O.K so lets find out what you know about dividing one number

by another

1) heres the first and best known ... it is called

'long division' by old people

lets start by dividing 12 by 7

"seven goes into twelve once remainder five"

so 12/7 = 1 rem 5

we use a number base 10 [decimal system] though this is not true of

the past where bases of 6 are quite common

among middle eastern cultures

anyway .. because we use a number base 10 we multiply

the remainder by 10 and try and divide it again

so multiply 5 by 10 in our example giving 50

then 50 divided by 7 = goes 7 times remainder 1

50/7 = 7 rem 1

the remainder 1 multiplied by 10 gives 10 and so on

the answer is .17142857142857142857..

this bit 142857 keeps repeating

and is effectively 10 divided by 7

[ill come back to the repeating sequence later on and explain why]

2) heres another form of division .. I've called this

'over division' because of the way you do it

lets start by dividing 10 by 7

normally you say "7 goes into 10 once" but what were going to do

is to make the number of sevens just bigger than the number each time

.. so for the number of sevens to be bigger than 10 we need 2 of them

7 times 2 = 10 remainder 4 (-4 actually)

so multiply the remainder by the base again 4 times 10 = 40

then 6 * 7 = 42 is the nearest number of sevens remainder 2

then 2 * 10 = 20 and so 3 * 7 = 21 rem 1 and were back where we started

to summarise

the resulting fraction is 263263263263

the 263 bit keeps repeating its self

so we've got this fraction but what does it mean ???

well it says my first approximation to 10/7 is 2 tenths

then i take 6 hundredths AWAY from it [this is .14]

then i add 3 thousandths back on to it [this is .143]

then i take 2 ten-thou. AWAY from it [this is .1428]

then i take 6 hundred-thou and add it [this is .14286]

etc ..

were doing just the same in 1)

the difference is that here the approximation to the number

zig zags toward it from above then below .. sandwiching the

result somewhere in between

the usual method 1) approximates from below getting closer and closer

Here i have to ask you .. DOES IT EVER GET THERE ???

It certainly gets very close .. but no matter how close it gets

theres always a disparity of 1 unit minimum and 8 maximum

HOWEVER ...

what happens if i use a number base of 7 ..

10 becomes 013 base 7 [thats no 49's ... one 7 ... and 3 units

i hope the 49 bit isn't confusing .. i put it in to highlight

how I'm using the 7 base instead of tens and hundreds]

OK so 10 is represented as 13 ..and 7 is 10

so the result of dividing 10 by 7 is 1.300000 ..

an exact rational number

This brings us to the relationship between the divisor and the number

base we are using

If the two are relatively prime .. i.e. they have no common divisor

then the sequence of division has to be an approximation where we can say

that it has a sequence at least 1 long and repeated

actually at most its = length of the divisor - 1

so for 7 a repeated sequence 7 - 1 i.e 6 digits long

for 11 it could be 10 , for 13 it could be 12 long

for 17 it could be 16 digits long before repeating and so on

there is a very obvious reason for this

the remainder of each division must obviously be less than the

divisor. If the remainder is repeated then the sequence will

obviously be repeated from that point, so it follows that the sequence

will be repeated before we have done the divisions 'divisor - 1' times

[note: if it was 'divisor times' we would have divided the number exactly

as 0 would be a remainder ]

example :-

dividing 10 by 7 gives remainders 3,2,6,4,5,1 .. then 3,2,6,4,5,1 etc

now

dividing 10 by 13 gives remainders 10,9,12,3,4,1, ...10,9,12,3,4,1,

so although we may say there are 12 remainders they repeat after

with a half period of 6

you probably have also noticed that the remainders have a sort of

sequence within that .. in 3,2,6,4,5,1 above

3 pairs with 4, 2 with 5 and 6 with 1 these being 3 numbers on from each other

The actual result of 10/7 ...142857... shares this property,

they sum to 9 and the entire number is rank 9

[see Ranks of numbers in the What The Artist Tried

.. its potentially a lot more powerful than the logic here ]

i.e. 142857 is divisible by 9

this is unfinished as we still have to discuss the lengths of sequences

ive got a table listing wether they are shorter than 'divisor - 1'

to put on .. from recollection all variants in period are possible

eg the sequence could repeat itself say 5 times where you would expect

it to be 'divisor - 1' long