Monday, January 28, 2013

Mathematics - bases taught in schools?

After my post regarding the new methods of multiplication and division I recalled an incident with the Bratii over the holiday period. As is often the case the entire family were discussing various topics one of which was Imperial vs. Metric measurements. My father was complaining regarding news and quiz questions that used the 'new' methods on the grounds that "no-one can understand them" this was quickly dismissed by presenting the point that it was not everyone merely himself and that this generation and the next would have no difficulty in imaging a "plume of dust 3km high". Also that this was far better for school work due to the reliance on base10 which was also used for basic mathematics. A point reinforced when I asked ho many feet there were in a mile and he came back with 1760 (which is yards) and then had to multiply by 3 in his head to provide an answer.

This led the conversation to the discussion of the various number bases used historically. Obvious base12 for inches into feet and pennies into shillings as well as base20 for shillings into pounds and others. Base10 seems only to have 'won' because that's the number of digits we normally possess. Although, it was pointed out, Base2 and Base16 is used in computers

"Why not just use Base2?" asked my mother "As that's the computer method"
"Because it's too large. If I wanted to represent 8b10 it would be as 1000b2" I replied
I then watched as both Bratus Pater and Bartus Major tilted their heads slightly in contemplation
"You're both working out if I've got that right" I laughed
Bratus Pater laughed and admitted to it but Major responded with "No! I've no idea what you're talking about"
This caused some wonderment and bemusement in both myself and his father. He tried some examples and neither Major nor Minor were any the wiser. I wondered if they'd been taught it under a different name and tried "Clock method" which was not understood by any including Bratus Pater until I explained it to him as modular arithmetic.

[Diversion - A simple method of base12 can be seen on an analogue clock (modular 12). Take a number such as 38b10 and travel around the clock subtracting 12 each time (similar to chunking) you go around 3 full times and need to travel 2 stops more (the modulus) the conversion becomes 3 mod 2 or 32b12]

At this point my father chimed in by asking how any of this was relevant or useful. Both myself and Bratus Pater (with our similar backgrounds) jumped in with the computer reference. We both explained that these (2 and 16) are the main bases that computers use; that although the majority of users have no requirement to ever know this it becomes more important at a lower programming level. Given that Bratus Major is moving towards an oil geology field, and that is a specialised subject, at some point he may well need to write or debug computer programmes to provide for something that cannot be obtained 'off the shelf'.

In terms of pure education I can now relate it to "gridding" as something that should be taught early on at the same time as basic base10 mathematics. Although base10 is the most used it appears to be taught in the same black box fashion as traditional multiplication. You start with the units and then move to the ten's the hundred's and so on. So what? Well why move up to hundred's? Consider if it were taught along side a base of 4.

We start with the units then move to the four's and then the sixteen's. Prompting the question why sixteen? So again in base10 we start with the unit's then the ten's then the ten-ten's then the ten-ten-ten's. In base4 again we start with the units and then move to the four's the four-four's and the four-four-four's. No need to get involved with exponents and powers just number multiplied by number.

To reiterate my father's question "Why does this matter?" Well it explains why the system works as it does and moreover demonstrates that the number 10 is not elevated to some special position. With a slight change in history we could all have been using base12 now.