［第01修訂版］不行了，今天必須要翻譯這篇文章了柏锄！

原標題：An Intuitive Guide To Exponential Functions & e

原文地址：https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

1 e has always bothered me — not the letter, but the mathematical constant. What does it really mean?
e 總是困擾著我 —— 不是這個字母本身蔽挠，而是作為自然常數(shù)的e蟹腾。它到究竟是什么東西坏挠？

2 Math books and even my beloved Wikipedia describe e using obtuse jargon:
數(shù)學(xué)課本，甚至我深愛的 Wikipedia 在解釋 e 時用的都是鳥語：

The mathematical constant e is the base of the natural logarithm.
數(shù)學(xué)常數(shù) e 就是自然對數(shù)的底屯阀。

3 And when you look up the natural logarithm you get:
而當你查看什么是自然對數(shù)時，你得到的結(jié)果是：

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.
自然對數(shù)轴术，曾經(jīng)叫雙曲對數(shù)难衰，是以 e 為底的對數(shù)，這里的 e 是一個無理數(shù)逗栽，它約為2.718281828459盖袭。

4 Nice circular reference there. It’s like a dictionary that defines labyrinthine with Byzantine: it’s correct but not helpful. What’s wrong with everyday words like “complicated”?
啊哈，這樣循環(huán)引用真的好嗎~ 這就像一本詞典用錯綜復(fù)雜 來解釋迷宮彼宠，然后用迷宮 來解釋錯綜復(fù)雜：是的鳄虱，這沒錯，然并卵凭峡。像復(fù)雜 這樣的日常詞語到底是怎么了拙已？

5 I’m not picking on Wikipedia — many math explanations are dry and formal in their quest for rigor. But this doesn’t help beginners trying to get a handle on a subject (and we were all a beginner at one point).
我不是挑Wikipedia的刺——為了嚴謹性，許多數(shù)學(xué)解釋都是格式化的摧冀、枯燥的悠栓。但這對于一個初學(xué)者并沒有多少幫助（在一定程度上我們都是初學(xué)者）。

6 No more! Today I’m sharing my intuitive, high-level insights about what e is and why it rocks. Save your rigorous math book for another time. Here’s a quick video overview of the insights:
所以按价，沒辦法惭适，今天不得不由我來分享我的關(guān)于e的直觀而且深刻的見解了。這次楼镐，你可以防好你嚴格的數(shù)學(xué)書癞志。下面是這個見解的視頻：

e is NOT Just a Number

e不僅僅是個數(shù)字

7 Describing e as “a constant approximately 2.71828…” is like calling pi “an irrational number, approximately equal to 3.1415…”. Sure, it’s true, but you completely missed the point.
說 e 是“一個近似值為 2.71828… 的數(shù)”無異于說 π是“一個近似值為 3.1415…無理數(shù)”。這完全正確框产，但完全不著要點凄杯。

8 π is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).
π是圓的周長和其半徑的比值错洁。它是所有圓周的固有的基本比例。它影響了圓戒突、球屯碴、圓柱等的周長、面積膊存、體積和表面積的計算导而。π很重要，它告訴我們所有的圓都是相關(guān)的隔崎，不是說三角函數(shù)（sin今艺，cos，tan）都是從圓導(dǎo)出的爵卒。

9 e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.
而e是所以連續(xù)增長過程的基本比率虚缎。e讓你得到一個簡單的增長比率（所有的變化發(fā)生在年底），并且找到復(fù)合連續(xù)增長產(chǎn)生的影響钓株。每一納秒实牡，你只要增長一點點。

10 e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.
e會出現(xiàn)在任何以指數(shù)連續(xù)增長的地方：人口轴合、放射性物質(zhì)的衰變创坞、利息的計算等等。甚至增長不平滑的jagged system也可以用e近似計算值桩。

11 Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of e (unit growth, perfectly compounded).
就像每個數(shù)字都可以成數(shù)字“1”（基本單位）的縮放版摆霉，每個圓都可以看成單位圓（半徑為1）的縮放版豪椿，每種增長率都可以看成e的縮放版（基本增長奔坟，完美復(fù)合的）。

12 So e is not an obscure, seemingly random number.e represents the idea that all continually growing systems are scaled versions of a common rate.
因此搭盾，e不是一個晦澀的咳秉、表面看來很隨意的一個數(shù)。e 代表著鸯隅，所有連續(xù)增長都是一個共同增長率的縮放版澜建。

Understanding Exponential Growth 什么是指數(shù)增長

13 Let's start by looking at a basic system that doubles after an amount of time. For example,
讓我們從一個基本的系統(tǒng)開始：隔一段時間就會翻倍的系統(tǒng)。例如：

• Bacteria can split and “doubles” every 24 hours
  細菌每24小時就會分裂并且數(shù)量加倍

• We get twice as many noodles when we fold them in half.
  做拉面時蝌以，如果我們從中間對折炕舵，我們就會得到兩倍的面條

• Your money doubles every year if you get 100% return (lucky!)
  你的錢每一年都可以翻倍，如果你每年都得到100%的回報的話

And it looks like this:
它看起來就像這樣：

2 times growth

14 Splitting in two or doubling is a very common progression. Sure, we can triple or quadruple, but doubling is convenient, so hang with me here.
分裂成兩個或者說翻倍是一種非常常見的過程跟畅。當然我們也能翻三倍或者四倍咽筋，但翻倍較為方便，所以我們就討論這種情況徊件。

15 Mathematically, if we have x splits then we get 2^x times as much stuff than when we started. With 1 split we have 2^1 or 2 times as much. With 4 splits we have 2^4= 16times as much. As a general formula:
計算可知奸攻，如果分裂x次蒜危，那我們得到的結(jié)果將是一開始的2^{x倍。分裂1次睹耐，結(jié)果是一開始的2}1倍辐赞，也就是2倍。分裂4次硝训，結(jié)果將是一開始的2^4倍响委，也就是16倍。通用的公式長這樣：

\displaystyle{ growth = 2^x }

17 Said another way, doubling is 100% growth. We can rewrite our formula like this:
換一種說法捎迫，翻倍其實就是增長率為100%的增長晃酒。這樣，我們我們可以把上面的公式改寫成下面的樣子：

\displaystyle{ growth = (1 + 100%)^x}

18 It’s the same equation, but we separate 2 into what it really is: the original value (1) plus 100%. Clever, eh?
公式還是同一個公式窄绒，但我們把2還原成了它的真實意思：原本的數(shù)量（1）和增長率100%贝次。哈哈，聰明吧彰导！

19 Of course, we can substitute any number (50%, 25%, 200%) for 100% and get the growth formula for that new rate. So the general formula for x periods of return is:
當然我們也可以用任何的數(shù)字（比如50%蛔翅，25，200%等等）來替換掉100%位谋，并且得到這個新增長率下的新公式山析。這樣，一段時間x下的通用公式如下：

\displaystyle{growth = (1 + return)^x}

20 This just means we use our rate of return, (1 + return), “x” times.
這僅僅意味著我們用我們的回報率作為增長率掏父，（1+回報率）笋轨，增長x次。

A Closer Look

21 Our formula assumes growth happens in discrete steps. Our bacteria are waiting, waiting, and then boom, they double at the very last minute. Our interest earnings magically appear at the 1 year mark. Based on the formula above, growth is punctuated and happens instantly. The green dots suddenly appear.
我們的公司有一個假設(shè)赊淑，就是增長是一次一次發(fā)生的爵政。我們的細菌要先長大，長大陶缺，在長大钾挟，等積累到一定大小的時候，“嘣”一下分裂成兩個饱岸。而我們感興趣的增長奇跡般的出現(xiàn)在剛好是一年的時候掺出。上面的公式中，增長是間斷發(fā)生的苫费，并且是立刻發(fā)生的汤锨，圖中綠色的圓點突然就出現(xiàn)了。

22 The world isn’t always like this. If we zoom in, we see that our bacterial friends split over time:
但這個世界并不總是如此百框。如果我們深入細致觀察闲礼，我們就能看到我們的細菌朋友隨時間是如何增長的。

2 times growth detail

23 Mr. Green doesn’t just show up: he slowly grows out of Mr. Blue. After 1 unit of time (24 hours in our case), Mr. Green is complete. He then becomes a mature blue cell and can create new green cells of his own.
小綠不是突然跳出來：他慢慢的從小藍里面長出來。經(jīng)過一個單位的時間（例如24小時）位仁，小綠才長成柑贞。然后他才“性成熟”，可以生育后代聂抢。

24 Does this information change our equation?
但這會對我們的公式造成任何改變嗎钧嘶？

Nope. In the bacteria case, the half-formed green cells still can’t do anything until they are fully grown and separated from their blue parents. The equation still holds.
NO。在細菌分裂這件事情上琳疏，才長出一半的綠細胞并不能做任何的事情有决，直到他們完全長大并從母體分離。上面那個公式仍然成立空盼。

Money Changes Everything

25 But money is different. As soon as we earn a penny of interest, that penny can start earning micro-pennies of its own. We don’t need to wait until we earn a complete dollar in interest — fresh money doesn’t need to mature.
但錢不一樣书幕。只要我們掙了一分的利息，這一分的利息就可以產(chǎn)生它的一厘的利息揽趾。我們不必等到利息長大成一元——新增加的錢不需要成熟台汇。

26 Based on our old formula, interest growth looks like this:
根據(jù)我們的“舊公式”，利息的增長像這樣：

interest rate growth

27 But again, this isn’t quite right: all the interest appears on the last day. Let’s zoom in and split the year into two chunks. We earn 100% interest every year, or 50% every 6 months. So, we earn 50 cents the first 6 months and another 50 cents in the last half of the year:
不過這一次又有一點不對：所有的利息都在最后一天才生成篱瞎。讓我們放大一下并且把一年分成兩部分苟呐。我們每年獲得100%的利息，或者六個月獲得50%的利息俐筋。因此我們前6個月將獲得50分的利息牵素，在后六個月獲得另外五十分的利息：

interest rate 6 months

28 But this still isn’t right! Sure, our original dollar (Mr. Blue) earns a dollar over the course of a year. But after 6 months we had a 50-cent piece, ready to go, that we neglected! That 50 cents could have earned money on its own:
但這仍然不對！的確我們的本金（小藍）經(jīng)過一年掙得了一塊錢的利息澄者。但經(jīng)過六個月我們笆呆，我們的到了50分利息，我們剛不小心忽略了它粱挡。這50分也能夠獲得它自己的利息：

compound interest

29 Because our rate is 50% per half year, that 50 cents would have earned 25 cents (50% times 50 cents). At the end of 1 year we’d have:
因為我們的半年利率是50%赠幕，因此那50分前經(jīng)過后半年會獲得25分的利息（50分×50%）。因此在一年結(jié)束的時候我們會得到：

• Our original dollar (Mr. Blue)
  我們一塊錢的本金（圖中的小藍）

• The dollar Mr. Blue made (Mr. Green)
  小藍的1美元利息（圖中的小綠）

• The 25 cents Mr. Green made (Mr. Red)
  呂先生勝出的25分利息（圖中的小紅）

30 Giving us a total of $2.25. We gained $1.25 from our initial dollar, even better than doubling!
這樣返還給我們的將是2.25元抱怔。我們從一元的本金里的到了1.25元的利息劣坊，比翻倍還要賺的多嘀倒！

31 Let’s turn our return into a formula. The growth of two half-periods of 50% is:
讓我們把我們的回報放回一個公式里。兩個半年50%的增長是這樣子的：

\displaystyle{growth = (1 + 100%/2)^{2} = 2.25}

Diving into Compound Growth

32 It’s time to step it up a notch. Instead of splitting growth into two periods of 50% increase, let’s split it into 3 segments of 33% growth. Who says we have to wait for 6 months before we start getting interest? Let’s get more granular in our counting.
Charting our growth for 3 compounded periods gives a funny picture:
是該xx的時候了。取代原來分割成兩個50%的增長階段的做法奸远，讓我們把利息的增長分割成三個33%的階段制肮。誰說我們必須要等六個月才能開始得到利息呢？讓我們把計算利息的次數(shù)增加一些碳胳。三次復(fù)合增長的圖象很有有趣：

4 month compound interest

33 Think of each color as shoveling money upwards towards the other colors (its children), at 33% per period:
設(shè)想每一段時間勇蝙，每種顏色都都以33%的比率向上面一種顏色（也就是它們的子代）送錢。

• Month 0:We start with Mr. Blue at $1.
  開始：我們以代表一美元的小藍開始挨约。

• Month 4:Mr. Blue has earned 1/3 dollar on himself, and creates Mr. Green, shoveling along 33 cents.
  4月末：小藍掙到了來自它自己的1/3美元味混，由此創(chuàng)造了小綠产雹，并把這三分之一美元給了小綠。

• Month 8:Mr. Blue earns another 33 cents and gives it to Mr. Green, bringing Mr. Green up to 66 cents. Mr. Green has actually earned 33% on his previous value, creating 11 cents (33% * 33 cents). This 11 cents becomes Mr. Red.
  8月末：小藍又掙到了另外33美分翁锡，并且把它給了小綠蔓挖，這時小綠就增長到了66美分。與此同時馆衔，小綠也在他上一次的基礎(chǔ)上增長了33%瘟判，掙了11美分（33美分*33%），并由此創(chuàng)造了小紅角溃。

• Month 12:Things get a bit crazy. Mr. Blue earns another 33 cents and shovels it to Mr. Green, bringing Mr. Green to a full dollar. Mr. Green earns 33% return on his Month 8 value (66 cents), earning 22 cents. This 22 cents gets added to Mr. Red, who now totals 33 cents. And Mr. Red, who started at 11 cents, has earned 4 cents (33% * .11) on his own, creating Mr. Purple.
  12月末：繼續(xù)下去就有點兒瘋狂了拷获。小藍又增長了33美分，并把它給了小綠减细，這使得小綠長成了完整的一美元匆瓜。與此同時，小綠在他上一次的基礎(chǔ)上增加了33%未蝌，也就是22美分（66美分*33%）陕壹，并且將其給了小紅，這樣小紅就達到了33美分树埠。與此同時小紅也在自己上一次的基礎(chǔ)上增加了33%糠馆，也就是4美分（11美分*33%），并由此創(chuàng)造了小紫怎憋。

34 Phew! The final value after 12 months is: 1 + 1 + .33 + .04 or about 2.37.
瞧又碌，經(jīng)過十二個月之后，我們將得到：1 + 1 + .33 + .04 ≈ 2.37绊袋。

35 Take some time to really understand what’s happening with this growth:
那我們花點時間來理解在這增長過程中到底發(fā)生了什么：

• Each color earns interest on itself and hands it off to another color. The newly-created money can earn money of its own, and on the cycle goes.
  每個色塊都掙得了自己的利息并且把他拱手送給了下一個顏色毕匀。而這些新掙的錢也可以以自己為基礎(chǔ)掙錢，并且送給下一個色塊兒癌别。后面以此類推皂岔。

• I like to think of the original amount (Mr. Blue) as never changing. Mr. Blue shovels money to create Mr. Green, a steady 33 every 4 months since Mr. Blue does not change. In the diagram, Mr. Blue has a blue arrow showing how he feeds Mr. Green.
  我喜歡把原來的那一美元（也就是小藍）看成從來沒有變。小蘭只是用自己掙得的錢創(chuàng)造了小綠展姐，并且把后面掙的錢都送給了它躁垛。在圖中，小藍用藍箭頭清晰的指出了它是如何“包養(yǎng)”小綠的圾笨。

• Mr. Green just happens to create and feed Mr. Red (green arrow), but Mr. Blue isn’t aware of this.
  小綠也像小藍一樣創(chuàng)造了小黃并且“包養(yǎng)”它（綠箭頭）教馆，而小藍對此事一無所知。*真是一個令人悲傷的包養(yǎng)故事……

• As Mr. Green grows over time (being constantly fed by Mr. Blue), he contributes more and more to Mr. Red. Between months 4-8 Mr. Green gives 11 cents to Mr. Red. Between months 8-12 Mr. Green gives 22 cents to Mr. Red, since Mr. Green was at 66 cents during Month 8. If we expanded the chart, Mr. Green would give 33 cents to Mr. Red, since Mr. Green reached a full dollar by Month 12.
  隨著時間的推移擂达，由于小藍的包養(yǎng)土铺，小綠持續(xù)的長大，可是它也把越來越多的錢花在了小紅身上。在4月末到8月末這段時間里悲敷，小綠給了小紅11美分究恤。在8月末到12月末這段時間，小綠又給了小紅22美分后德。因為在八月末的時候小綠只有66美分丁溅。如果我們將這個圖標繼續(xù)隱身小綠又將會給小紅33美分，因為小綠在12月末的時候已經(jīng)是一個完整的一美元了探遵。

36 Make sense? It’s tough at first — I even confused myself a bit while putting the charts together. But see that each dollar creates little helpers, who in turn create helpers, and so on.
懂了嗎窟赏？開頭的時候的確很難——開始我把這些表格放在一起的時候，我甚至把自己都弄得有點糊涂了箱季。但后來就明白了涯穷，那只是每一美元都在制造小幫手，小幫手又在制造小幫手藏雏，如此下去而已拷况。

37 We get a formula by using 3 periods in our growth equation:
我們的到了將一個周期分為三個階段的增長公式：

\displaystyle{growth = (1 + 100%/3)^3 = 2.37037...}

38 We earned $1.37, even better than the $1.25 we got last time!
我們掙到了1.37美元，比上次的1.25美元還要好掘殴！

Can We Get Infinite Money? 我們可以得到無限多的錢嗎赚瘦？

39 Why not take even shorter time periods? How about every month, day, hour, or even nanosecond? Will our returns skyrocket?
Our return gets better, but only to a point. Try using different numbers of n in our magic formula to see our total return:
為什么不把每個階段的時間都縮短呢？以每個月或每一天或每個小時奏寨，甚至每豪秒為一個階段怎么樣呢起意？我們得到的回報將像火箭一樣猛漲嗎？這次只是做夢了病瞳，我們能得到更好的回報揽咕，但會有一個極限。你可以試試將不同的n代入我們的增長公式里 看看最后的總回報：

n          (1 + 1/n)^n
-----------------------
1          2
2          2.25
3          2.37
5          2.488
10         2.5937
100        2.7048
1,000      2.7169
10,000     2.71814
100,000    2.718268
1,000,000  2.7182804
...

40 The numbers get bigger and converge around 2.718. Hey… wait a minute… that looks like e!
數(shù)字在變大并停留在2.718左右套菜。等等亲善，這怎么這么像e？

41 Yowza. In geeky math terms, e isdefinedto be that rate of growth if we continually compound 100% return on smaller and smaller time periods:
沒錯逗柴，在枯燥得令人乏味的數(shù)學(xué)術(shù)語里面蛹头，嗯被定義為xxx：

\displaystyle{growth = e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n}

42 This limit appears to converge, and there are proofs to that effect. But as you can see, as we take finer time periods the total return stays around 2.718.
這個極限最終會收斂于一個值，這個已經(jīng)被證明過戏溺。正如你所見渣蜗，不管我們把時間分的多么多么的短，我們得到的總回報最終停留在了2.718附近于购。

But what does it all mean?

43 The number e (2.718…) is the maximum possible result when compounding 100% growth for one time period. Sure, you started out expecting to grow from 1 to 2 (that’s a 100% increase, right?). But with each tiny step forward you create a little dividend that starts growing on its own. When all is said and done, you end up with e (2.718…) at the end of 1 time period, not 2. e is the maximum, what happens when we compound 100% as much as possible.
數(shù)字e（2.718）是一個周期以100%的復(fù)合增長率增長的最大結(jié)果袍睡。當然開始的時候你只是希望從1增長到2（那就是100%的增長率嘛）知染。但隨著一小步一小步的推進肋僧，你不但獲得了每一小步的紅利，而且這個紅利本身也要繼續(xù)產(chǎn)生紅利。當整個增長周期結(jié)束后嫌吠，你得到的是e（2.718）止潘，而不僅僅是2。e是以100%復(fù)合增長得到的最大的結(jié)果辫诅。

44 So, if we start with $1.00 and compound continuously at 100% return we get 1e. If we start with $2.00, we get 2e. If we start with $11.79, we get 11.79e.
因此如果我們開始的時候有1美元凭戴，那么一100%的增長率連續(xù)復(fù)合增長，那返回給我們的將是1e美元炕矮。如果能開始兩美元么夫，那反正給我們的就是2e美元。如果我們開始的時候有11.79美元肤视，那返還給我們的將是11.79e美元档痪。

45 e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process. You might not always reach the speed limit, but it’s a reference point: you can write every rate of growth in terms of this universal constant.
e就像速度的極限（就像光速c），他表示在連續(xù)的增長過程中邢滑，你究竟可能獲得多大的增長快慢腐螟。你可能不是總能達到極限速度，但它至少是一個參考值：你可以根據(jù)這個宇宙常數(shù)寫出任意增長率的增長困后。

46 (Aside: Be careful about separating the increase from the final result. 1 becoming e (2.718…) is an increase (growth rate) of 171.8%. e, by itself, is the final result you observe after all growth is taken into account (original + increase)).
（注意：小心區(qū)分增加量和最后結(jié)果乐纸。1變成了e（2.718），增加量（增長率）是171.8%摇予。而e是包含增加量的最后結(jié)果汽绢。

What about different rates? 如果增長率不同呢？

47 Good question. What if we grow at 50% annually, instead of 100%? Can we still use e?
問得好侧戴！如果我們的年增長率只有50%而不是100%庶喜，我們?nèi)匀荒苡胑嗎？

48 Let’s see. The rate of 50% compound growth would look like this:
請看救鲤，年增長率為50%的復(fù)合增長會是這樣：

\displaystyle{\lim_{n\to\infty} \left( 1 + \frac{.50}{n} \right)^n}

49 Hrm. What can we do here? Remember, 50% is the total return, and n is the number of periods to split the growth into for compounding. If we pick n=50, we can split our growth into 50 chunks of 1% interest:
恩久窟，這次我們怎么處理？記住50%是總回報本缠，n是我們?yōu)榱双@得復(fù)合增長而將這個增長過程分割的次數(shù)斥扛。如果取n=50，我們就可以把我的整個生產(chǎn)過程分割成五十段丹锹，每一段的增長率都為1%稀颁。

\displaystyle{\left( 1 + \frac{.50}{50} \right)^{50} = \left( 1 + .01 \right)^{50}}

50 Sure, it’s not infinity, but it’s pretty granular. Now imagine we also divided our regular rate of 100% into chunks of 1%:
當然這不是無限的，不過已經(jīng)很細了￠故颍現(xiàn)在設(shè)想我們同樣把增長率為100%的整個過程分解為（100段）增長率為1%的小段匾灶。

\displaystyle{e \approx \left( 1 + \frac{1.00}{100} \right)^{100} = \left( 1 + .01 \right)^{100}}

51 Ah, something is emerging here. In our regular case, we have 100 cumulative changes of 1% each. In the 50% scenario, we have 50 cumulative changes of 1% each.
哈，似乎有點兒線索了租漂。通常我們有100段每段增長率為1%的連續(xù)積累的增長阶女，在總增長率為50%的情況下颊糜，我們有50段這樣的增長。

Different exponential rates

52 What is the difference between the two numbers? Well, it’s just half the number of changes:
第二個數(shù)比起第一個數(shù)有什么區(qū)別呢秃踩？僅僅是增長的小段數(shù)少了一半罷了衬鱼。

\displaystyle{\left( 1 + .01 \right)^{50} = \left( 1 + .01 \right)^{100/2} = \left( \left( 1 + .01 \right)^{100}\right)^{1/2} = e^{1/2} }

53 This is pretty interesting. 50 / 100= .5, which is the exponent we raise e to. This works in general: if we had a 300% growth rate, we could break it into 300 chunks of 1% growth. This would be triple the normal amount for a net rate of e^3.
有趣的是，50/100=.5憔杨，這個0.5正好是數(shù)字e的指數(shù)鸟赫。而且這是通用的：如果我們的總增長率是300%，我們可以把整個增長的過程分割成300小段消别，每一段增長率為1%的小段抛蚤。最終，這將以e^3的倍數(shù)使本金增翻三倍寻狂。

54 Even though growth can look like addition (+1%), we need to remember that it’s really a multiplication (x 1.01). This is why we use exponents (repeated multiplication) and square roots (e^1/2 means “half” the number of changes, i.e. half the number of multiplications).
盡管這個增長過程我們可以看成加法（每次增加1%）霉颠，我們也要記住它實際上是乘法（×1.01）。這也是我們?yōu)槭裁匆弥笖?shù)（重復(fù)相乘）和平方根（e^(1/2)的意思是荆虱，將增長的次數(shù)減半蒿偎，將乘的次數(shù)減半）。

55 Although we picked 1%, we could have chosen any small unit of growth (.1%, .0001%, or even an infinitely small amount!). The key is that for any rate we pick, it’s just a new exponent on e:
前面我們?nèi)〉氖?%怀读，其實我們也可以取更小的增長率（.1%诉位，0.0001%，甚至無限小的數(shù)）菜枷。核心是苍糠，不管我們?nèi)《啻蟮脑鲩L率，這只是e的一個新指數(shù)而已啤誊。

\displaystyle{growth = e^{rate}}

What about different times?

56 Suppose we have 300% growth for 2 years. We’d multiply one year’s growth (e^3) by itself:
設(shè)想我們用300%的增長率岳瞭，增長整整兩年。我們就用他自己去乘以一年的增長率（e^3）蚊锹。

\displaystyle{growth = \left(e^{3}\right)^{2} = e^{6}}

58 And in general:
通式是：

\displaystyle{growth = \left(e^{rate}\right)^{time} = e^{rate \cdot time}}

Because of the magic of exponents, we can avoid having two powers and just multiply rate and time together in a single exponent.
由于指數(shù)函數(shù)的神奇之處瞳筏，我們可以避免有兩個指數(shù)。我們只需要將增長率和時間乘在一起作為一個指數(shù)就可以了牡昆。

The big secret: e merges rate and time. 大秘密：e融合的增長率和時間姚炕。

59 This is wild! e^x can mean two things:
這真是太野蠻啦！e^x可以有兩種理解：

• x is the number of times we multiply a growth rate: 100% growth for 3 years is e^3
  x可以是我們乘以增長率的次數(shù)：以100%（復(fù)合）增長3年就是e^3
• x is the growth rate itself: 300% growth for one year is e^3
  x也可以是增長率本身：以300%的增長率（復(fù)合）增長一年就是e^3

Won’t this overlap confuse things? Will our formulas break and the world come to an end?

60 It all works out.When we write:
其實沒問題丢烘。當我們寫下：

\displaystyle{e^x}

\displaystyle{x = rate \cdot time}

61 Let me explain. When dealing with continuous compound growth, 10 years of 3% growth has the same overall impact as 1 year of 30% growth (and no growth afterward).
舉個例子。當遇到連續(xù)復(fù)合增長時播瞳，以3%的增長率復(fù)合增長十年掸刊，其結(jié)果和以30%的復(fù)合增長率增長一年的結(jié)果是一樣的。

• 10 years of 3% growth means 30 changes of 1%. These changes happen over 10 years, so you are growing continuously at 3% per year.
  以3%的增長率復(fù)合增長10年赢乓，等于30次1%的復(fù)合增長忧侧。這30次復(fù)合增長在十年才完成石窑。因此，你在以每年3%的增長率連續(xù)增長苍柏。

• 1 period of 30% growth means 30 changes of 1%, but happening in a single year. So you grow for 30% a year and stop.
  一次30%的復(fù)合增長也等于30次1%的復(fù)合增長尼斧，只是在一年完成姜贡。因此试吁，你以每年30%的復(fù)合增長率一年，然后就停了楼咳。

62 The same “30 changes of 1%” happen in each case.The faster your rate (30%) the less time you need to grow for the same effect (1 year). The slower your rate (3%) the longer you need to grow (10 years).
兩種情況都是“30%次1%復(fù)合增長”熄捍。同樣的結(jié)果，增長率越快母怜，用的時間就越短余耽。反之，用的時間就越長苹熏。

63 But in both cases, the growth is e^.30 = 1.35 in the end. We’re impatient and prefer large, fast growth to slow, long growth but e shows they have the same net effect.
但是兩種情況下碟贾，最終的增長都是e^.3 = 1.35。我們是不耐心的轨域，我們喜歡大的快的增長勝過慢的長的袱耽，但e告訴我們這些因素疊加的效果是一樣的。

So, our general formula becomes:
所以我們的通用公式變?yōu)椋?/p>

\displaystyle{growth = e^x = e^{rt}}

If we have a return of r for t time periods, our net compound growth is e^(rt). This even works for negative and fractional returns, by the way.
如果回報率是r干发，經(jīng)歷的時間是t朱巨，那么復(fù)合增長就是e^(rt)。

Example Time!

64 Examples make everything more fun. A quick note: We’re so used to formulas like 2^x and regular, compound interest that it’s easy to get confused (myself included). Read more about simple, compound and continuous growth.
例子可以讓一切理論變得有趣枉长。速記：公式比如2^x冀续、常規(guī)和復(fù)合利息是如此的常見，因此也很容易混淆（我也是）必峰。閱讀更多關(guān)于簡單復(fù)合連續(xù)增長洪唐。

65 These examples focus on smooth, continuous growth, not the jumpy growth that happens at yearly intervals. There are ways to convert between them, but we’ll save that for another article.
這些粒子將著重于平滑連續(xù)增長，而不是那種間斷的一年一次的增長吼蚁。他們兩者是可以相互轉(zhuǎn)換的桐罕，不過這將被留在另一篇文章里講。

Example 1: Growing crystals
例1：晶體的形成

66 Suppose I have 300kg of magic crystals. They’re magic because they grow throughout the day: I watch a single crystal, and in the course of 24 hours it sheds off its own weight in crystals. (The baby crystals start growing immediately at the same rate, but I can’t track that — I’m watching how much the original sheds). How much will I have after 10 days?
設(shè)想我有300kg魔法水晶桂敛。他們的魔法之初在于他們這樣天都在生長：我觀察一塊水晶功炮，？术唬？薪伏？。（剛產(chǎn)生的水晶會以相同的速率立刻開始增長粗仓，但我無法跟蹤——我正在觀察有多少原始的水晶嫁怀？设捐？？）塘淑。10天后我能得到多少水晶呢萝招？

67 Well, since the crystals start growing immediately, we want continuous growth. Our rate is 100% every 24 hours, so after 10 days we get: 300 · e^(1 · 10) = 6.6 million kg of our magic gem.
Well，因為水晶是直接開始生長的存捺，我們想要的是連續(xù)增長槐沼。復(fù)合增長率是每天100%，因此10天后我們將得到：300 · e^(1 · 10) = 6600000kg捌治。

68 This can be tricky: notice the difference between the input rate and the total output rate. The “input” rate is how much a single crystal changes: 100% in 24 hours. The net output rate is e (2.718x) because the baby crystals grow on their own.
注意輸入增長率和總輸出增長率的區(qū)別岗钩。“輸入”增長率是單個的水晶如何變化：每24小時增長100%肖油，復(fù)合輸出增長率是e（2.718x）兼吓，那是因為紫水晶也要在它們自己的基礎(chǔ)上增長。

69 In this case we have the input rate (how fast one crystal grows) and want the total result after compounding (how fast the entire group grows because of the baby crystals). If we have the total growth rate and want the rate of a single crystal, we work backwards and use the natural log.
這種情況下我們已知輸入增長率（也就是單個的水晶如何增長）森枪，欲求復(fù)合增長后的總增長率（也就是包含“子”水晶的整個族群的增長量）视搏。如果已知總增長率，欲求單個水晶的增長率县袱，我們可以運用自然對數(shù)函數(shù)逆向計算浑娜。

Example 2: Maximum interest rates
例2：最大利率

70 Suppose I have $120 in an account with 5% interest. My bank is generous and gives me the maximum possible compounding. How much will I have after 10 years?
設(shè)我的銀行賬戶中有120美元，利息是5%显拳。銀行很慷慨棚愤，并且想要給我最可能大的復(fù)合增長率，十年后我有多少錢杂数？

71 Our rate is 5%, and we’re lucky enough to compound continuously. After 10 years, we get$120 · e^(.05 · 10) = $197.85. Of course, most banks aren’t nice enough to give you the best possible rate. The difference between your actual return and the continuous one is how much they don’t like you.
利息是5%宛畦，很幸運的是連續(xù)復(fù)合增長。10年后揍移，我們將得到：$120 · e^(.05 · 10) = $197.85次和。不過大多數(shù)銀行可不想給你這么高的利率。你的實際回報和連續(xù)增長的回報取決于他們有多喜歡你那伐。

Example 3: Radioactive decay
例3：放射性衰變

72 I have 10kg of a radioactive material, which appears to continuously decay at a rate of 100% per year. How much will I have after 3 years?
一種放射性物質(zhì)踏施，其每年的連續(xù)復(fù)合衰變率是100%，問10kg的該物質(zhì)經(jīng)過3年后還剩多少罕邀？

Zip? Zero? Nothing? Think again.
0畅形？沒有了？再想想诉探。

73 Decaying continuously at 100% per year is the trajectory we start off with. Yes, we do begin with 10kg and expect to “l(fā)ose it all” by the end of the year, since we’re decaying at 10 kg/year.
每年連續(xù)衰變100%只是在一開始的時候日熬。是的，開始的時候的確有10kg肾胯，并且預(yù)期到年底的時候會全部“消失”竖席，因為是以10kg/年的速度衰變的耘纱。

74 We go a few months and get to 5kg. Half a year left? Nope! Now we’re losing at a rate of 5kg/year, so we have another full year from this moment!
經(jīng)過幾個月剩下5kg。還剩半年就衰變完嗎毕荐？不束析，現(xiàn)在衰變的速率是5kg/年，因此此時又有一年來衰變憎亚！

75 We wait a few more months, and get to 2kg. And of course, now we’re decaying at a rate of 2kg/year, so we have a full year (from this moment). We get 1 kg, have a full year, get to .5 kg, have a full year — see the pattern?
又過幾個月员寇，剩下2kg。同樣虽填，這時的衰變速率變成了2kg/年丁恭。因此曹动，此刻起斋日，又有1年來衰變。又過幾個月墓陈，剩下1kg恶守，但還有1年的時間供它衰變。又過幾個月贡必，剩下.5kg兔港，但還有1年的時間供它衰變——看出其中的套路來了嗎？

76 As time goes on, we lose material, but our rate of decay slows down. This constantly changing growth is the essence of continuous growth & decay.
隨著時間推移仔拟，放射性物質(zhì)減少了衫樊，但是衰變的速度也減慢了。不間斷變化增長是連續(xù)增長和衰變的核心要點利花。

77 After 3 years, we’ll have 10 · e^(-1 · 3)= .498 kg. We use a negative exponent for decay — we want a fraction (1/e^(rt)) vs a growth multiplier (e^(rt)). [Decay is commonly given in terms of "half life" -- we'll talk about converting these rates in a future article.]
3年后科侈，有10 · e^(-1 · 3)= .498 kg。在處理衰變問題時炒事，我們用到了負指數(shù)——我們要分數(shù)（1/e^{(rt)）而不是增長乘數(shù)（e}(rt)）臀栈。［衰變通常用到的術(shù)語是“半衰期”——我們會在將來的文章中講到這些比率的轉(zhuǎn)換問題。］

More Examples
更多例子

78 If you want fancier examples, try the Black-Scholes option formula (notice e used for exponential decay in value) or radioactive decay. The goal is to see e^(rt) in a formula and understand why it’s there: it’s modeling a type of growth or decay.
如果你想要更精彩的例子挠乳，試一下布萊克-斯科爾斯期權(quán)公式或者放射性衰變权薯。目的是了解公式中的e^(rt)，而且理解為什么在那里：它為增長或衰變建立模型睡扬。

79 And now you know why it’s “e”, and not pi or some other number: e raised to “r*t” gives you the growth impact of rate r and time t.
現(xiàn)在你知道了為什么他是“e”盟蚣，而不是π或者其他的數(shù)字：e的（r*t）次方告訴你增長率r和時間t的影響。

There’s More To Learn 還有更多可以學(xué)

80 My goal was to:
我的目標是：

• Explain why e is important:It’s a fundamental constant, like pi, that shows up in growth rates.
  解釋e為什么很重要：和π一樣卖怜，它也是重要的基本常數(shù)屎开，它在增長率的問題里面就會出現(xiàn)。

• Give an intuitive explanation:e lets you see the impact of any growth rate. Every new “piece” (Mr. Green, Mr. Red, etc.) helps add to the total growth.
  給出一個直觀的解釋：e讓你了解任何增長率的影響韧涨，每一塊兒新的（小綠牍戚、小紅等）都在為總增長做貢獻侮繁。

• Show how it’s used: e^x lets you predict the impact of any growth rate and time period.
  展示它的應(yīng)用：e^x可以讓你預(yù)測任何增長率和時間帶來的影響。

• Get you hungry for more: In the upcoming articles, I’ll dive into other properties of e.
  激發(fā)你的好奇心：在接下來的文章里如孝，我將深入講解e的其它特性宪哩。

81 This article is just the start — cramming everything into a single page would tire you and me both. Dust yourself off, take a break and learn about e’s evil twin, the natural logarithm.
這篇文章是一個開始——把所有的東西都塞滿一張頁面上會讓你和我感到疲憊的。起身活動活動第晰，休息一下锁孟，然后學(xué)一學(xué)e的魔鬼雙胞胎——自然對數(shù)。