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  • How high can you count on your fingers?

    用手指頭數數能數到多大?

  • It seems like a question with an obvious answer.

    問題的答案似乎顯而易見

  • After all, most of us have ten fingers,

    畢竟大部分的人都有十根手指頭

  • or to be more precise,

    或著更精確一點

  • eight fingers and two thumbs.

    八根手指及兩根拇指

  • This gives us a total of ten digits on our two hands,

    兩隻手總共十個數字

  • which we use to count to ten.

    能讓我們算到 10

  • It's no coincidence that the ten symbols we use in our modern numbering system

    這也難怪現代數字系統用的十個符號

  • are called digits as well.

    也叫數字

  • But that's not the only way to count.

    但是這不是數數的唯一方法

  • In some places, it's customary to go up to twelve on just one hand.

    在某些地方用一隻手 數到 12 是很平常的事

  • How?

    怎麼數?

  • Well, each finger is divided into three sections,

    這麼說吧,每根手指 都可以分成三個指節

  • and we have a natural pointer to indicate each one, the thumb.

    我們還有一個天生的指標 能指出每個指節,就是拇指

  • That gives us an easy way to count to twelve on one hand.

    這樣我們很容易 就能用一隻手數到 12

  • And if we want to count higher,

    如果還想算到更大的數字

  • we can use the digits on our other hand to keep track of each time we get to twelve,

    我們還能用另一隻手 來記我們算了幾次 12

  • up to five groups of twelve, or 60.

    總共可以算五次 12,就是 60

  • Better yet, let's use the sections on the second hand

    更棒的還在後面 我們還可以用第二隻手的指節

  • to count twelve groups of twelve, up to 144.

    數十二次 12,總共 144

  • That's a pretty big improvement,

    這進步很大吧

  • but we can go higher by finding more countable parts on each hand.

    但是還可以算到更大 只要找出每一隻手可以拿來數的部分

  • For example, each finger has three sections and three creases

    舉例來說,每根手指 都有三個指節及三個皺褶

  • for a total of six things to count.

    這樣總共可以算到 6

  • Now we're up to 24 on each hand,

    現在每一隻手可以算到 24

  • and using our other hand to mark groups of 24

    再用另一隻手去數 總共算了幾次 24

  • gets us all the way to 576.

    我們就可以算到 576

  • Can we go any higher?

    還能再算大一點嗎?

  • It looks like we've reached the limit of how many different finger parts

    手掌能拿來精確算數的部分

  • we can count with any precision.

    好像都用完了

  • So let's think of something different.

    來想點別的吧

  • One of our greatest mathematical inventions

    人類偉大的數學發明之一

  • is the system of positional notation,

    就是位置記法這套系統

  • where the placement of symbols allows for different magnitudes of value,

    字符的位置決定數值大小

  • as in the number 999.

    就像 999 這個數字

  • Even though the same symbol is used three times,

    雖然同一個字符用了三次

  • each position indicates a different order of magnitude.

    每個字符的位置 都代表不同的數量級

  • So we can use positional value on our fingers to beat our previous record.

    所以我們能用手指的位置值來創新高

  • Let's forget about finger sections for a moment

    先把手指指節忘了吧

  • and look at the simplest case of having just two options per finger,

    來看最簡單的情況 每根手指只有兩個選擇

  • up and down.

    上或下

  • This won't allow us to represent powers of ten,

    這不能讓我們算十的次方

  • but it's perfect for the counting system that uses powers of two,

    對二的次方計數系統卻很完美

  • otherwise known as binary.

    也就是所謂的二進位

  • In binary, each position has double the value of the previous one,

    二進位中每個位置 都比前一個位置大兩倍

  • so we can assign our fingers values of one,

    所以我們可以把手指的值記為 1

  • two,

    2

  • four,

    4

  • eight,

    8

  • all the way up to 512.

    一直到 512

  • And any positive integer, up to a certain limit,

    在某個限度前的每一個正整數

  • can be expressed as a sum of these numbers.

    都可以用這些數字的總和來表現

  • For example, the number seven is 4+2+1.

    譬如 7 就是 4+2+1

  • so we can represent it by having just these three fingers raised.

    所以我們可以伸出 這幾根手指來表現這個數字

  • Meanwhile, 250 is 128+64+32+16+8+2.

    250 則是 128+64+32+16+8+2

  • How high can we go now?

    現在我們能算到多大?

  • That would be the number with all ten fingers raised, or 1,023.

    就到十根指頭都伸出來為止 即 1,023

  • Is it possible to go even higher?

    還可以再更大嗎?

  • It depends on how dexterous you feel.

    那就要看你的手指有多靈活了

  • If you can bend each finger just halfway, that gives us three different states -

    如果你手指只能彎一半 那就給我們三個不同的狀態

  • down,

  • half bent,

    彎一半

  • and raised.

    伸出來

  • Now, we can count using a base-three positional system,

    現在我們能用 3 為基數的位置系統

  • up to 59,048.

    算到 59,048

  • And if you can bend your fingers into four different states or more,

    如果你能把手指 彎成四種以上不同的狀態

  • you can get even higher.

    你就能算到更大的數字

  • That limit is up to you, and your own flexibility and ingenuity.

    上限取決於你及你的靈活度 和有多心靈手巧

  • Even with our fingers in just two possible states,

    即使我們只能把手指彎成兩個狀態

  • we're already working pretty efficiently.

    我們也已經很有效率了

  • In fact, our computers are based on the same principle.

    事實上我們的電腦就是 基於同一個原理運作

  • Each microchip consists of tiny electrical switches

    每個微晶片都有很小的電路開關

  • that can be either on or off,

    可以是開或關

  • meaning that base-two is the default way they represent numbers.

    代表二進位為電腦表現數字的預設法

  • And just as we can use this system to count past 1,000 using only our fingers,

    我們能夠用手指系統算超過 1,000

  • computers can perform billions of operations

    電腦卻能執行數十億的運算

  • just by counting off 1's and 0's.

    只用 1 和 0就行了

How high can you count on your fingers?

用手指頭數數能數到多大?

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A2 初級 中文 TED-Ed 數字 位置 數到 數數 次方

【TED-Ed】你以為手指只能數到十?其實可以更多! (How high can you count on your fingers? (Spoiler: much higher than 10) - James Tanton)

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    wangjiechin 發佈於 2017 年 05 月 28 日
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