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

3,472,087 views

2016-12-15 ・ TED-Ed


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

3,472,087 views ・ 2016-12-15

TED-Ed


Please double-click on the English subtitles below to play the video.

00:06
How high can you count on your fingers?
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It seems like a question with an obvious answer.
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After all, most of us have ten fingers,
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or to be more precise,
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eight fingers and two thumbs.
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This gives us a total of ten digits on our two hands,
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which we use to count to ten.
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It's no coincidence that the ten symbols we use in our modern numbering system
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are called digits as well.
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But that's not the only way to count.
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In some places, it's customary to go up to twelve on just one hand.
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How?
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Well, each finger is divided into three sections,
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and we have a natural pointer to indicate each one, the thumb.
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That gives us an easy to way to count to twelve on one hand.
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And if we want to count higher,
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we can use the digits on our other hand to keep track of each time we get to twelve,
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up to five groups of twelve, or 60.
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Better yet, let's use the sections on the second hand
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to count twelve groups of twelve, up to 144.
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That's a pretty big improvement,
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but we can go higher by finding more countable parts on each hand.
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For example, each finger has three sections and three creases
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for a total of six things to count.
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Now we're up to 24 on each hand,
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and using our other hand to mark groups of 24
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gets us all the way to 576.
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Can we go any higher?
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It looks like we've reached the limit of how many different finger parts
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we can count with any precision.
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So let's think of something different.
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One of our greatest mathematical inventions
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is the system of positional notation,
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where the placement of symbols allows for different magnitudes of value,
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as in the number 999.
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Even though the same symbol is used three times,
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each position indicates a different order of magnitude.
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So we can use positional value on our fingers to beat our previous record.
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Let's forget about finger sections for a moment
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and look at the simplest case of having just two options per finger,
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up and down.
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This won't allow us to represent powers of ten,
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but it's perfect for the counting system that uses powers of two,
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otherwise known as binary.
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In binary, each position has double the value of the previous one,
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so we can assign our fingers values of one,
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two,
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four,
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eight,
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all the way up to 512.
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And any positive integer, up to a certain limit,
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can be expressed as a sum of these numbers.
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For example, the number seven is 4+2+1.
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so we can represent it by having just these three fingers raised.
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Meanwhile, 250 is 128+64+32+16+8+2.
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How high an we go now?
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That would be the number with all ten fingers raised, or 1,023.
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Is it possible to go even higher?
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It depends on how dexterous you feel.
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If you can bend each finger just halfway, that gives us three different states -
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down,
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half bent,
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and raised.
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Now, we can count using a base-three positional system,
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up to 59,048.
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And if you can bend your fingers into four different states or more,
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you can get even higher.
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That limit is up to you, and your own flexibility and ingenuity.
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Even with our fingers in just two possible states,
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we're already working pretty efficiently.
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In fact, our computers are based on the same principle.
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Each microchip consists of tiny electrical switches
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that can be either on or off,
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meaning that base-two is the default way they represent numbers.
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And just as we can use this system to count past 1,000 using only our fingers,
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computers can perform billions of operations
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just by counting off 1's and 0's.
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