The big, famous constant, e! Okay, it's one of the famous mathematical constants,

偉大而著名的 e ！好吧，著名的數學常數之一

One of the most important, goes along with pi, and I don't know, golden ratio, and square root of two,

圓周率 (π)、黃金比例 (φ)、二的平方根 (√2) 以外，

Constants in maths that are the most important constants, and e is one of those constants.

e 是最重要的數學常數之一

So e is an irrational number, and it's equal to... 2.718281828, something, something, ...

e 是一個無理數 (Irrational)， 並等於 2.718281828……

The problem with e, is it's not defined by geometry.

e 有一個問題是：它並非定義自幾何學

Now pi, is a something that is defined by geometry, right, it's the ratio of a circle's circumference and it's diameter.

圓周率 (π)，是用幾何學定義： 它是「圓周除以直徑」

And it's something the ancient Greeks knew about. And a lot of mathematical constants go back to the ancient Greeks,

這是古希臘人所知的， 而眾多數學常數都可追溯至古希臘

but e is different. e is not based on a shape, it's not based on geometry.

但 e 不是，它不是建基於形狀或者幾何

It's a mathematical constant that is related to growth, and rate of change, but why is it related to growth and rate of change?

它是一個關係到增長和變化率的常數， 但為何與兩者有關呢？

So let's look at the original problem where e was first used.

來看看首次用到 e 的問題

So we're going to go back to the seventeenth century, and this is Jacob Bernoulli, and he was interested in compound interest, so, earning interest on your money.

我們回到 17 世紀，雅各布·白努利 (Jacob Bernoulli) 對複息 (compund interest) 十分有興趣，也就是賺取利息

So imagine you've got one pound in the bank. And you have a very generous bank and they're gonna offer you 100 percent interest every year.

想像你的銀行帳戶有一元，而銀行慷慨地提供一百厘 (100%) 利息

Wow, thanks alot, bank!

嘩！衷心感激銀行！

So, 100 percent interest, so it means after one year,

所以，百厘利息，一年之後

you'll have two pounds.

你就會有兩元

So you've earned one pound interest and you've got your original pound.

在你原有的一元以上，你賺取了一元利息

So, you now have two pounds.

所以你現在有兩元

What if I offered you instead fifty percent interest, every six months?

那麼，如果是每半年複合 (compounded half-yearly) 一次，每次五十厘 (50%) 利息？

Now is that better or worse?

那會變好，抑或變差？

Well, let's think about it.

先想一想

Ok, you're starting with one pound and then I'm going to offer you fifty percent interest every six months.

一元起步，然後每六個月有五十厘利息

So after six months, you now have one pound, fifty

所以六個月後，你擁有 1.5 元

and then you wait another six months and you're earning fifty percent interest on your total,

再坐等六個月，再賺五十厘利息

which is another seventy-five p.

即是，額外的 0.75 元

and you add that on to what you had so it's two pounds twenty-five

計算總和，你現擁有 2.25 元

Better!

結果變得更好！

It's better. So what happens if I do this more regularly?

如果複合 (compound) 再頻密一些，又會如何？

What if I do it every month?

假如每月？

I offer you one-twelfth interest every month

每個月， 1/12 利息？

Is that better?

會否更好？

So, let's think about that.

想一想

So after the first month, it's gonna be multiplied by this.

因此，第一個月，本金 (Principal) 會乘以…

One plus one-twelfth.

(1 + 1/12)

So one-twelfth, that's your interest and then you're adding that onto the original pound that you've got.

也就是說，把利息 (1/12) 加上你的本金

So, you do that, that's your first month,

這就是你的首月結餘 (balance)

then for your second month

然後在第二個月

you take that and multiply it again

乘首月結餘以…

by the same value.

(1 + 1/12)

and your third month you would multiply it again,

第三個月再把上月結餘乘以 (1+1/12)

and again.

諸如此類…

you actually do that twelve times in a year.

一年中總共計算十二次利息

So in a year,

所以一年內，

you'd raise that to a power twelve,

會有本金的 (1+1/12) 的 12 次方

and you would get two pounds sixty-one.

即是 2.61 元

So it's actually better. In fact, the more frequent your interest is

可見結果改善了。 實際上，複合愈頻密，

the better the results.

結果會更好

Let's start with every week. So if we do it for every week, how much better is that?

再來每星期，又會好多少？

What I'm saying is you're earning one over fifty two interest every week. And then after the end of the year

即是每週賺 1/52 複息。所以一年後，

you got fifty two weeks and you would have two pounds sixty nine.

即 52 週後，你將有 2.69 元

So it's getting better and better and better.

結果愈來愈好

In general, you might be able to see a pattern happening here. In general it would look like this:

一般，你應該已經看到規律：

You'd be multiplying by one plus one over n, to the power n. Hopefully you can see that pattern happening.

你相乘以 (1+1/n) 的 n 次方。 希望你早已發現此

So here n is equal to twelve if you do it every month, fifty two if you do it every week.

每月複合，n 等於 12 每週複合，n 等於 52

If you did it every day, it'd be one pound multiplied by [one plus] one over three hundred and sixty five

每日複合，就是乘以 (1 + 1/365) 的…

to the power three hundred and sixty five. And that's equal to two pounds, seventy one.

的 365 次方，並等於 2.71 元

Right, and so it would get better if you did it every second, or every nanosecond.

嗯，如果每秒甚至每納秒 (Nanosecond) 複合，更加好

What if I could do it continuously?

那麼連續地呢？

Every instant I'm earning interest. Continuous interest. What does that look like?

也就是，每一刻都賺取利息，連續地複合利息。 看來如何？

That means if I take this formula here

這代表，利用此公式，

one plus one over n to the n, I'm gonna n tend to infinity.

(1 + 1/n)^n，而 n 趨向無限

That would be continuous interest. Now what is that? What is that value?

那就是連續複利。答案數值是甚麼？

And that's what Bernoulli wanted to know.

這就是白努利 (Jacob Bernoulli) 想知的

He didn't work it out. He knew it was between two and three. So fifty years later, Euler worked it out.

但他未能計出，只知道是介乎二和三。 而五十年後，歐拉 (Euler) 計出了

Euler, he works everything out.

歐拉理通了所有事

[Brady]: Him or Gauss?

[Brady]: 他和高斯 (Gauss)?

[James]: It's either Euler or Gauss. Say Euler or Gauss, you're probably going to be right.

[James]: 歐拉或者高斯，任何一者，你很大機會沒有錯

And the value was 2.718281828459... and so on.

而得出數值是 2.718281828459……

[Brady]: We were pretty close when we were doing it daily, weren't we? It was already two seventy one at daily.

[Brady]: 當我們計算每日複合時，得出的 2.71 已經十分相近，不是嗎？

[James]: You're right, You're right. We were getting closer, weren't we?

[James]: 的而且確。我們正接近…

We were getting close and closer to this value. So already we're quite close to it.

我們愈來愈接近這個數值，而且已頗接近它

If you did it forever though, of course you would have this irrational number.

如果不停計下去，你就會得到這個無理數

Now Euler called this e. He didn't name it after himself, although it is now known as the Euler constant.

而歐拉稱它為 e。儘管現又被稱為「歐拉常數」(Euler's constant)，他沒有以自己命名

[Brady]: Why'd he call it e then?

[Brady]: 他為甚麼稱之為 e？

[James]: It was just a letter. He might've used a, b, c, and d already for something else.

[James]: 那只是一個字母，他可能已經在其他地方用了 a、b、c、d

Right? So you use the next one.

所以他用了 e

[Brady]: Bit of a coincidence!

[Brady]: 有點巧合！

[James]: It's a lovely coincidence! I fully believe that he's not being a jerk here, naming it after himself.

[James]: 的確十分巧合！可是，我深信他沒有以自己命名

But it's a lovely coincidence that it's e for Euler's number.

歐拉常數 (Euler's Constant) 是 e 純屬巧合

[Brady]: Would you have called it g if you discovered it?

[Brady]: 如果是你發現，你會否命名為 g？

[James]: I would not have called it g. No, I would've hoped somebody else would've called it g

[James]: 我不會叫它作 g。我寧願其他人叫它作 g

and then I would have accepted that.

然後我再接受這稱呼

Euler proved that this was irrational.

歐拉證明了 e 是無理 (Irrational)

He found a formula for e which was a new formula. Not this one here, a different formula.

他找到了一條 e 的新公式。不是這條，而是另外一條

And it showed that it was irrational. I'll quickly show you that.

那條公式證明 e 是無理。我很快地展示給你看

He found that e was equal to two plus one over

他發現，e 等於 2 加 [如影片所示]

one plus one over two plus one over one plus one over one plus one over four plus one over one plus one over

[如影片所示]

one plus one over six... and this goes on forever.

[如影片所示]…而這分數延續下去

This is a fraction that goes on forever, continuous fraction. But you can see it goes on forever

這分數是一個連分數 (Continued fraction)，而且不斷延續

Because there's a pattern, and that pattern does hold.

因為它有一個可成立的規律

You got two, one one four, one one six, one one eight.

2，1、1、4，1、1、6，1、1、8

So you can see that pattern goes on forever, and if the fraction goes on forever

這規律無盡，所以這個連分數亦無盡頭

it means it's an irrational number.

所以是一個無理數

If it didn't go on forever, it would terminate, and if you terminate you can write it as a fraction.

如果它有盡頭，你就可以寫作一個分數，那就是有理數

And he also worked out the value for e. He did it up to eighteen decimal places.

歐拉亦計算出 e 的數值，高達十八個小數位

To do that, he had a different formula to do that, I'll show you that one.

他用了另一公式得出這數值。讓我展示一下這公式

And this time, he worked out e was equal to one plus

而今次，他找出 e 等於 1 加

one over one factorial plus one over two factorial plus one over three factorial

1/1! (階乘 Factorial) 加 1/2! 加 1/3!

plus one over four factorial... and this is something that's going on forever.

加 1/4!…如此類推並無限延續

It's a nice formula, if you're happy with factorials. Factorials means you're multiplying all the numbers

如果你明白階乘 (Factorial)，這公式十分好用。 階乘就是從 1 乘至…

up to that value. So if it was four factorial, it'd be four times three times two times one.

該數字。例如四的階乘 (4!)，就是 4×3×2×1

Okay, why is e a big deal? It's because e is the natural language of growth.

那麼， 為甚麼 e 重要呢？ 因為它可用作說明增長 (Growth)

And I'll show why. Okay, let's draw a graph y equals e to the x.

我解釋一下。先畫出 y=e^x 的圖

So we're taking powers of e. So over here at zero, this would cross at one.

這曲線代表了 e 的 x 次方， 而當 x 等於零，這線穿過了一

So if you took a point on this graph, the value at that point is e to the power x.

線上任意一點皆等於 e 的 x 次方的數值 (e^x)

And this is why it's important. The gradient at that point, the gradient of the curve

而它之所以重要，該點的斜率…

at that point is e to the x. And the area under the curve which means the area under the curve

正好等於 e^x。而曲線下面積 (Area under the curve)…

all the way down to minus infinity is e to the x.

從負無限 (-∞) 至該點的曲線下面積，也等於 e^x

And it's the only function that has that property.

而且只有這函數 (Function) 有此特質

So it has the same value, gradient, and area at every point along the line.

線上每點的數值、斜率、曲線下面積都是一樣

So at one, the value is e because it's e to the power one. The value is 2.718, the gradient is 2.718

當 x=1，數值是 e^1=e=2.718， 斜率亦是 2.718，

and the area under the curve is 2.718. The reason this is important then, because it's unique

曲線下面積亦是 2.718。 這很重要，因為…

in having this property as well, it becomes the natural language of calculus.

這特質是它獨有，所以自然成為了微積分 (Calculus) 重要的一部分

And calculus is the maths of rate of change and growth and areas, maths like that.

而微積分就是變化率 (Rate of change) 、增長、面積等的數學

And if you're interested in those things, if you write it in terms of e, then the maths becomes much simpler.

如果你對這些東西有興趣，用 e 計算較簡單

Because if you don't write it in terms of e, you get lots of nasty constants

因為如果不用 e，就要用大量繁複的常數

and the maths is really messy. If you're trying to deliberately avoid using e,

然後就變得十分繁複。如果你故意為之，

you're making it hard for yourself. It's the natural language of growth.

就是自找麻煩。增長很自然地會用到 e

And of course e is famous for bringing together all the famous mathematical constants with this formula,

而且用以下公式，e 把各著名數學常數放到一起

Euler's formula, which is e to the i pi plus one equals zero.

歐拉公式 (Euler's formula)， 即是 e^(iπ) + 1 =0

So there we have all the big mathematical constants in one formula brought together.

用一條公式，我們把所有鼎鼎有名的數學常數放在一起

We've got e, we've got i, square root of minus one, we've got pi of course, we've got one and zero

我們有 e，有虛數 (即 i=√-1)，有圓周率 (π)，還有 1 和 0

and they bring them all together in one formula

並帶到同一公式中

which is often voted as the most beautiful formula in mathematics.

並經常被譽為數學中最美麗的公式

I've seen it so often, I'm kinda jaded to it, don't put that in the video.

我經常見到它，已經有點厭悶，千萬不要公開這番話

[Brady]: Sometimes here on Numberphile we can make more videos than we'd otherwise be able to

[Brady]: Numberphile 可以製作這麼多優質影片，

thanks to excellent sponsors. And today we'd like to thank "The Great Courses Plus,"

全靠眾贊助商支持。 今日我們想感謝 "The Great Courses Plus"

which is a fabulous service. If you like Numberphile videos, you're really going to like this.

它的服務和內容一流，如果你喜歡 Numberphile 的影片，你一定會喜歡它

They've got an incredible array of lessons, something like over seven thousand videos

它有大量課堂，大概超過七千段影片

from world experts, top professors around the world on all sorts of subjects from photography

並由世界各地的頂尖專家和教授，教導各式科目，如攝影、

to paleontology to prime numbers. No matter what you're into, you're gonna find something.

古生物學 (Paleontology)、質數 (Prime numbers)，應有盡有

What I've been looking at just recently on the site is paleontology, and one of my favorite subjects,

我最近在上面學習古生物學，及我最喜愛的科目之一，

Egyptology. They've got some great videos on those. Now plans on The Great Courses Plus

古埃及學。它有不少優質影片。 The Great Courses Plus 的收費方案是…

start from fourteen dollars ninety nine a month, but there's a special offer where you can go and get

每月 $14.99 起， 但現有一個特別優惠別給你：

a whole month, unlimited access to all the videos for free!

免費，無限，任看影片足一個月

Go to TheGreatCoursesPlus.com/numberphile

馬上到 TheGreatCoursesPlus.com/numberphile

and have a free month's access, check it all out. If it's something you like, maybe you'll sign on for more.

並取得一個月任看優惠。 如果你喜歡它的服務，才再購買更多

I think there's a good chance you will. There's the link on the screen and also there's a link

我相信你很大機會喜歡它。 屏幕上有它的網址，另外在…

in the video description. Why not give it a click? Just to go and have a look.

影片描述中也有。為何不點下去，去看一看如何？

It's a great way of showing your support for Numberphile

這是一個顯露你對我們 Numberphile 的支持的好方法

and showing the people at The Great Courses Plus that you came from here.

並令 The Great Courses Plus 的人員知道你來自我們這裏

Our thanks again to them for supporting this video.

我們再次感激他們對本影片的幫助

Okay, I'm gonna go for e, e.

Okay, I'm gonna go for e, e.