Subject: repeating decimals
I understand that with repeating decimals (those with a pattern), the number of digits repeated is put into fraction form with the same number of 9's
ex. 0.4444 = 4/9 0.145145145 = 145/999
How can I explain why this is so? Some inquisitive 8th graders are anxious
to find out!
Suppose that the number you have is 0.4444... then 10 times this number is 4.4444... Hence subtracting the number from 10 times itself gives 9 times the number. But 4.444... - 0.444... = 4. Thus 9 times the number is 4 and hence the number is 4/9.
Th second example is similar. I don't know if your students have seen any algebra but it is easier to say with some algebraic notation.
1000x = 145.145145...
Thus 1000x - x = 145.145145... - 0.145145145...
and hence 999x = 145 so
x = 145/999
Now, working with decimals, this always works and always gives 10k - 1 where k is how many places it takes for a first repetition.
Like many things we actually do in algebra, this depends on the visual appearance of the writing of decimals - and the sense of 'shift' to make most of the number look the same before and after. How we write, what things look like, has a lot to do with 'algebraic thinking' and 'good notation'. I tell my students that algebra is 'cosmetics'. Changing the way something looks without changing what is underneath (no 'surgery')!
In principle, if you go into work further on in high school,
this is related to the methods for any 'geometric series'
Penny and Walter