to find the decimal expansions of 1 3 2 ,,c the blocks of t 7, 7, 7, 71 , 4. (a)
ID: 3210377 • Letter: T
Question
to find the decimal expansions of 1 3 2 ,,c the blocks of t 7, 7, 7, 71 , 4. (a) Use long division What do you observe about the patterns of the blocks i in the given Wha repeating digits? (b) Convert each of th fractions without reducing these fractions to lowest terms from t) to make the following conclusions about the vert each of the repeating decimals in (a number 1-142857: (i) 999999, and (ii) the first 6 multiples of are whole numbers yclically , ie context to these fascinating sor of obtained from itself by permuting the digits of "c is a divisor of whichare 428571-285714, etc. (Note: Part (c) could be done independently of (a) and (b), of course, but parts (a) and (b) give facts. It seems unlikely that without getting the decimal expansion of the proper fractions with denominator 7, these facts about 142857 would have been discovered. For more information, see [Kalman].)Explanation / Answer
We have 1/7 =0.142857142857...
The repitition is 142857
2/7 = 2(0.142857...) and so on.
b) To get back fraction
Let x =0.142857142857....
1000000x = 142857.142857.....
Subtract to get
999999x = 142857
x = 142857/999999 = 15873/111111
(i) If theta is a divisor of 999,999 then theta = 7
. Then fraction = 1/7, 2/7 we get back the original fractions.
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