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  • How high can you count on your fingers?


  • It seems like a question with an obvious answer.


  • After all, most of us have ten fingers, or to be more precise, eight fingers and two thumbs.


  • 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.


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


  • 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?


  • Well, each finger is divided into three sections, and we have a natural pointer to indicate each one, the thumb.


  • 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.


  • 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?


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


  • So let's think of something different.


  • 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.


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


  • 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.


  • 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.


  • 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.


  • 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.


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

    250 則是 128+64+32+16+8+2。

  • How high can we go now?


  • 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?


  • It depends on how dexterous you feel.


  • If you can bend each finger just halfway, that gives us three different states - down, half bent, and raised.


  • 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.


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


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


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


  • 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.


  • 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。

How high can you count on your fingers?


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