A Real Example: Understanding e
实例:理解e
Understanding the number e has been a major battle. e appears all of science, and has numerous definitions, yet rarely clicks in a natural way. Let’s build some insight around this idea. The following section will have several equations, which are simply ways to describe ideas. Even if the equation is gibberish, there’s a plain-english idea behind it.
理解数字e一直以来都是困难的。e出现在科学的各个分支中，有无数个定义，但却很少以一种自然的方式出现。让我们来建立对它的直观。下面的部分有几个等式，可以用来简单描述一些概念。哪怕等式看上去像胡扯，我们仍然能发现它背后的简明理解。
Here’s a few popular definitions of e:
一些e的定义(图在后面):
The first step is to find a theme. Looking at e’s history, it seems it has something to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.
首先是找到实质。看看e的历史，它总是和增长与利率有关。e是在做商业计算时被发现的，而不是抽象的数学概念所以利率，或称增长，也许是它的核心。
Let’s look at the first definition, in the upper left. The key jump, for me, was to realize how much this looked like the formula for compound interest. In fact, it is the interest formula when you compound 100% interest for 1 unit of time, compounding as fast as possible.
首先看看它最初的定义。在这个图的左上角。对我而言，关键是看出这个东西与复利计算公式有几分相似。实际上，这就是你以100％的利率在单位时间内获得的利息公式。
Definition 1: Define e as 100% compound growth at the smallest increment possible.
定义1:(我没法描述这句，丢人.jpg→是山寨gpt给的翻译)将e定义为在最小增量上以100%的复利增长。
Let’s look at the second definition: an infinite series of terms, getting smaller and smaller. What could this be?
看看第二个定义:无穷的多项式，而且越来越小。这又是个什么?
\displaystyle{e = {1 \over 0!} + {1 \over 1!} + {1 \over 2!} + {1 \over 3!} + \cdots}
After noodling this over using the theme of “interest” we see this definitions shows the components of compound interest. Now, insights don’t come instantly — this insight might strike after brainstorming “What could 1 + 1 + 1/2 + 1/6 + …” represent when talking about growth?”
用之前我们找到的"利率"的概念来艰难地理解这个，它显示了这概念的组成。但是灵感还没有来。也许在头脑风暴后就会来了:"谈论增长时，1 + 1 + 1/2 + 1/6 + …代表什么?"
Well, the first term (1 = 1/0!, remembering that 0! is 1) is your principal, the original amount. The next term (1 = 1/1!) is the “direct” interest you earned — 100% of 1. The next term (0.5 = 1/2!) is the amount of money your interest made (“2nd level interest”). The following term (.1666 = 1/3!) is your “3rd-level interest” — how much money your interest’s interest earned!
好吧，首项(0！=1)是你的本金。第二项是你"直接"的利息。100％的1。第三项是你的利息的利息。第四项是你利息的利息！
Money earns money, which earns money, which earns money, and so on — the sequence separates out these contributions.There’s much more to say, but that’s the “growth-focused” understanding of that idea.
钱生钱，钱生钱生钱——这个多项式展现了它们各自的贡献。这里还能作文章，但是这个概念的核心理解就是"增长"。
Definition 2: Define e by the contributions each piece of interest makes
定义2:e是每一份利息做出的贡献。
Neato.
好。
Now to the 3rd, and shortest definition. What does it mean? Instead of thinking “derivative” (which turns your brain into equation-crunching mode), think about what it means. The feeling of the equation. Make it your friend.
现在是第三个，最短的定义。它是什么?别再想导数了，你的大脑会进入方程计算模式。想想它的意义。方程的感觉。让它成为你的朋友。
\displaystyle{\frac{d}{dx}Blah = Blah}
It’s the calculus way of saying “Your rate of growth is equal to your current amount”. Well, growing at your current amount would be a 100% interest rate, right? And by always growing it means you are always calculating interest — it’s another way of describing continuously compound interest!
它以微积分的形式讲述了"你的增长速率等于你现在的量"。以你现在的量增长，当然就是100％的利率吧?而且它永远在增长，你就得永远计算利息——另一种描述复利的方式！
Definition 3: Define e as a function that always grows at 100% of your current value
定义3:定义e为一个永远以你当前的量翻倍的方式增长的函数
Nice — e is the number where you’re always growing by exactly your current amount (100%), not 1% or 200%.
好——e是精确描述你以当前的量的100％增长的数字，不是1％也不是200％。
Time for the last definition — it’s a tricky one. Here’s my interpretation: Instead of describing how much you grew, why not say how long it took?
来到最后一个定义——这个很麻烦。这是我的理解:为什么不用增长的时间(次数）来替代增长量?
If you’re at 1 and growing at 100%, it takes 1 unit of time to get from 1 to 2. But once you’re at 2, and growing 100%, it means you’re growing at 2 units per unit time! So it only takes 1/2 unit of time to go from 2 to 3. Going from 3 to 4 only takes 1/3 unit of time, and so on.
你从1开始，增长率100％，需要一次增长才能到2。但一旦你增长到2，你就能每次增长2！所以只需要二分之一的次数增长到3。从3到4只需要三分之一的次数，以此类推。
The time needed to grow from 1 to A is the time from 1 to 2, 2 to 3, 3 to 4… and so on, until you get to A. The first definition defines the natural log (ln) as shorthand for this “time to grow” computation.
从1到A的次数是从1到2，2到3，3到4…直到增长到A。第一个定义就将自然对数㏑以缩写的方式表示了增长次数。
ln(a) is simply the time to grow from 1 to a. We then say that “e” is the number that takes exactly 1 unit of time to grow to. Said another way, e is is the amount of growth after waiting exactly 1 unit of time!
lnA正是从1增长到A所需的次数。我们就可以说，e是一次增长会增长出的量。换句话说，e就是一次增长的数量的极限！
Definition 4: Define the time needed to grow continuously from 1 to a as ln(a). e is the amount of growth you have after 1 unit of time.
定义4:(和上面的表述一样，偷个懒)
Whablamo! These are four different ways to describe the mysterious e. Once we have the core idea (“e is about 100% continuous growth”), the crazy equations snap into place — it’s possible to translate calculus into English. Math is about ideas!
哇，四种方法描述神秘的e！只要我们掌握了核心观点，疯狂的方程式就原型毕露了——可以将微积分翻译成人话。数学是有关思想的！ |
网站是 https://betterexplained.com/
