9.1Insight: Sometimes Infinity Can Be Measured

Here’s a quick brainteaser for you.两个朋友相距10英里，以每小时5英里的速度向对方移动。A mosquito files quickly between them, touching one person, then the other, on and on, until the friends high-five and the mosquito is squished.

假设蚊子以每小时20英里的速度前进。Can you figure out how far it flew before its demise?

Yikes.This one is tricky: once the mosquito leaves the first person, touches the second, and turns around… the first person has moved closer!We have an infinite number of ever-diminishing distances to add up.这个问题似乎很难解决，对吧?

Well, how about this reasoning: from the perspective of the people walking, they’re going to walk for an hour total.After all, they start 10 miles apart, and the gap shrinks at 10 miles per hour (5mph + 5mph).Therefore, the mosquito must be flying for an hour, and go 20 miles.

哇!Did we just find the outcome of a process with an infinite number of steps?I think so!

9.2Splitting A Whole Into Infinite Parts

It’s time to turn our step-by-step thinking into overdrive.Can we think about a finite shape being split into infinite parts?

• In the beginning of the course, we saw a circle could be split into rings.有多少?Well, an infinite number!
• A number line can be split into aninfinitenumber of neighboring points.How many decimals would you say there are between 1.0 and 2.0?
• 一只蚊子走过的路可以看作是一个整体，也可以看作是一个被细分成无数段的旅程。

When we have two viewpoints (the mosquito, and the walkers), we can pick the one that’s easier to work with.在这种情况下，步行者的整体观点更简单。With the circle, it’s easier to think about the rings themselves.It’s nice to have both options available.

Here’s another example: can you divide a cake into 3 equal portions, by only cutting intoquarters?

It’s a weird question… but possible!把整个蛋糕切成四等份。Share 3 pieces and leave 1.Cut the remaining piece into quarters.分享3块，留下1块。不断重复这个过程:在每个步骤中，每个人都得到了平等的份额，剩下的蛋糕也会被平均分配。Wouldn’t this plan maintain an even split among 3 people?

我们看到了无限x射线和延时视觉背后的直觉:放大，把一个整体变成无限序列。At first, we might think dividing something into infinite parts requires each part to be nothing.But, that’s not right: the number line can be subdivided infinitely, yet there’s a finite gap between 1.0 and 2.0.

9.3Two Fingers Pointing At The Same Moon

Why can we understand variations of the letter A, even when pixelated?

Even though the rendering is different, we see theidea being pointed to.All three versions, from perfectly smooth to jagged, create the same letter A in our heads (or, are you unable to read words when written out with rectangular pixels?).An infinite sequence can point to the same result we’d find if we took it all at once.

在微积分中，有关于如何找到无限步的结果的详细规则。而且，有些序列是无法计算出来的。But, for this primer, we’ll deal with functions that behave nicely.

We’re used to jumping betweenfiniterepresentations of the same idea (5 = V = IIIII).Now we’re seeing we can convert between a finite andinfiniterepresentation of an idea, similar to\( \frac{1}{3} = .333\ldots = .3 \ + \ .03 \ + \ .003 \ + \ \ldots \).

当我们把一个圆变成一个环形三角形时，我们说:“我们的圆中无限多的圆环可以变成无限多的板，组成一个三角形。”And the resulting triangle is easy to measure.”

Today’s goal isn’t to become an expert on infinity.It’s to intuitively appreciate a few practical conclusions:

• Infinitely many parts can combined to a finite result, if they decrease fast enough
• A process with limited (but improving) precision can point to the same result as one with infinite precision

In Calculus terms, this means the conclusions drawn from our finite (but growing) sequence of steps can be trusted¹.