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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?

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