Wednesday, 26 September 2012

Duodecimal finger counting

This is a second attempt to explain how to count in base twelve on your fingers.  The former attempt was said to be confusing because it was too "three dimensional".  I don't understand this criticism, so i've redone the video in the possibly vain hope that it's clearer.

The point is this.  We have ten digits on our hands and as a result we use ten symbols for our numbers:  0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.  As a result, we have a number system with at least two drawbacks which make arithmetic harder and lead people to feel that maths as such is more difficult than it really needs to be.  Both of these arise from the awkwardness of the number ten.

Ten divides into two and five, and of course itself and one, and nothing else.  This means that the number of regular patterns in the multiplication table in numbers written using this system is rather limited.  There are the repeating pattern in the five times' table, the even numbers of the twos, the downward counting of the nines, the repetition of the elevens and the adding a zero of the tens.  That's it.

The second problem is associated with decimal fractions.  Just up to ten, the fractions associated with the following numbers can never be exact:  3, 6, 7, 9, 11 and an infinite number of further numbers.  Each of these is recurring:

This makes things difficult unnecessarily.

One suggestion is to use hexadecimal - base sixteen.  This has much to recommend it but i have personally chosen the duodecimal base instead.  Moreover, i would advocate the universal use of the duodecimal system where possible.  This is the system which uses twelve symbols instead of ten, and as a result most recurring decimals and hard-to-discern or absent patterns are eliminated from arithmetic.  Twelve divides into two, three, four and six as well as itself and one.  This gives each of those times tables a pattern.  It also gives eight and nine times tables a pattern because they are multiples of factors.  Eleven has a pattern akin to nine in the decimal system and even thirteen has a pattern.  All that's needed is to use two extra symbols, and for typographical ease i use A and B, though there are other choices.

Another advantage is the absence of recurring "duodecimals".  In this system, 2, 3, 4 and 6 have no recurring digits.  Hence fractions are also easier.

One objection frequently made to this is that we tend to have fewer than a dozen digits on our hands.  This is not a problem because what we do have on our hands is a dozen finger bones.  This video shows how to count using those finger bones, and it is in fact possible to count to 144 using this system, and to do various forms of arithmetic using easy finger manipulations - addition is demonstrated here, but other arithmetic operations are also easier.

Moreover, i believe the decimal number system should be abandoned for weights and measures and money, for the same reason, and replaced with duodecimal.

As it stands, we have a disabling numerical notation that teaches the lie that arithmetic and maths are hard and to be hated and feared.  Get rid of decimal and that fear and hatred will go away.  Do it now.