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Displaying Large Repetends on Small Calculators

Date: 05/31/2002 at 13:02:45
From: Jack Carter
Subject: repeating decimal form of rational number

The approximation 

  1/7 = .142857142857142857

where the '142857' is the 6-digit repetend, is easy to display on 
a calculator, but 1/17 has a 16-digit repetend.
How can I display this repetend on a calculator?

Is there a general procedure for the ratio of two integers a/b?

Date: 05/31/2002 at 16:43:09
From: Doctor Peterson
Subject: Re: repeating decimal form of rational number

Hi, Jack.

Obviously you can't display all 16 or more digits on an 8-digit 
display; but you can get, say, 6 or 8 digits at a time with a 
little effort.

Imagine dividing by hand; at each step you would have a remainder 
(less than 17) and if you stop at some point, you can continue on 
to the next digits knowing only that remainder.

So all you have to do is find the remainder after however many 
digits you can see, and then continue from there. For example, if 
you have the digits

    1/17 = 0.0588235

which is the same as saying that

    10000000/17 = 0588235,

the remainder at this point is

    10000000 - 0588235*17 = 10000000 - 9999995 = 5

so you can find the next set of digits by dividing

    5000000/17 = 294117

This gives us the digits

    0.0588235 294117

You can repeat, using the remainder from the last division:

    5000000 - 294117*17 = 5000000 - 4999989 = 11

    11000000/17 = 647058

so we are now up to

    0.0588235 294117 647058

and we've started repeating. (To verify that it is actually 
repeating, you can find the remainder at the start and end of the 
suspected repetend, and check that they are the same.)

Here are some related answers from our archives:


- Doctor Peterson, The Math Forum
Associated Topics:
High School Calculators, Computers
High School Number Theory
Middle School Fractions

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