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Bases: Doing arithmetic in counting systems other than denary (tens)




One of the demo lessons used many times in China, sometimes to a very large audience, was to demonstrate counting on one hand in binary. I learned to do this so as to keep track of my place during long rests in orchestral music (this happens to brass players especially) without needing to put my instrument down. I still use this. I shared this with very many people, including the band I played with in Cornwall.

Your thumb is the most mobile finger. Up is one, down is zero.
Your first finger, next to the thumb is a two when up.
Your middle finger is a four. If showing just four, you might want to hold the hand sideways so as not to annoy the PC brigade.
Your ring finger is an eight. Many find this hard to move without the other fingers moving.
Your little finger is a sixteen. This moves quite easily on its own for most of us.

Fingers can be combined.  Looking at you right hand palm up, if only the thumb is up I can write that as 00001.

Counting to seven is 00001 00010 00011 00100 00101 00110 00111    and you can have fun by remembering to put 00100 and 00110 sideways for very similar reasons.
Counting eight to fifteen is the same, but with your ring finger up. Some of us need to find a way to control that finger, perhaps by puttting all fingers on a surface like a desk or a knee. Counbting from sixteen upwards is as the previosu fifteen but with the little finger up too.  And up to thirty one.

If you were to use the other hand too, you could just pass a thousand  (that is, 2^10 -1) becasue the eleventh finger would be 2 10 of course.

On the occasions I have wanted to count significantly further than 31 I have counted in base three, ternary, by having a half-way position for each finger, number d here as 1 and 2.
Using just three fingers, we begin 001 002 010 011 012 020 021 022 100 … and the whole hand would count to two hundred and forty two – just on the one hand.

I have done this for such a long time that reading the numbers has become easy, and reading the numbers off someone else’s hand is not difficult. When we watched tv quiz shows as a family, we would keep the score on a hand, using multiples of five – University Challenge, for example. One hand reached 155 and we often scored just more than this, giving rise to the shout “Secondhand!” in between disputed answers. I have even attemped doing this ‘lesson' in Mandarin.

As an introduction to the idea of counting in somethigin that is not tens, it works well.

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