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

翻译人员: Cissy Yun 校对人员: Roger NI

How high can you count on your fingers?

你用手指最高能数到多少？

It seems like a question with an obvious answer.

这个问题答案好像很明显

After all, most of us have ten fingers,

毕竟，大多数的我们都有十根手指

or to be more precise,

或者更精确的说

eight fingers and two thumbs.

八根手指和两个大拇指

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

这是我们两只手一共有的十位数

which we use to count to ten.

我们用来数到十

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

我们在现代编号系统使用十个符号并不是巧合

are called digits as well.

这称为十进制

But that's not the only way to count.

但是这不是唯一的计数方法

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

有些地方，他们有种方法，可以用一只手数到12

How?

怎么做？

Well, each finger is divided into three sections,

每根手指分为3节

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

我们有个可以指示每一个的天然指针，拇指

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

这让我们很容易用一只手数到12

And if we want to count higher,

而且如果我们想数到更高

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

我们可以用另一只手来帮助我们记录数到12的次数

up to five groups of twelve, or 60.

提高到5组12 也就是60

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

更厉害的是，我们可以在第二只手上用同样的方法

to count twelve groups of twelve, up to 144.

得到12组个12 可以数到144

That's a pretty big improvement,

这是个很大的进步

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

但我们可以在每只手上找到更多计数部分来提高数字

For example, each finger has three sections and three creases

例如 每根手指有3节和3条折痕

for a total of six things to count.

一共6个部分可用于计数

Now we're up to 24 on each hand,

现在我们把每只手提高到了24

and using our other hand to mark groups of 24

再用另一只手记录多少组24

gets us all the way to 576.

这样我们把数字提高到576

Can we go any higher?

我们还能数到更高吗？

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

看来我们已经达到手指分区的极限

we can count with any precision.

得到计数最大的精度

So let's think of something different.

所以让我们换个方式思考

One of our greatest mathematical inventions

我们最伟大的数学发明之一

is the system of positional notation,

位值制计数法

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

在不同的位置表示不同的值

as in the number 999.

例如数字999

Even though the same symbol is used three times,

尽管同样的数字用了三次

each position indicates a different order of magnitude.

但每个位置表示不同的数量级

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

所以我们可以给手指的位置赋值去打破我们之前的记录

Let's forget about finger sections for a moment

让我们先忘了手指分节

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

来看最简单的情况 每根手指有两个选项

up and down.

伸开和收起

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

这就不适用以十为进率

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

但非常符合以二为进率的计数系统

otherwise known as binary.

称为二进制

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

二进制中，每个位置都是前一个值的两倍

so we can assign our fingers values of one,

所以我们可以给手指赋值为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,

在一定范围内的任何正整数

can be expressed as a sum of these numbers.

都可以表示为这些数字之和

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

例如 数字7为 4+2+1

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

所以我们伸出这三根手指表示它

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

同理 250为 128+64+32+16+8+2

How high an we go now?

现在我们能数到多少了？

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

伸出全部十根手指来表示 1,023

Is it possible to go even higher?

还有没有可能更高了？

It depends on how dexterous you feel.

这取决于你有多灵巧

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

如果你每根手指都能做到弯曲一半 我们就有了三种不同的状态

down,

收起

half bent,

一半

and raised.

伸出

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

现在 我们基于这三种状态系统计数

up to 59,048.

可以达到 59,048

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

如果你的手指可以弯曲四种或者更多不同状态

you can get even higher.

你还可以达到更高

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

这个极限取决于你 和你的灵活性和创造性

Even with our fingers in just two possible states,

即使我们手指只有两种状态

we're already working pretty efficiently.

我们也已经做的很棒了

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

事实上 我们的电脑正是基于同样的原则

Each microchip consists of tiny electrical switches

每个芯片都由微小的电子开关组成

that can be either on or off,

它们的打开或关闭

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

是默认基于二的表示数字的方式

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

刚刚我们用这种方法 仅用手指就数超过1,000

computers can perform billions of operations

计算机可以执行数十亿的操作

just by counting off 1's and 0's.

只是通过计算1和0