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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

  • 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。

e to the pi i equals negative 1

e^(PI i) = -1

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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
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