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Sunday, 4 February 2007

Counting on your Fingers

In my post on bases of number systems I asked, rhetorically, "Why ten?" Why do we use base ten rather than another base? It's worth noting that this is not just a cultural thing - most cultures use 5, 10, or 20 as a base for their way of counting. The obvious answer is that we have five fingers on each hand; ten fingers total; twenty fingers and toes. This is supported by the fact that, in some cultures, the word for 'five' is related to the word for 'hand'. Indeed, in English, the word 'digit' can also mean 'finger'. People have been counting on their fingers for a long time.

The only exception that I know of, off the top of my head, is the Babylonians. Their system was essentially in base 60. There is an advantage to this.

Why is the five times table easy? 5, 10, 15, 20, 25... it's a nice pattern. The reason the pattern exists is because 5 x 2 = 10. As a direct result of this, every element in the sequence of multiples of five has the same final digit as the element two steps back. Similarly, every element in the sequence of multiples of two has the same final digit as the element five steps back (check this if it doesn't seem immediately obvious). Times tables are easier for numbers which are factors of the base you are using. Indeed, multiplication in general is easier for numbers which are factors of the base you are using. For instance, a quick way of multiplying by five is to multiply by ten and then divide by 2. And fractions? 1/5 = 0.2 and 1/2 = 0.5, but if you want to divide by three or seven or nine - by anything which is not a multiple of 2 or 5 - you're going to end up with an infinite (recurring) decimal expansion. In fact, six is a multiple of 2 but 1/6 still has an infinite decimal expansion because the fact that it is also a multiple of 3 mucks things up.

I hope you can see that using a base that had a large number of factors would give you considerable advantages. Ten is pretty useless in this regard actually. Short of using a prime number (which has no factors apart from the obvious factorisation into one times the number itself) we really couldn't have done much worse. Sixty, on the other hand... sixty is rather large, but let me show you why the Babylonians liked it:

60
= 2 x 30
= 3 x 10
= 4 x 15
= 5 x 12
= 6 x 10

2, 3, 4, 5 and 6 all go nicely into 60, and bring some other numbers along for the ride. That's why the Babylonians used it.

Closer to our usual ten, twelve would have been pretty good:

12
= 2 x 6
= 3 x 4

2, 3, 4, and 6. For a small number like 12, that's about as good as you could hope for. Why couldn't we have had six fingers?

Actually...

There's a lot to be said for counting on your fingers in base six! Look at it this way. In base 10 we have 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. On our hands, we have only five fingers, but we can also make a zero by putting up no fingers. So we have 'digits' 0, 1, 2, 3, 4, 5, representable by holding up that number of fingers on a single hand. We count like this:

ONE: hold up one finger (on the right hand, say)
TWO: hold up two fingers on the right hand
THREE: three fingers on the right hand
...
FIVE: five fingers on the right hand
SIX: zero fingers on the right hand, one finger on the left

Can you see how this works? In order to recover the original number, we multiply the number of fingers on the left hand by six, and add that to the number of fingers on the right hand (multiplied by one). This is similar to the way 23 is equal to the digit on the left multiplied by ten, added to the digit on the right (multiplied by one). We are in base six rather than base ten. We continue:

SEVEN: one finger on the right hand, one finger on the left
EIGHT: two fingers on the right hand, one finger on the left
...
TWELVE: zero fingers on the right hand, two fingers on the left

The highest number possible has five fingers on both hands. That's five times six, plus five = 35. 6 x 6 = 36 plays a similar role in this system to the one played by 10 x 10 = 100 in the usual one. We would need a third hand ("digit") to represent it.

Base 6 would be better than base 10 in some ways. 6 is smaller than 10, but it has the same number of factors. The only number less than 6 that isn't a multiple of a factor of 6 is 5. Maybe we were created by a being that knew more about maths than about how to create people that would be good enough at maths to notice from the first that 6 is a good base to use for counting on your fingers. I am obliged to confess that I had been counting on my fingers in base five for years before a friend of mine pointed out that six would be more sensible!

1 comment:

Cairnarvon said...

If you count in binary you can get to 1,023 using just your fingers. Conversion can be a bit tricky, but if you're dealing with computers a lot, it's really handy to be able to do.