Name: 數學趣事 - 測量距離 (Math Antics - Measuring Distance)
Uploaded: 2025-03-22T04:51:14.000Z
Duration: 10 min 58 s

嗨，我是羅布。歡迎來到數學趣談。

形容詞最高級

In a previous video, we learned about the most common units for measuring distances.

在上一段視頻中，我們瞭解了測量距離的最常用組織、部門。

現在進行式

In this video, we're going to talk about how you actually use some of those units when making a measurement in real life.

在本視頻中，我們將講述在實際生活中如何使用這些組織、部門進行測量。

Making measurements is something you would typically learn how to do in science class, but since measurement uses math to get the job done, it's often taught in math class too.

進行測量通常是在科學課上學習的內容，但由於測量是用數學來完成的，所以數學課上也經常會講到。

Suppose we're given an object like this pencil, and we're asked to measure its length in centimeters.

假設我們得到了一個像這支鉛筆一樣的物體，要求我們用釐米來測量它的長度。

To do that, we'd need some method or device that will tell us how many centimeters long the pencil is.

要做到這一點，我們需要某種方法或裝置來告訴我們鉛筆有幾釐米長。

Remembering that a centimeter is roughly the width of a pinky finger,

記住，一釐米大約是小指的寬度、

I could just use my finger as that device and see how many finger widths it takes to get from one end of the pencil to the other, but something a little more accurate would be nice.

我可以用我的手指作為這個裝置，看看從鉛筆的一端到另一端需要多少手指寬度，但如果能更精確一點就更好了。

這時候，一把尺子就派上用場了。

I, the great King Rob, ruler of all the land, declare that the span of my royal hand shall henceforth be the measure of all things in my kingdom, great or small.

我，偉大的國王羅布，全境的統治者，宣佈從今往後，我的王室之手的跨度將成為我王國一切事物的尺度，無論大小。

In math and science, a ruler is a flat piece of material that has markings on it that correspond to standard units of distance.

在數學和科學中，尺子是一塊平面材料，上面有與標準距離組織、部門相對應的標記。

For example, this ruler has a series of markings that correspond to inches on one side and centimeters on the other.

例如，這把尺子有一系列標記，一邊對應英寸，另一邊對應釐米。

That's cool, we'll just ignore the inches side and use the centimeters side for this measurement.

沒關係，我們可以忽略英寸這一邊，使用釐米這一邊進行測量。

We start by moving the ruler so that the zero centimeter mark is aligned as closely as we can with one end of our pencil.

我們首先移動直尺，使零釐米標記儘可能與鉛筆的一端對齊。

Then we'll see where the other end of the pencil lies on the scale of centimeters.

然後，我們再看看鉛筆的另一端在釐米刻度上的位置。

Looking at the numbers, you can see that the other end lines up nicely with the 19th centimeter mark.

從數字上可以看出，另一端與 19 釐米的標記非常吻合。

But what if the pencil gets sharpened and then used and sharpened and used again so that it gets shorter?

但是，如果鉛筆削了再用，用了再削，削了再用，鉛筆變短了怎麼辦？

Now if we re-measure it with our ruler, you'll see that we have a small problem.

現在，如果我們用尺子重新測量一下，你會發現我們有一個小問題。

The tip of the pencil doesn't line up with any of the centimeter marks anymore.

鉛筆尖不再與任何釐米標記對齊。

It lies somewhere between the 17 and 18 centimeter marks.

它位於 17 釐米和 18 釐米之間。

We could just say that it's between 17 and 18 centimeters long, but it would be nice if we could be a little more accurate than that.

我們可以說它的長度在 17 釐米到 18 釐米之間，但如果能比這更精確一點就更好了。

Being more accurate means making a measurement that is closer to the true value.

更精確意味著測量結果更接近真實值。

Fortunately, most rulers divide the space between each centimeter mark into ten equal parts that represent millimeters which are exactly one-tenth of a centimeter.

幸運的是，大多數直尺將每個釐米標記之間的空間抽成十等份，代表毫米，而毫米正好是十分之一釐米。

Because the millimeter marks are so much smaller, they don't have numbers on them.

因為毫米標記小得多，所以上面沒有數字。

But if you look closely, you'll be able to count that there are nine smaller lines that divide the centimeter into ten equal parts.

但如果你仔細觀察，就會發現有九條小線將釐米抽成了十等份。

The middle of these nine lines is usually a little longer than the rest so that it's easier to tell where the halfway point is.

這九條線的中間部分通常比其他部分稍長，這樣就更容易分辨出中點的位置。

Using these subdivision marks, we can get a more accurate measurement of the length of our sharpened pencil.

利用這些細分標記，我們可以更精確地測量削好的鉛筆的長度。

Do you see how the pencil's tip almost lines up with the third subdivision line that comes right after the 17th centimeter mark?

你看到鉛筆尖幾乎與第 17 釐米標記後的第三條細分線對齊了嗎？

That means that the length of the pencil is 17 centimeters plus 3 millimeters, or 17.3 centimeters.

也就是說，鉛筆的長度是 17 釐米加 3 毫米，即 17.3 釐米。

Well, that's a pretty close measurement, but remember the tip of the pencil didn't line up exactly with the third subdivision line.

嗯，測量結果很接近，但請記住，鉛筆尖與第三條細分線並不完全一致。

Our ruler doesn't have markings smaller than a millimeter, so it will be hard for us to make a measurement more accurate than that.

我們的尺子上沒有小於一毫米的標記，所以我們很難進行比這更精確的測量。

For example, it looks like the pencil tip goes past the 3 millimeter mark by a very small amount… maybe just a tenth of a millimeter.

例如，鉛筆尖似乎超過了 3 毫米的標記，但幅度很小......可能只有十分之一毫米。

So we could estimate that its length is closer to 17.31 centimeters.

是以，我們可以估計它的長度接近 17.31 釐米。

If you really did need a more accurate measurement, you'd need to use a better measurement device that could provide that level of accuracy.

如果您真的需要更精確的測量，您就需要使用更好的測量設備來提供這種精確度。

For example, calipers and micrometers are devices that can measure distances as accurate as a tenth or even a hundredth of a millimeter.

例如，卡尺和千分尺是可以測量精確到十分之一甚至百分之一毫米距離的設備。

And certain measurement techniques using lasers can achieve even higher accuracies, down to extremely small units like nanometers.

某些使用脈衝光的測量技術甚至可以達到更高的精度，小到納米這樣的極小組織、部門。

Those more advanced types of measurements are beyond the scope of this video, but hopefully they'll help you realize something fundamental about the nature of measurement.

這些更高級的測量類型超出了本視頻的範圍，但希望它們能幫助你認識到測量的本質。

You can't measure the EXACT value of something.

你無法衡量某樣東西的確切價值。

There's always a limit to the accuracy you can achieve based on your measurement device.

測量設備所能達到的精度總是有限的。

Accuracy is basically how close a measured value is to the true value.

準確度基本上是指測量值與真實值的接近程度。

And in measurement, the idea is to get as close as possible, or at least as close as you need for your purposes.

而在測量中，我們的想法是儘可能地接近，或至少接近到您所需的目的。

Luckily, I didn't really need to know the length of this pencil down to the nearest tenth of a millimeter.

幸運的是，我並不需要知道這支鉛筆的長度精確到十分之一毫米。

In fact, I didn't really need to know its length at all, since I'm just gonna write with it.

事實上，我根本不需要知道它的長度，因為我只想用它來寫作。

Okay, now that you understand what accuracy is, and we've made a measurement that was accurate to the nearest centimeter as well as one that was accurate to the nearest millimeter, let's try making a measurement with the non-metric side of our ruler, which measures inches.

好了，既然你已經明白了什麼是精確度，而且我們已經進行了精確到釐米和精確到毫米的測量，那麼讓我們試著用尺子的非度量面進行測量吧，它測量的是英寸。

Suppose we want to know the length of this toothbrush in inches.

假設我們想知道這把牙刷的長度，組織、部門是英寸。

To measure that, we first line up one end of the toothbrush with the start of the inches scale on the ruler, which represents zero inches.

測量時，我們首先將牙刷的一端與尺子上英寸刻度的起點對齊，這代表零英寸。

Then we see where the other end of the toothbrush lies on that scale.

然後，我們再看看牙刷的另一端在天平上的位置。

Notice it's somewhere between 7 and 8 inches.

Can we get a more accurate measurement than that?

我們能得到比這更精確的測量結果嗎？

Fortunately, as was the case with centimeters on the other side of the ruler, inches are usually subdivided into fractions of an inch also.

幸運的是，就像尺子另一面的釐米一樣，英寸通常也被細抽成英寸的幾分之一。

On this particular ruler, each inch is subdivided into 8 equal parts.

在這把特殊的尺子上，每英寸被細分為 8 個相等的部分。

That means our toothbrush is not quite as long as 7 3⁄8 of an inch, but it's a little longer than 7 2⁄8 of an inch, which would be equivalent to 7 ¼ inches.

也就是說，我們的牙刷沒有 7 3⁄8 英寸長，但比 7 2⁄8 英寸長一點，相當於 7 ¼ 英寸。

Whoa, whoa, whoa! What are you talking about with all these fractions?

哇哦 哇哦 哇哦你說的這些分數是什麼意思？

I thought this video was about measurement, not fractions!

我以為這段視頻是關於測量的，而不是分數！

Well yes, but when things don't line up exactly with a particular unit, to get more accuracy, you need to use fractions of that unit.

嗯，是的，但當事情與某個組織、部門不完全一致時，為了獲得更高的精確度，就需要使用該組織、部門的分數。

The metric system makes that look easy because things are always divided by 10, so the fractions match up really nicely with our base 10 decimal system.

公制看起來很簡單，因為事物總是除以 10，所以分數與我們的基數 10 十進制系統非常匹配。

But English or American units have traditionally been divided up differently, so the fractions you use for them are a little bit trickier.

但英式或美式組織、部門的劃分方式歷來不同，是以使用的分數也比較麻煩。

So… you're saying this is all America's fault?

所以......你是說這都是美國的錯？

It's just a traditional system of units that goes way back in history to the days of monarchies, so I guess it's probably some king's fault.

這只是一種傳統的組織、部門制度，其歷史可以追溯到君主制時代，所以我猜這可能是某個國王的錯。

Anyway, it's true that the traditional way of subdividing inches is kinda messy when compared to the metric system, so it will help if we take a closer look.

總之，與公制相比，傳統的英寸細分方法確實有點亂，所以我們仔細看看會有幫助。

Basically, there's two ways that inches are commonly subdivided.

基本上，英寸通常有兩種細分方式。

One is based on dividing by 10, and the other is based on dividing by 2.

一個是除以 10，另一個是除以 2。

We'll start with the system that's based on dividing by 10 because that sounds a lot like the metric system, doesn't it?

我們先從除以 10 的系統開始，因為這聽起來很像公制，不是嗎？

It turns out that an inch can be divided up in a metric-like way even though an inch is NOT a metric unit. Here's how that works.

原來，儘管一英寸不是公制組織、部門，卻可以用類似公制組織、部門的方法來分割一英寸。下面是具體的操作方法。

You start with an inch and then divide it into 10 equal parts.

從一英寸開始，然後將其平均抽成 10 份。

Each mark represents a tenth of an inch, so you can express the fractional parts easily with decimal digits, just like you do with the metric system.

每個標記代表十分之一英寸，是以您可以像使用公制系統一樣，用小數位輕鬆表示小數部分。

For example, if a measurement came out to be 1 and 2 tenths inches, you'd just say it's 1.2 inches.

例如，如果測量結果是十分之二英寸，你就會說是 1.2 英寸。

Or if a measurement came out to be 5 and 8 tenths inches, you'd just say it's 5.8 inches.

或者，如果測量結果是十分之五又十分之八英寸，你就說是 5.8 英寸。

And when you need more accuracy, you can keep subdividing by 10 so that you'd get hundredths of an inch, thousandths of an inch, ten thousandths of an inch, and so on.

當你需要更高的精度時，你可以繼續用 10 細分，這樣你就可以得到百分之一英寸、千分之一英寸、萬分之一英寸，以此類推。

This way of dividing up inches is commonly used in American engineering since it has many of the benefits of the metric system even though it's based on inches.

這種以英寸為組織、部門的劃分方法在美國的工程設計中很常用，因為它具有公制的許多優點，儘管它是以英寸為組織、部門的。

The other way of dividing up inches, which is still commonly used in American construction, is to divide them up by two. Here's how that system works.

美國建築業仍然普遍使用的另一種英寸劃分方法是將英寸除以 2。以下是該系統的工作原理。

You start with an inch and then divide it into two equal parts.

從一英寸開始，然後將其抽成兩等份。

That means you can now measure to an accuracy of half an inch.

這意味著您現在可以將測量精度提高到半英寸。

Then you divide those half inches by two so you can measure to an accuracy of a quarter of an inch.

然後將半英寸除以二，這樣就能精確到四分之一英寸。

Then you divide those quarter inches by two so you can measure to an accuracy of an eighth of an inch.

然後將四分之一英寸除以二，這樣就能精確到八分之一英寸。

Dividing by two again lets you measure to an accuracy of a sixteenth of an inch.

再除以 2，測量精度就能達到十六分之一英寸。

And dividing by two again lets you measure to an accuracy of a thirty-second of an inch, and so on.

再除以 2，測量精度就能達到 30 秒英寸，以此類推。

The bases of these fractional parts of an inch are all different, but they're all powers of two.

這些小數部分的基數各不相同，但都是 2 的冪。

It's actually a really logical system if you think about it.

仔細想想，這其實是一個非常合理的系統。

But it has the disadvantage that it's harder to convert to decimal values when you need them.

但它有一個缺點，就是在需要時較難轉換成十進制數值。

It's also harder to add and subtract measurements if the fractions don't have the same base.

如果分數的基數不一樣，加減測量也比較困難。

When things are divided by ten, it's easy because we use decimal number places that are specifically designed for counting fractions like tenths, hundredths and thousandths.

當事情被十整除時，很容易，因為我們使用的小數位是專門用來計算十分之一、百分之一和千分之一等分數的。

But we don't have number places for halves, quarters and eighths.

但我們沒有二分之一、四分之三和八分之一的數字位置。

Instead, it's very common for people who use a lot of these traditional fractions of an inch to just memorize some of the most common equivalent decimal values and use a calculator to convert the rest.

相反，對於經常使用這些傳統的英寸分數的人來說，只需記住一些最常見的等效十進制值，然後使用計算器來轉換其餘的值是很常見的。

For example, they might memorize things like one-half equals 0.5, one-quarter equals 0.25, and one-eighth equals 0.125.

例如，他們可能會記住二分之一等於 0.5，四分之一等於 0.25，八分之一等於 0.125。

Now that you know the two main ways of dividing up inches, let's go back to our toothbrush example.

現在您已經知道了兩種主要的英寸劃分方法，讓我們回到牙刷的例子。

If we use a ruler that has divisions of a sixteenth of an inch, you can see that its length is about 7 and 5 sixteenths of an inch.

如果我們使用一把以十六分之一英寸為組織、部門的尺子，你可以看到它的長度大約是十六分之七又五英寸。

That should be plenty accurate for brushing my teeth!

這對我刷牙來說應該足夠準確了！

So now you know the basics of how you use a ruler to make measurements, but you may need to brush up on your fractions to be successful at it.

現在，你已經知道了如何使用尺子進行測量的基本方法，但要想成功地使用它，你可能還需要補習分數。

Last of all, I want to briefly introduce you to some low-tech devices that are commonly used to measure longer distances.

最後，我想向大家簡單介紹一些常用於測量較長距離的低科技設備。

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數學趣事 - 測量距離 (Math Antics - Measuring Distance)

stick

equivalent

basically

practice