When students first start learning about fractions and decimals, they see that
1/2 = .5 1/3 = .3333333333... 1/4 = .25 1/5 = .2 1/6 = .1666666666...
Some of the decimals are simple and others repeat, but evidently in a pretty straightforward way. Then they see
1/7 = .142857142857142857...
Where did that come from? Continuing a bit further
1/8 = .125 1/9 = .1111111111... 1/10 = .1 1/11 = .0909090909... 1/12 = .0833333333...
Has sanity returned? But then
1/13 = 0.076923076923...
Something is going on here, obviously. Rational numbers either terminate or repeat endlessly in blocks.
Given the block of digits abcd, for example, the fraction abcd/9999 will repeat as .abcdabcdabcd….
This is why 1/9 = .11111111... and similarly 2/9 = .22222222.... Also, 1/3 = 3/9 = .33333333....
In case you didn’t realize it, .99999999... = 9/9 = 1.
If you have n digits that repeat in a block, you divide the block by n 9s to get the fraction corresponding to the repeating decimal.
Now you can do designer repeating decimals. Love the sequence 54321? Then 54321/99999 = .5432154321....
Going back to one of the examples we saw before .142857142857142857... = 142857/999999. But then reducing fractions we have 142857/999999 = 1/7, as expected.
What about values like .0833333333...? This is
.0833333333... = .08 + .0033333333... = 8/100 + .33333333... /100 = 8/100 + (3/9)/100 = 9*8/9*100 + 3/9*100 = 75/900 = 1/12
So given a decimal with a final repeating section, you can get back to the fraction that has the decimal equivalence.
For fun, find the fractions that correspond to the decimals:
.001001001....202202202...1.2341234...7.54667667667...
Oh fun! I’d not seen the 9′s trick as a way to get the pattern into rational form. Makes it all look simple. So if we can delimit the repeating part, we know how to recover the exact rational form.
Dennis, the use of the 9s is an artifact of using base 10 and long division.
Now we just have to sit back and wait for the “.9999… is not equal to 1 !” nuts to find you.
Barry, I think we’re safe. Intel says that it’s close enough for extremely large values of 0. There’s a lot to be said for pragmatism.