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, or to be more precise, eight fingers and two thumbs.

畢竟大部分的人都有十根手指，或者更精確一點，八根手指及兩根拇指

This gives us a total of ten digits on our two hands, which we use to count to ten.

兩隻手總共十個數字能讓我們算到 10。

It's no coincidence that the ten symbols we use in our modern numbering system are called digits as well.

這也難怪現代數字系統用的十個符號也叫數字。

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

這麼說吧，每根手指都可以分成三個指節，而我們還有一個天生自帶的指標，也就是拇指，能指出每個指節。

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

如此，我們很容易就能用一隻手數到 12。

And if we want to count higher, we can use the digits on our other hand to keep track of each time we get to twelve, up to five groups of twelve, or 60.

如果想算到更大的數字，我們還能用另一隻手來記算了幾次 12，總共可以算五次 12，也就是到 60。

Better yet, let's use the sections on the second hand to count twelve groups of twelve, up to 144.

更棒的還在後面，我們還可以用第二隻手的指節數十二次 12，總共 144。

That's a pretty big improvement, but we can go higher by finding more countable parts on each hand.

這進步很大吧！但還可以算到更大，只需要找出每隻手可以拿來數的所有部分。

For example, each finger has three sections and three creases for a total of six things to count.

舉例來說，每根手指都有三個指節及三個皺褶，這樣總共可以算到 6。

Now we're up to 24 on each hand, and using our other hand to mark groups of 24 gets us all the way to 576.

現在每一隻手就可以算到 24，再用另一隻手去數總共算了幾次 24，我們就可以算到 576。

Can we go any higher?

還能再算更多嗎？

It looks like we've reached the limit of how many different finger parts we can count with any precision.

手掌能拿來精確算數的部分好像都用完了。

So let's think of something different.

我們換個方式思考吧！

One of our greatest mathematical inventions is the system of positional notation, where the placement of symbols allows for different magnitudes of value, as in the number 999.

人類偉大的數學發明之一就是位置記法這套系統，字符的位置決定數值大小，就像 999 這個數字。

Even though the same symbol is used three times, each position indicates a different order of magnitude.

雖然同一個字符用了三次，但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挑戰新極限。

Let's forget about finger sections for a moment and look at the simplest case of having just two options per finger, up and down.

先把手指指節忘了吧，來看最簡單的情況，每根手指只有兩個位置選擇：上或下。

This won't allow us to represent powers of ten, but it's perfect for the counting system that uses powers of two, otherwise known as binary.

這不能讓我們算十的次方，但對二的次方計數系統卻十分完美，也就是所謂的二進位。

In binary, each position has double the value of the previous one, so we can assign our fingers values of one, two, four, eight, all the way up to 512.

二進位中每個位置都比前一個位置大兩倍，所以我們可以把手指的值記為 1、2、4、8，一直到 512。

And any positive integer, up to a certain limit, can be expressed as a sum of these numbers.

在某個限度前的每一個正整數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的數了？

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

十根指頭都伸出來的數即為 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 - down, half bent, and raised.

如果你手指只能彎一半，那就可以有三個不同的狀態：折起、折一半跟伸出來。

Now, we can count using a base-three positional system, up to 59,048.

現在我們能用 3 為基數的位置系統算到 59,048。

And if you can bend your fingers into four different states or more, you can get even higher.

如果你能把手指彎成四種以上不同的狀態，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、你的靈活度和你有多心靈手巧。

Even with our fingers in just two possible states, we're already working pretty efficiently.

即使我們只能把手指彎成兩個狀態，也已經很有效率了。

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

事實上電腦就是基於同一個原理運作的。

Each microchip consists of tiny electrical switches that can be either on or off, meaning that base-two is the default way they represent numbers.

每個微晶片都有很小的電路開關可以控制開或關來代表二進位，而這就是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, computers can perform billions of operations just by counting off 1's and 0's.

就像我們能夠用手指系統算到超過 1,000，電腦能執行數十億的運算，只靠運算 1 和 0。