How high can you count on your fingers? (Spoiler: much higher than 10) - James Tanton

譯者: Regina Chu 審譯者: Songzhe Gao

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 to 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,

eight,

4

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 an 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,

下

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