［第01修订版］不行了，今天必须要翻译这篇文章了！

原标题：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 总是困扰着我 —— 不是这个字母本身，而是作为自然常数的e。它到究竟是什么东西？

2 Math books and even my beloved Wikipedia describe e using obtuse jargon:
数学课本，甚至我深爱的 Wikipedia 在解释 e 时用的都是鸟语：

The mathematical constant e is the base of the natural logarithm.
数学常数 e 就是自然对数的底。

3 And when you look up the natural logarithm you get:
而当你查看什么是自然对数时，你得到的结果是：

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.
自然对数，曾经叫双曲对数，是以 e 为底的对数，这里的 e 是一个无理数，它约为2.718281828459。

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，这样循环引用真的好吗~ 这就像一本词典用错综复杂 来解释迷宫，然后用迷宫 来解释错综复杂：是的，这没错，然并卵。像复杂 这样的日常词语到底是怎么了？

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——为了严谨性，许多数学解释都是格式化的、枯燥的。但这对于一个初学者并没有多少帮助（在一定程度上我们都是初学者）。

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，没办法，今天不得不由我来分享我的关于e的直观而且深刻的见解了。这次，你可以防好你严格的数学书。下面是这个见解的视频：

e is NOT Just a Number

e不仅仅是个数字

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.
说 e 是“一个近似值为 2.71828… 的数”无异于说 π是“一个近似值为 3.1415…无理数”。这完全正确，但完全不着要点。

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是圆的周长和其半径的比值。它是所有圆周的固有的基本比例。它影响了圆、球、圆柱等的周长、面积、体积和表面积的计算。π很重要，它告诉我们所有的圆都是相关的，不是说三角函数（sin，cos，tan）都是从圆导出的。

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是所以连续增长过程的基本比率。e让你得到一个简单的增长比率（所有的变化发生在年底），并且找到复合连续增长产生的影响。每一纳秒，你只要增长一点点。

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会出现在任何以指数连续增长的地方：人口、放射性物质的衰变、利息的计算等等。甚至增长不平滑的jagged system也可以用e近似计算。

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).
就像每个数字都可以成数字“1”（基本单位）的缩放版，每个圆都可以看成单位圆（半径为1）的缩放版，每种增长率都可以看成e的缩放版（基本增长，完美复合的）。

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，e不是一个晦涩的、表面看来很随意的一个数。e 代表着，所有连续增长都是一个共同增长率的缩放版。

Understanding Exponential Growth 什么是指数增长

13 Let's start by looking at a basic system that doubles after an amount of time. For example,
让我们从一个基本的系统开始：隔一段时间就会翻倍的系统。例如：

• Bacteria can split and “doubles” every 24 hours
  细菌每24小时就会分裂并且数量加倍

• We get twice as many noodles when we fold them in half.
  做拉面时，如果我们从中间对折，我们就会得到两倍的面条

• Your money doubles every year if you get 100% return (lucky!)
  你的钱每一年都可以翻倍，如果你每年都得到100%的回报的话

And it looks like this:
它看起来就像这样：

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.
分裂成两个或者说翻倍是一种非常常见的过程。当然我们也能翻三倍或者四倍，但翻倍较为方便，所以我们就讨论这种情况。

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:
计算可知，如果分裂x次，那我们得到的结果将是一开始的2^{x倍。分裂1次，结果是一开始的2}1倍，也就是2倍。分裂4次，结果将是一开始的2^4倍，也就是16倍。通用的公式长这样：

\displaystyle{ growth = 2^x }

17 Said another way, doubling is 100% growth. We can rewrite our formula like this:
换一种说法，翻倍其实就是增长率为100%的增长。这样，我们我们可以把上面的公式改写成下面的样子：

\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?
公式还是同一个公式，但我们把2还原成了它的真实意思：原本的数量（1）和增长率100%。哈哈，聪明吧！

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:
当然我们也可以用任何的数字（比如50%，25，200%等等）来替换掉100%，并且得到这个新增长率下的新公式。这样，一段时间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.
这仅仅意味着我们用我们的回报率作为增长率，（1+回报率），增长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.
我们的公司有一个假设，就是增长是一次一次发生的。我们的细菌要先长大，长大，在长大，等积累到一定大小的时候，“嘣”一下分裂成两个。而我们感兴趣的增长奇迹般的出现在刚好是一年的时候。上面的公式中，增长是间断发生的，并且是立刻发生的，图中绿色的圆点突然就出现了。

22 The world isn’t always like this. If we zoom in, we see that our bacterial friends split over time:
但这个世界并不总是如此。如果我们深入细致观察，我们就能看到我们的细菌朋友随时间是如何增长的。

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.
小绿不是突然跳出来：他慢慢的从小蓝里面长出来。经过一个单位的时间（例如24小时），小绿才长成。然后他才“性成熟”，可以生育后代。

24 Does this information change our equation?
但这会对我们的公式造成任何改变吗？

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。在细菌分裂这件事情上，才长出一半的绿细胞并不能做任何的事情，直到他们完全长大并从母体分离。上面那个公式仍然成立。

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.
但钱不一样。只要我们挣了一分的利息，这一分的利息就可以产生它的一厘的利息。我们不必等到利息长大成一元——新增加的钱不需要成熟。

26 Based on our old formula, interest growth looks like this:
根据我们的“旧公式”，利息的增长像这样：

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:
不过这一次又有一点不对：所有的利息都在最后一天才生成。让我们放大一下并且把一年分成两部分。我们每年获得100%的利息，或者六个月获得50%的利息。因此我们前6个月将获得50分的利息，在后六个月获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:
但这仍然不对！的确我们的本金（小蓝）经过一年挣得了一块钱的利息。但经过六个月我们，我们的到了50分利息，我们刚不小心忽略了它。这50分也能够获得它自己的利息：

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:
因为我们的半年利率是50%，因此那50分前经过后半年会获得25分的利息（50分×50%）。因此在一年结束的时候我们会得到：

• Our original dollar (Mr. Blue)
  我们一块钱的本金（图中的小蓝）

• The dollar Mr. Blue made (Mr. Green)
  小蓝的1美元利息（图中的小绿）

• The 25 cents Mr. Green made (Mr. Red)
  吕先生胜出的25分利息（图中的小红）

30 Giving us a total of $2.25. We gained $1.25 from our initial dollar, even better than doubling!
这样返还给我们的将是2.25元。我们从一元的本金里的到了1.25元的利息，比翻倍还要赚的多！

31 Let’s turn our return into a formula. The growth of two half-periods of 50% is:
让我们把我们的回报放回一个公式里。两个半年50%的增长是这样子的：

\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:
是该xx的时候了。取代原来分割成两个50%的增长阶段的做法，让我们把利息的增长分割成三个33%的阶段。谁说我们必须要等六个月才能开始得到利息呢？让我们把计算利息的次数增加一些。三次复合增长的图象很有有趣：

4 month compound interest

33 Think of each color as shoveling money upwards towards the other colors (its children), at 33% per period:
设想每一段时间，每种颜色都都以33%的比率向上面一种颜色（也就是它们的子代）送钱。

• Month 0:We start with Mr. Blue at $1.
  开始：我们以代表一美元的小蓝开始。

• Month 4:Mr. Blue has earned 1/3 dollar on himself, and creates Mr. Green, shoveling along 33 cents.
  4月末：小蓝挣到了来自它自己的1/3美元，由此创造了小绿，并把这三分之一美元给了小绿。

• 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：小蓝又挣到了另外33美分，并且把它给了小绿，这时小绿就增长到了66美分。与此同时，小绿也在他上一次的基础上增长了33%，挣了11美分（33美分*33%），并由此创造了小红。

• 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：继续下去就有点儿疯狂了。小蓝又增长了33美分，并把它给了小绿，这使得小绿长成了完整的一美元。与此同时，小绿在他上一次的基础上增加了33%，也就是22美分（66美分*33%），并且将其给了小红，这样小红就达到了33美分。与此同时小红也在自己上一次的基础上增加了33%，也就是4美分（11美分*33%），并由此创造了小紫。

34 Phew! The final value after 12 months is: 1 + 1 + .33 + .04 or about 2.37.
瞧，经过十二个月之后，我们将得到：1 + 1 + .33 + .04 ≈ 2.37。

35 Take some time to really understand what’s happening with this growth:
那我们花点时间来理解在这增长过程中到底发生了什么：

• 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.
  每个色块都挣得了自己的利息并且把他拱手送给了下一个颜色。而这些新挣的钱也可以以自己为基础挣钱，并且送给下一个色块儿。后面以此类推。

• 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.
  我喜欢把原来的那一美元（也就是小蓝）看成从来没有变。小兰只是用自己挣得的钱创造了小绿，并且把后面挣的钱都送给了它。在图中，小蓝用蓝箭头清晰的指出了它是如何“包养”小绿的。

• Mr. Green just happens to create and feed Mr. Red (green arrow), but Mr. Blue isn’t aware of this.
  小绿也像小蓝一样创造了小黄并且“包养”它（绿箭头），而小蓝对此事一无所知。*真是一个令人悲伤的包养故事……

• 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.
  随着时间的推移，由于小蓝的包养，小绿持续的长大，可是它也把越来越多的钱花在了小红身上。在4月末到8月末这段时间里，小绿给了小红11美分。在8月末到12月末这段时间，小绿又给了小红22美分。因为在八月末的时候小绿只有66美分。如果我们将这个图标继续隐身小绿又将会给小红33美分，因为小绿在12月末的时候已经是一个完整的一美元了。

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.
懂了吗？开头的时候的确很难——开始我把这些表格放在一起的时候，我甚至把自己都弄得有点糊涂了。但后来就明白了，那只是每一美元都在制造小帮手，小帮手又在制造小帮手，如此下去而已。

37 We get a formula by using 3 periods in our growth equation:
我们的到了将一个周期分为三个阶段的增长公式：

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

38 We earned $1.37, even better than the $1.25 we got last time!
我们挣到了1.37美元，比上次的1.25美元还要好！

Can We Get Infinite Money? 我们可以得到无限多的钱吗？

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:
为什么不把每个阶段的时间都缩短呢？以每个月或每一天或每个小时，甚至每豪秒为一个阶段怎么样呢？我们得到的回报将像火箭一样猛涨吗？这次只是做梦了，我们能得到更好的回报，但会有一个极限。你可以试试将不同的n代入我们的增长公式里 看看最后的总回报：

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!
数字在变大并停留在2.718左右。等等，这怎么这么像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:
没错，在枯燥得令人乏味的数学术语里面，嗯被定义为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.
这个极限最终会收敛于一个值，这个已经被证明过。正如你所见，不管我们把时间分的多么多么的短，我们得到的总回报最终停留在了2.718附近。

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.
数字e（2.718）是一个周期以100%的复合增长率增长的最大结果。当然开始的时候你只是希望从1增长到2（那就是100%的增长率嘛）。但随着一小步一小步的推进，你不但获得了每一小步的红利，而且这个红利本身也要继续产生红利。当整个增长周期结束后，你得到的是e（2.718），而不仅仅是2。e是以100%复合增长得到的最大的结果。

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.
因此如果我们开始的时候有1美元，那么一100%的增长率连续复合增长，那返回给我们的将是1e美元。如果能开始两美元，那反正给我们的就是2e美元。如果我们开始的时候有11.79美元，那返还给我们的将是11.79e美元。

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就像速度的极限（就像光速c），他表示在连续的增长过程中，你究竟可能获得多大的增长快慢。你可能不是总能达到极限速度，但它至少是一个参考值：你可以根据这个宇宙常数写出任意增长率的增长。

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：小心区分增加量和最后结果。1变成了e（2.718），增加量（增长率）是171.8%。而e是包含增加量的最后结果。

What about different rates? 如果增长率不同呢？

47 Good question. What if we grow at 50% annually, instead of 100%? Can we still use e?
问得好！如果我们的年增长率只有50%而不是100%，我们仍然能用e吗？

48 Let’s see. The rate of 50% compound growth would look like this:
请看，年增长率为50%的复合增长会是这样：

\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，这次我们怎么处理？记住50%是总回报，n是我们为了获得复合增长而将这个增长过程分割的次数。如果取n=50，我们就可以把我的整个生产过程分割成五十段，每一段的增长率都为1%。

\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%:
当然这不是无限的，不过已经很细了。现在设想我们同样把增长率为100%的整个过程分解为（100段）增长率为1%的小段。

\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，似乎有点儿线索了。通常我们有100段每段增长率为1%的连续积累的增长，在总增长率为50%的情况下，我们有50段这样的增长。

Different exponential rates

52 What is the difference between the two numbers? Well, it’s just half the number of changes:
第二个数比起第一个数有什么区别呢？仅仅是增长的小段数少了一半罢了。

\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，这个0.5正好是数字e的指数。而且这是通用的：如果我们的总增长率是300%，我们可以把整个增长的过程分割成300小段，每一段增长率为1%的小段。最终，这将以e^3的倍数使本金增翻三倍。

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).
尽管这个增长过程我们可以看成加法（每次增加1%），我们也要记住它实际上是乘法（×1.01）。这也是我们为什么要用指数（重复相乘）和平方根（e^(1/2)的意思是，将增长的次数减半，将乘的次数减半）。

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:
前面我们取的是1%，其实我们也可以取更小的增长率（.1%，0.0001%，甚至无限小的数）。核心是，不管我们取多大的增长率，这只是e的一个新指数而已。

\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:
设想我们用300%的增长率，增长整整两年。我们就用他自己去乘以一年的增长率（e^3）。

\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.
由于指数函数的神奇之处，我们可以避免有两个指数。我们只需要将增长率和时间乘在一起作为一个指数就可以了。

The big secret: e merges rate and time. 大秘密：e融合的增长率和时间。

59 This is wild! e^x can mean two things:
这真是太野蛮啦！e^x可以有两种理解：

• x is the number of times we multiply a growth rate: 100% growth for 3 years is e^3
  x可以是我们乘以增长率的次数：以100%（复合）增长3年就是e^3
• x is the growth rate itself: 300% growth for one year is e^3
  x也可以是增长率本身：以300%的增长率（复合）增长一年就是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:
其实没问题。当我们写下：

\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).
举个例子。当遇到连续复合增长时，以3%的增长率复合增长十年，其结果和以30%的复合增长率增长一年的结果是一样的。

• 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%的增长率复合增长10年，等于30次1%的复合增长。这30次复合增长在十年才完成。因此，你在以每年3%的增长率连续增长。

• 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%的复合增长也等于30次1%的复合增长，只是在一年完成。因此，你以每年30%的复合增长率一年，然后就停了。

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).
两种情况都是“30%次1%复合增长”。同样的结果，增长率越快，用的时间就越短。反之，用的时间就越长。

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.
但是两种情况下，最终的增长都是e^.3 = 1.35。我们是不耐心的，我们喜欢大的快的增长胜过慢的长的，但e告诉我们这些因素叠加的效果是一样的。

So, our general formula becomes:
所以我们的通用公式变为：

\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.
如果回报率是r，经历的时间是t，那么复合增长就是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.
例子可以让一切理论变得有趣。速记：公式比如2^x、常规和复合利息是如此的常见，因此也很容易混淆（我也是）。阅读更多关于简单复合连续增长。

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.
这些粒子将着重于平滑连续增长，而不是那种间断的一年一次的增长。他们两者是可以相互转换的，不过这将被留在另一篇文章里讲。

Example 1: Growing crystals
例1：晶体的形成

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?
设想我有300kg魔法水晶。他们的魔法之初在于他们这样天都在生长：我观察一块水晶，？？？。（刚产生的水晶会以相同的速率立刻开始增长，但我无法跟踪——我正在观察有多少原始的水晶？？？）。10天后我能得到多少水晶呢？

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，因为水晶是直接开始生长的，我们想要的是连续增长。复合增长率是每天100%，因此10天后我们将得到：300 · e^(1 · 10) = 6600000kg。

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.
注意输入增长率和总输出增长率的区别。“输入”增长率是单个的水晶如何变化：每24小时增长100%，复合输出增长率是e（2.718x），那是因为紫水晶也要在它们自己的基础上增长。

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.
这种情况下我们已知输入增长率（也就是单个的水晶如何增长），欲求复合增长后的总增长率（也就是包含“子”水晶的整个族群的增长量）。如果已知总增长率，欲求单个水晶的增长率，我们可以运用自然对数函数逆向计算。

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?
设我的银行账户中有120美元，利息是5%。银行很慷慨，并且想要给我最可能大的复合增长率，十年后我有多少钱？

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%，很幸运的是连续复合增长。10年后，我们将得到：$120 · e^(.05 · 10) = $197.85。不过大多数银行可不想给你这么高的利率。你的实际回报和连续增长的回报取决于他们有多喜欢你。

Example 3: Radioactive decay
例3：放射性衰变

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?
一种放射性物质，其每年的连续复合衰变率是100%，问10kg的该物质经过3年后还剩多少？

Zip? Zero? Nothing? Think again.
0？没有了？再想想。

73 Decaying continuously at 100% per year is the trajectory we start off with. Yes, we do begin with 10kg and expect to “lose it all” by the end of the year, since we’re decaying at 10 kg/year.
每年连续衰变100%只是在一开始的时候。是的，开始的时候的确有10kg，并且预期到年底的时候会全部“消失”，因为是以10kg/年的速度衰变的。

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!
经过几个月剩下5kg。还剩半年就衰变完吗？不，现在衰变的速率是5kg/年，因此此时又有一年来衰变！

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?
又过几个月，剩下2kg。同样，这时的衰变速率变成了2kg/年。因此，此刻起，又有1年来衰变。又过几个月，剩下1kg，但还有1年的时间供它衰变。又过几个月，剩下.5kg，但还有1年的时间供它衰变——看出其中的套路来了吗？

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.
随着时间推移，放射性物质减少了，但是衰变的速度也减慢了。不间断变化增长是连续增长和衰变的核心要点。

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，有10 · e^(-1 · 3)= .498 kg。在处理衰变问题时，我们用到了负指数——我们要分数（1/e^{(rt)）而不是增长乘数（e}(rt)）。［衰变通常用到的术语是“半衰期”——我们会在将来的文章中讲到这些比率的转换问题。］

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，试一下布莱克-斯科尔斯期权公式或者放射性衰变。目的是了解公式中的e^(rt)，而且理解为什么在那里：它为增长或衰变建立模型。

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.
现在你知道了为什么他是“e”，而不是π或者其他的数字：e的（r*t）次方告诉你增长率r和时间t的影响。

There’s More To Learn 还有更多可以学

80 My goal was to:
我的目标是：

• Explain why e is important:It’s a fundamental constant, like pi, that shows up in growth rates.
  解释e为什么很重要：和π一样，它也是重要的基本常数，它在增长率的问题里面就会出现。

• 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.
  给出一个直观的解释：e让你了解任何增长率的影响，每一块儿新的（小绿、小红等）都在为总增长做贡献。

• Show how it’s used: e^x lets you predict the impact of any growth rate and time period.
  展示它的应用：e^x可以让你预测任何增长率和时间带来的影响。

• Get you hungry for more: In the upcoming articles, I’ll dive into other properties of e.
  激发你的好奇心：在接下来的文章里，我将深入讲解e的其它特性。

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.
这篇文章是一个开始——把所有的东西都塞满一张页面上会让你和我感到疲惫的。起身活动活动，休息一下，然后学一学e的魔鬼双胞胎——自然对数。