Euler's formula, e to the pi i equals negative 1.

根據歐拉公式，e 對 pi i 等於負 1。

As an anniversary of sorts, I want to revisit that same idea.

作為週年紀念，我想重溫一下同樣的想法。

For one thing, I've always wanted to improve on the presentation, but I wouldn't rehash an old topic if there wasn't something new to teach.

首先，我一直想改進演講，但如果沒有新東西可教，我也不會老調重彈。

The idea underlying that video was to take certain concepts from a field in math called group theory and show how they give Euler's formula a much richer interpretation than a mere association between numbers.

該視頻的基本思想是利用數學領域中一個名為群論的領域中的某些概念，展示它們是如何賦予歐拉公式比單純的數字之間的聯繫更豐富的解釋。

And two years ago, I thought it might be fun to use those ideas without referencing group theory itself or any of the technical terms within it.

兩年前，我覺得在不引用群論本身或其中任何專業術語的情況下使用這些想法可能會很有趣。

But I've come to see that you all actually quite like getting into the math itself, even if it takes some time.

但我發現，你們其實都很喜歡鑽研數學本身，即使這需要一些時間。

So here, two years later, let's you and me go through an introduction to the basics of group theory, building up to how Euler's formula comes to life under this light.

是以，兩年後的今天，我們將在這裡一起學習群論的基礎知識，瞭解歐拉公式是如何在群論的光照下栩栩如生地展現出來的。

If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.

如果你只想快速瞭解歐拉公式，如果你對矢量微積分很熟悉，我會在螢幕上顯示一個特別簡短的解釋，你可以停下來思考一下。

If it doesn't make sense, don't worry about it, it's not needed for where we're going.

如果沒道理，也不用擔心，我們要去的地方不需要它。

The reason I want to put out this group theory view, though, is not because I think it's a better explanation.

不過，我之所以想提出這種群體理論觀點，並不是因為我認為它是一種更好的解釋。

Heck, it's not even a complete proof, it's just an intuition really.

哎呀，這甚至不是一個完整的證明，它只是一種直覺。

It's because it has the chance to change how you think about numbers, and how you think about algebra.

因為它有機會改變你對數字的看法，改變你對代數的看法。

You see, group theory is all about studying the nature of symmetry.

你看，群論就是研究對稱性的本質。

For example, a square is a very symmetric shape, but what do we actually mean by that?

例如，正方形是一種非常對稱的形狀，但我們實際上是什麼意思呢？

One way to answer that is to ask about what are all the actions you can take on the square that leave it looking indistinguishable from how it started?

回答這個問題的一種方法是，你可以在廣場上採取哪些行動，讓廣場看起來與開始時沒有區別？

For example, you could rotate it 90 degrees counterclockwise, and it looks totally the same to how it started.

例如，你可以將它逆時針旋轉 90 度，它看起來和開始時完全一樣。

You could also flip it around this vertical line, and again, it still looks identical.

你也可以圍繞這條垂直線翻轉，同樣，看起來還是一模一樣。

In fact, the thing about such perfect symmetry is that it's hard to keep track of what action has actually been taken, so to help out I'm going to stick on an asymmetric image here.

事實上，如此完美的對稱性會讓人很難記住到底採取了什麼行動，所以為了方便起見，我要在這裡貼上一張不對稱的圖片。

We call each one of these actions a symmetry of the square, and all of the symmetries together make up a group of symmetries, or just group for short.

我們把其中的每一個動作都稱為正方形的對稱性，所有的對稱性加在一起就是一組對稱性，簡稱為 "組"。

This particular group consists of 8 symmetries.

這個特殊的組由 8 個對稱性組成。

There's the action of doing nothing, which is one that we count, plus 3 different rotations, and then there's 4 ways you can flip it over.

有什麼都不做的動作，這是我們計算的一個動作，再加上 3 種不同的旋轉，然後還有 4 種翻轉的方式。

And in fact, this group of 8 symmetries has a special name.

事實上，這 8 組對稱有一個特殊的名字。

It's called the dihedral group of order 8.

它被稱為 8 階二面群。

And that's an example of a finite group, consisting of only 8 actions, but a lot of other groups consist of infinitely many actions.

這是一個有限群組的例子，它只由 8 個動作組成，但很多其他群組都由無限多個動作組成。

Think of all possible rotations, for example, of any angle.

想想所有可能的旋轉，例如任何角度的旋轉。

Maybe you think of this as a group that acts on a circle, capturing all of the symmetries of that circle, that don't involve flipping it.

也許你可以把它想象成一個作用於圓的組，它捕捉到了圓的所有對稱性，而不涉及圓的翻轉。

Here, every action from this group of rotation lies somewhere on the infinite continuum between 0 and 2π radians.

在這裡，這組旋轉的每個動作都位於 0 和 2π 弧度之間的無限連續體上的某個位置。

One nice aspect of these actions is that we can associate each one of them with a single point on the circle itself, the thing being acted on.

這些動作的一個好處是，我們可以將每一個動作與圓本身上的一個點（即被動作的事物）聯繫起來。

You start by choosing some arbitrary point, maybe the one on the right here.

你可以先任意選擇一個點，也許就是右邊這個點。

Then every circle symmetry, every possible rotation, takes this marked point to some unique spot on the circle, and the action itself is completely determined by where it takes that spot.

然後，每一次圓的對稱，每一次可能的旋轉，都會把這個標記點帶到圓上的某個獨特位置，而動作本身則完全取決於它把這個點帶到哪裡。

This doesn't always happen with groups, but it's nice when it does, because it gives us a way to label the actions themselves, which can otherwise be pretty tricky to think about.

這種情況並不總是發生在群組中，但如果發生了就很好，因為它給我們提供了一種給行動本身貼標籤的方法，否則思考起來會非常棘手。

The study of groups is not just about what a particular set of symmetries is, whether that's the 8 symmetries of a square, the infinite continuum of symmetries of the circle, or anything else you dream up.

群的研究並不僅僅是關於一組特定的對稱性，無論是正方形的 8 個對稱性，還是圓的無限連續對稱性，或是你夢想的任何其他對稱性。

The real heart and soul of the study is knowing how these symmetries play with each other.

研究的真正核心和靈魂在於瞭解這些對稱性是如何相互影響的。

On the square, if I rotate 90 degrees and then flip around the vertical axis, the overall effect is the same as if I had just flipped over this diagonal line.

在正方形上，如果我旋轉 90 度，然後繞縱軸翻轉，整體效果與我在這條對角線上翻轉是一樣的。

So in some sense, that rotation plus the vertical flip equals that diagonal flip.

是以，從某種意義上說，旋轉加上垂直翻轉等於對角線翻轉。

On the circle, if I rotate 270 degrees and then follow it with a rotation of 120 degrees, the overall effect is the same as if I had just rotated 30 degrees to start with.

在圓形上，如果我旋轉 270 度，然後再旋轉 120 度，那麼整體效果與我一開始就旋轉 30 度是一樣的。

So in this circle group, a 270 degree rotation plus a 120 degree rotation equals a 30 degree rotation.

是以，在這個圓組中，270 度旋轉加上 120 度旋轉等於 30 度旋轉。

And in general, with any group, any collection of these sorts of symmetric actions, there's a kind of arithmetic where you can always take two actions and add them together to get a third one by applying one after the other.

一般來說，對於任何一組、任何一組對稱動作的集合，都有一種算術方法，你總是可以把兩個動作加在一起，通過一個接一個的應用，得到第三個動作。

Or maybe you think of it as multiplying actions, it doesn't really matter.

或者你可以把它看作是乘法行動，這並不重要。

The point is that there is some way to combine the two actions to get out another one.

問題的關鍵在於，有某種方法可以將這兩個動作結合起來，從而得出另一個動作。

That collection of underlying relations, all associations between pairs of actions and the single action that's equivalent to applying one after the other, that's really what makes a group a group.

基礎關係的集合、成對動作之間的所有關聯，以及相當於一個接一個應用的單個動作，這才是一個組之所以成為一個組的真正原因。

It's actually crazy how much of modern math is rooted in, well, this, in understanding how a collection of actions is organized by this relation, this relation between pairs of actions and the single action you get by composing them.

事實上，現代數學有很多內容都源於這種關係，即一對動作和一個動作之間的關係。

Groups are extremely general, a lot of different ideas can be framed in terms of symmetries and composing symmetries.

群是非常普遍的，很多不同的想法都可以用對稱和組成對稱來概括。

And maybe the most familiar example is numbers, just ordinary numbers.

最熟悉的例子可能就是數字，普通的數字。

And there are actually two separate ways to think about numbers as a group, one where composing actions is going to look like addition, and another where composing actions will look like multiplication.

實際上，有兩種不同的方法可以把數字看作一個組，一種是把動作組合起來，看起來就像加法，另一種是把動作組合起來，看起來就像乘法。

It's a little weird, because we don't usually think of numbers as actions, we usually think of them as counting things.

這有點奇怪，因為我們通常不會把數字當成行動，我們通常把數字當成數數的東西。

But let me show you what I mean.

不過，還是讓我來告訴你我的意思吧。

Think of all the ways you can slide a number line left or right along itself.

想一想，你能用什麼方法讓數線沿其自身向左或向右滑動？

This collection of all sliding actions is a group, what you might think of as the group of symmetries on an infinite line.

所有滑動動作的集合就是一個群，你可以把它想象成無限直線上的對稱群。

And in the same way that actions from the circle group could be associated with individual points on that circle, this is another one of those special groups where we can associate each action with a unique point on the thing it's actually acting on.

同樣，圓組中的操作可以與圓上的各個點相關聯，這也是另一個特殊的組，我們可以將每個操作與它實際作用對象上的一個獨特點相關聯。

So just follow where the point that starts at 0 ends up.

是以，只需跟蹤從 0 開始的點的最終位置即可。

For example, the number 3 is associated with the action of sliding right by 3 units.

例如，數字 3 與向右滑動 3 個組織、部門的動作有關。

The number –2 is associated with the action of sliding 2 units to the left, since that's the unique action that drags the point at 0 over to the point at –2.

數字 -2 與向左滑動 2 個組織、部門的動作有關，因為這是唯一能將 0 點拖到 -2 點的動作。

The number 0 itself, well, that's associated with the action of just doing nothing.

數字 "0 "本身與 "什麼都不做 "有關。

This group of sliding actions, each one of which is associated with a unique real number, has a special name, the additive group of real numbers.

這組滑動作用（每個滑動作用都與一個唯一的實數相關）有一個特殊的名稱，即實數加法群。

The reason the word additive is in there is because of what the group operation of applying one action followed by another looks like.

之所以用 "加法 "這個詞，是因為在應用一個動作後，又應用另一個動作，這樣的群體操作看起來像什麼。

If I slide right by 3 units, and then I slide right by 2 units, the overall effect is the same as if I slid right by 3 plus 2, or 5 units.

如果我向右滑動 3 個組織、部門，然後再向右滑動 2 個組織、部門，整體效果與我向右滑動 3 加 2 或 5 個組織、部門是一樣的。

Simple enough, we're just adding the distances of each slide, but the point here is that this gives an alternate view for what numbers even are.

很簡單，我們只是將每張幻燈片的距離相加，但重點是，這提供了另一種視角，讓我們瞭解數字到底是什麼。

They are one example in a much larger category of groups, groups of symmetries acting on some object, and the arithmetic of adding numbers is just one example of the arithmetic that any group of symmetries has within it.

數字加法運算只是任何一組對稱運算中的一個例子。

We could also extend this idea, instead asking about the sliding actions on the complex plane.

我們還可以擴展這一想法，詢問複平面上的滑動動作。

The newly introduced numbers i, 2i, 3i, and so on on this vertical line would all be associated with vertical sliding motions, since those are the actions that drag the point at 0 up to the relevant point on that vertical line.

在這條垂直線上，新引入的數字 i、2i、3i 等都與垂直滑動運動有關，因為這些都是將 0 點拖到垂直線上相關點的動作。

The point over here at 3 plus 2i would be associated with the action of sliding the plane in such a way that drags 0 up and to the right to that point, and it should make sense why we call this 3 plus 2i.

這裡的 3 加 2i 點與滑動平面的動作有關，滑動平面會將 0 向上、向右拖動到該點，這也就解釋了為什麼我們稱之為 3 加 2i。

That diagonal sliding action is the same as first sliding by 3 to the right, and then following it with a slide that corresponds to 2i, which is 2 units vertically.

這個對角線滑動的動作等同於先向右滑動 3，然後再滑動 2i，也就是垂直方向滑動 2 個組織、部門。

Similarly, let's get a feel for how composing any two of these actions generally breaks down.

同樣，讓我們來了解一下其中任何兩個動作一般是如何組成的。

Consider this slide by 3 plus 2i action, as well as this slide by 1 minus 3i action, and imagine applying one of them right after the other.

考慮一下這個滑動 3 加 2i 的動作，以及這個滑動 1 減 3i 的動作，想象一下緊接著應用其中一個動作的情景。

The overall effect, the composition of these two sliding actions, is the same as if we had slid 3 plus 1 to the right, and 2 minus 3 vertically.

這兩個滑動動作構成的整體效果，與我們向右滑動 3 加 1 和垂直滑動 2 減 3 的效果相同。

Notice how that involves adding together each component.

請注意，這需要將每個部分相加。

So composing sliding actions is another way to think about what adding complex numbers actually means.

是以，組合滑動操作是思考複數加法實際含義的另一種方法。

This collection of all sliding actions on the 2D complex plane goes by the name the additive group of complex numbers.

二維複數平面上所有滑動作用的集合被稱為複數的加法群。

Again, the upshot here is that numbers, even complex numbers, are just one example of a group, and the idea of addition can be thought of in terms of successively applying actions.

同樣，這裡的結果是，數字，甚至是複數，只是組的一個例子，而加法的概念可以從連續應用動作的角度來思考。

But numbers, schizophrenic as they are, also lead a completely different life as a completely different kind of group.

但是，數字雖然精神分裂，但作為一個完全不同的群體，也過著完全不同的生活。

Consider a new group of actions on the number line, all ways you can stretch or squish it, keeping everything evenly spaced and keeping that number 0 fixed in place.

在數線上考慮一組新的動作，你可以通過各種方式拉伸或擠壓數線，保持所有動作的間距均勻，並將數字 0 固定在原位。

Yet again, this group of actions has that nice property where we can associate each action in the group with a specific point on the thing it's acting on.

然而，這組動作又有一個很好的特性，那就是我們可以將這組中的每個動作與它所作用的事物上的某個特定點聯繫起來。

In this case, follow where the point that starts at the number 1 goes.

在這種情況下，請跟隨從數字 1 開始的點移動。

There is one and only one stretching action that brings that point at 1 to the point at 3, for instance, namely stretching by a factor of 3.

例如，有且只有一個拉伸動作能將 1 處的點拉伸到 3 處的點，即拉伸 3 倍。

Likewise, there is one and only one action that brings that point at 1 to the point at ½, namely squishing by a factor of ½.

同樣，有且僅有一個動作能把 1 處的點帶到 ½ 處的點，即把它壓扁 ½ 倍。

I like to imagine using one hand to fix the number 0 in place, and using the other to drag the number 1 wherever I like, while the rest of the number line just does whatever it takes to stay evenly spaced.

我喜歡想象用一隻手將數字 0 固定在原位，另一隻手隨意拖動數字 1，而數字線的其餘部分則不惜一切代價保持均勻的間距。

In this way, every single positive number is associated with a unique stretching or squishing action.

這樣，每個正數都與獨特的拉伸或擠壓動作相關聯。

Notice what composing actions look like in this group.

請注意這一組的作曲動作。

If I apply the stretch by 3 action, and then follow it with the stretch by 2 action, the overall effect is the same as if I had just applied the stretch by 6 action, the product of the two original numbers.

如果我應用 3 的拉伸操作，然後再應用 2 的拉伸操作，那麼整體效果與我應用 6 的拉伸操作（兩個原始數字的乘積）是一樣的。

And in general, applying one of these actions followed by another corresponds with multiplying the numbers they are associated with.

一般來說，在應用其中一個操作後再應用另一個操作，就相當於將它們相關聯的數字相乘。

In fact, the name for this group is the multiplicative group of positive real numbers.

事實上，這個群的名稱是正實數乘法群。

So multiplication, ordinary familiar multiplication, is one more example of this very general and very far-reaching idea of groups, and the arithmetic within groups.

是以，乘法，我們熟悉的普通乘法，就是這種非常普遍、非常深遠的群組思想以及群組內算術的又一個例子。

And we can also extend this idea to the complex plane.

我們還可以將這一想法擴展到複平面。

Again, I like to think of fixing 0 in place with one hand, and dragging around the point at 1, keeping everything else evenly spaced while I do so.

同樣，我喜歡用一隻手固定 0 的位置，然後圍繞 1 的點拖動，在拖動的過程中保持其他部分間距均勻。

But this time, as we drag the number 1 to places that are off the real number line, we see that our group includes not only stretching and squishing actions, but actions that have some rotational component as well.

但這次，當我們把數字 1 拖到偏離實數線的位置時，我們會發現，我們這組數字不僅包括拉伸和擠壓動作，還包括一些旋轉動作。

The quintessential example of this is the action associated with that point at i, one unit above 0.

最典型的例子就是與 0 上一個組織、部門的 i 點相關的動作。

What it takes to drag the point at 1 to that point at i is a 90 degree rotation.

將 1 處的點拖動到 i 處的點需要旋轉 90 度。

So the multiplicative action associated with i is a 90 degree rotation.

是以，與 i 相關的乘法作用是旋轉 90 度。

And notice, if I apply that action twice in a row, the overall effect is to flip the plane 180 degrees.

注意，如果我連續兩次執行該操作，整體效果就是將平面翻轉 180 度。

And that is the unique action that brings the point at 1 over to negative 1.

這就是把 1 點變成負 1 點的獨特作用。

So in this sense, i times i equals negative 1, meaning the action associated with i, followed by that same action associated with i, has the same overall effect as the action associated with negative 1.

是以，從這個意義上說，i 乘以 i 等於負 1，這意味著與 i 相關的行動，以及與 i 相關的同一行動，其總體效果與與負 1 相關的行動相同。

As another example, here's the action associated with 2 plus i, dragging 1 up to that point.

再比如，下面是 2 加 i 的相關動作，將 1 拖到該點。

If you want, you could think of this as broken down as a rotation by 30 degrees, followed by a stretch by a factor of square root of 5.

如果你願意，可以把它理解為旋轉 30 度，然後拉伸 5 的平方根。

In general, every one of these multiplicative actions is some combination of a stretch or a squish, an action associated with some point on the positive real number line, followed by a pure rotation, where pure rotations are associated with points on this circle, the one with radius 1.

一般來說，每一個乘法運算都是拉伸或擠壓的組合，都是與正實數線上的某個點相關的運算，然後是純旋轉，其中純旋轉與這個半徑為 1 的圓上的點相關。

This is very similar to how the sliding actions in the additive group could be broken down as some pure horizontal slide, represented with points on the real number line, plus some purely vertical slide, represented with points on that vertical line.

這與加法組中的滑動動作可以分解為一些純粹的水準滑動（用實數線上的點表示）和一些純粹的垂直滑動（用垂直線上的點表示）非常相似。

That comparison of how actions in each group break down is going to be important, so remember it.

比較每組行動的細分情況非常重要，請務必牢記。

In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.

在每一箇中，你都可以把任何動作分解為一些純粹的實數動作，然後是複數特有的動作，無論是加法運算組的垂直滑動，還是乘法運算組的純粹旋轉。

So that's our quick introduction to groups.

這就是我們對小組的簡單介紹。

A group is a collection of symmetric actions on some mathematical object, whether that's a square, a circle, the real number line, or anything else you dream up.

群是某個數學對象上對稱作用的集合，無論是正方形、圓形、實數線，還是你夢想的任何其他對象。

Every group has a certain arithmetic, where you can combine two actions by applying one after the other, and asking what other action from the group gives the same overall effect.

每個組都有一定的運算法則，你可以將兩個動作組合起來，一個接一個地應用，並詢問該組中還有什麼動作能帶來相同的整體效果。

Numbers, both real and complex numbers, can be thought of in two different ways as a group.

數字（包括實數和複數）作為一組數，可以有兩種不同的理解方式。

They can act by sliding, in which case the group arithmetic looks like ordinary addition, or they can act by stretching, squishing, rotating actions, in which case the group arithmetic looks like multiplication.

它們可以通過滑動進行運算，在這種情況下，分組運算看起來就像普通的加法運算；它們也可以通過拉伸、擠壓、旋轉等動作進行運算，在這種情況下，分組運算看起來就像乘法運算。

And with that, let's talk about exponentiation.

說到這裡，讓我們來談談指數。

Our first introduction to exponents is to think of them in terms of repeated multiplication, right?

我們對指數的最初認識，是從重複乘法的角度來思考的，對嗎？

I mean, the meaning of something like 2³ is to take 2x2x2, and the meaning of something like 2⁵ is 2x2x2x2x2.

我的意思是，2³ 的意思是取 2x2x2，而 2⁵ 的意思是 2x2x2x2。

And a consequence of this, something you might call the exponential property, is that if

其結果就是，如果

I add two numbers in the exponent, say 2³ plus 5, this can be broken down as the product of 2³ times 2⁵.

我在指數中加上兩個數字，比如 2³ 加上 5，這可以分解為 2³ 乘以 2⁵ 的乘積。

And when you expand things, this seems reasonable enough, right?

當你把事情擴大時，這似乎就足夠合理了，對嗎？

But expressions like 2½, or 2-1, and much less 2i, don't really make sense when you think of exponents as repeated multiplication.

但是，如果把指數看作是重複乘法，像 2½ 或 2-1，更不用說 2i 這樣的表達式就不太合理了。

What does it mean to multiply two by itself half of a time, or negative one of a time?

2 乘以自身的一半或負數 1 是什麼意思？

So we do something very common throughout math, and extend beyond the original definition, which only makes sense for counting numbers, to something that applies to all sorts of numbers.

是以，我們做了一件在數學中很常見的事情，將原本只對計數數字有意義的定義擴展到適用於各種數字的定義。

But we don't just do this randomly.

但我們並不是隨意這樣做的。

If you think back to how fractional and negative exponents are defined, it's always motivated by trying to make sure that this property, 2²x±, equals 2²x²y, still holds.

如果你回想一下分數和負指數是如何定義的，就會發現它總是試圖確保 2²x± 等於 2²x²y 這一性質仍然成立。

To see what this might mean for complex exponents, think about what this property is saying from a group theory light.

要想知道這對復指數意味著什麼，請從群論的角度思考這一性質的含義。

It's saying that adding the inputs corresponds with multiplying the outputs, and that makes it very tempting to think of the inputs not merely as numbers, but as members of the additive group of sliding actions, and to think of the outputs not merely as numbers, but as members of this multiplicative group of stretching and squishing actions.

這就是說，輸入的加法與輸出的乘法相對應，這就很容易讓人把輸入不僅僅看作是數字，而是看作是滑動動作的加法組的成員，把輸出不僅僅看作是數字，而是看作是拉伸和擠壓動作的乘法組的成員。

Now it is weird and strange to think about functions that take in one kind of action and spit out another kind of action, but this is something that actually comes up all the time throughout group theory, and this exponential property is very important for this association between groups.

現在想想，函數吸收一種作用，然後吐出另一種作用，是一件很奇怪的事情，但實際上這在整個群論中經常出現，而這種指數性質對於群之間的聯繫非常重要。

It guarantees that if I compose two sliding actions, maybe a slide by negative 1, and then a slide by positive 2, it corresponds to composing the two output actions, in this case squishing by 2 to the negative 1, and then stretching by 2².

它保證，如果我組合兩個滑動動作，可能是負 1 的滑動，然後是正 2 的滑動，這就相當於組合了兩個輸出動作，在這種情況下，負 1 壓縮 2，然後拉伸 2²。

Mathematicians would describe a property like this by saying that the function preserves the group structure, in the sense that the arithmetic within a group is what gives it its structure, and a function like this exponential plays nicely with that arithmetic.

數學家在描述這樣的性質時會說，這個函數保留了組的結構，也就是說，組內的算術運算賦予了組的結構，而像這樣的指數函數很好地利用了這種算術運算。

Functions between groups that preserve the arithmetic like this are really important throughout group theory, enough so that they've earned themselves a nice fancy name, homomorphisms.

在整個群論中，保留算術的群間函數非常重要，以至於它們為自己贏得了一個好聽的名字--同態。

Now think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.

現在想想，這一切對於將複數平面中的加法群與複數平面中的乘法群聯繫起來意味著什麼。

We already know that when you plug in a real number to 2 to the x, you get out a real number, a positive real number in fact.

我們已經知道，當你把一個實數 2 放入 x 時，你會得到一個實數，事實上是一個正實數。

So this exponential function takes any purely horizontal slide and turns it into some pure stretching or squishing action.

是以，這個指數函數將任何純粹的水準滑動轉化為某種純粹的拉伸或擠壓動作。

So wouldn't you agree that it would be reasonable for this new dimension of additive actions, slides up and down, to map directly into this new dimension of multiplicative actions, pure rotations?