in place

停到那個數字上，而乘數把零點固定下來

• and brings the point for one to the point for the number while keeping

而把那個點帶到這個數字同時保持

• everything evenly spaced.

一切距離都均等。

• This can now include rotating along with some stretching and shrinking.

這現在可以包括沿的一些伸展和收縮轉動。

• All the actions of the real numbers still apply, sliding side to side and stretching,

實數的所有運算仍然適用，向傍邊的滑動和拉伸，

• but now we have a whole host of new actions.

但現在我們有一大堆新的運算。

• For instance, take this point here. We call it "i". As an adder,

舉例來說，在這裡拿了這個點。我們稱之為它的“i”。作為一個加數，

• it slides the plane up, and as a multiplier, it turns it a quarter of the way around

它在平面向上滑動，而作為一個乘數，它把它轉過四分之一的圓周。

• Since multiplying it by itself gives -1, which is to say

由於自身乘以-1相乘，這就是說

• applying this action twice is the same as the action of -1

應用此動作兩次是一樣的 的作用-1

• as a multiplier,

為乘數，

• it is the square root of -1. All adding is some combination of sliding sideways

它是-1的平方根。所有增加的是 側向滑動的一些組合

• and sliding up or down, and all multiplication is some combination of

並向上或向下滑動起來，而所有 乘法的某種組合

• stretching and rotating.

拉伸和旋轉。

• Since we already know that e to the x turns slide side to side into stretches,

既然我們已經知道，電子的 X變為滑動一邊到另一邊成伸展，

• the most natural thing you might expect his for to turn this new dimension of adders,

你所期望的最自然不過的事 他對把加法器這一新的概念，

• slides up and down, directly into the new dimension of multipliers,

上下滑動，直接進入新 乘法器的尺寸，

• rotations. In terms points on the plane, this would mean e to the x takes

旋轉。在上表明的觀點 面，在x採用這意味著è

• points on this vertical line

指出在這個垂直線

• which correspond to adders that slide the plane up and down, and puts them on the

這對應於滑動加法器 飛機上下，並把它們放在

圈半徑一個

• which corresponds with the multipliers that rotate the plane. The most natural way

其與乘法器對應 旋轉的飛機。最自然的方式

• you could imagine doing this

你能想像這樣做

• is to wrap the line around the circle without stretching your squishing it

是環繞了一圈線 沒有伸展你壓扁它

• which would mean it takes a length of two pi to go completely around the circle,

這將意味著它需要的長度 2 PI去完全繞了一圈，

• since by definition this is the ratio the circumference of a circle to its radius.

因為根據定義，這是比 圓到其半徑的圓周。

• This means going up pi would translate to going exactly half way around the circle.

這意味著往上走PI將轉化 兜兜轉轉一圈正好一半。

• When in doubt, if there's a natural way to do things,

如果有疑問，如果有一種自然的方式 做的事情，

• this is exactly what e to the x will do, and this case is no exception.

這正是e將X將 做的，這種情況下也不例外。

• If you want to see a full justification for why e to the x behaves this way,

如果你想看到一個完整的理由 為什麼e將X的行為這種方式，

• see this additional video here. So there you have it,

在這裡看到這個額外的視頻。因此，有 你擁有了它，

• this function e to the x take the adder pi i to the multiplayer -1.

此函數e於x取加法器PI i到多人-1。

e to the pi i equals negative 1

e^(PI i) = -1

B2 中高級 中文 美國腔 次方 函數 加法 數字 拉伸 伸展

# 理解e到pi i (Understanding e to the pi i)

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jeffrey 發佈於 2021 年 01 月 14 日