In your example 7834095.56/143.8 it doesn't divide evenly (as a terminating decimal). When do you stop calculating, and in what form is the anser to be given?
I contemplated getting up a calculator and trying it, but didn't, because I reckoned that if it didn't give a terminating decimal it would be even more interesting to get the child in question to ask such questions!
When I use a calculator I get 54479.10682 (a 10 digit approximation, not exact). But when you use long division, do you know a way of getting that the first digit of the quotient is 5 straight away without some trial and error (and subsequently the other digits the same way). Suppose, when you ask how many 10000's of 143.8 go into 7834095.56, if you carelessly guess 4 of them instead of 5, and subtract those off, you'll realise you haven't subtracted off enough 10000's of 143.8 and you have to subtract off one more before proceeding to the next digit of the quotient. The only way of avoiding this that I can see is to know the multiples of 143.8 up to 9*143.8, but I'd be curious to here other ideas. Can it be made purely mechanical. Of course it would be easier if we all used base 2.
Well, it *is* purely mechanical: you just gave the procedure yourself. Binary admittedly easier, if longer, though!
This does relate to various interesting issues (that might make good discussion points with interested children) though. For example, we can get an approximation of the answer by using approximations of the arguments, and we can bound the error, but nevertheless, knowing when we have been precise enough to be sure about the first digit can be tricky. If you don't have access to full information about the arguments, it can be more than tricky, and one can go off into study of the on-line computable functions (which do not include division, IIRR)...
Balanced ternary notation is a cute and accessible thing to talk about in this context...
(Disclaimer: none of this is remotely my field, but I think I kept this vague enough not to be wrong...)
