The Period of 61/97
Date: 01/12/2001 at 08:46:22 From: Celia Subject: period of 61/97 I need to know the period of 61/97 (if there is one). I've calculated it to the 31st decimal place, but can't tell if it is a finite or infinite and periodic... Is there any way I can figure it out? Thank you! Celia
Date: 01/12/2001 at 15:52:02 From: Doctor Rick Subject: Re: period of 61/97 Hi, Celia. I can tell you that the period is finite; more particularly, that it is no greater than 97 digits. You've done at least 1/3 of the work. How do I know that? If you've done the calculations by long division, then you've seen a sequence of remainders: 0.628865979... ----------------- 97 ) 61.000000000... 58 2 ---- 2 80 1 94 ---- 860 776 --- 840 776 --- 640 582 --- 580 485 --- 950 873 --- 770 679 --- 910 873 --- 37 The partial remainders are 28, 86, 84, 64, 58, 95, 77, 91, 37, ... The first time you get a remainder that has shown up before, the next quotient digit will be the same as it was the first time, and the next remainder will be the same as it was the first time, and so on - in other words, the decimal begins repeating at that point. Now we can apply the "pigeonhole principle." How many different remainders can there be? The remainder must be between 0 and 96. In fact, if it's 0 then the decimal will terminate, and we know that won't happen. (If you don't know why, I'd be glad to tell you.) There are thus only 96 different possible remainders. When we've gotten 97 digits and 97 remainders, we can be sure that one of them has shown up before. You can imagine 96 "pigeonholes" labeled with the possible remainders. We have 97 numbers (the actual remainders) to put in them; at least two of the numbers must go into the same pigeonhole. Therefore, the decimal must begin repeating by the 97th digit. If you're doing the calculation by a calculator (such as the calculator built into Windows) that handles up to 31 digits but no more, you can use this trick. Calculate 25 digits (so we have some digits to spare and we won't overtax the calculator in what follows): 61/97 = 0.6288659793814432989690721 Then find the partial remainder after the last digit so far. Here's how to do it. Multiply the decimal by 97, subtract this from 61, and multiply by 1E25. I get a partial remainder of 63. Next, divide the remainder by 97. You will get a continuation of the decimal expansion of 61/97. I get: 63/97 = 0.64948453608247422680412371134021 When I calculated 61/97, I got 32 digits: 61/97 = 0.62886597938144329896907216494845 Note that the last 7 digits, 6494845, match the first 7 digits of the continuation. This is a check on the accuracy of my method. The full decimal expansion so far is thus: 61/97 = 0.62886597938144329896907216494845649484536082474226804 12371134021 You can continue in this way as far as you need in order to find where the decimal begins repeating. Have fun! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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