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

I'm about to give

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。

all at once. But instead, forget that you already know anything about addition

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，

but you shouldn't think about this infinite sum when trying to make sense of it.

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

circle with radius one

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。