Repeating DecimalsDate: 01/24/2001 at 12:11:39 From: LoraAnn Barclift Subject: Repeating decimals By looking at the denominator, how can you determine if the decimal representation of a fraction will terminate or repeat? By observation I have concluded that the fraction will repeat if the denominator is prime or a multiple of 3. Is there a theory in number theory that justifies this conclusion? Date: 01/24/2001 at 16:19:27 From: Doctor Greenie Subject: Re: Repeating decimals Hi, LoraAnn - You have done some nice work with your observations; however, your conclusion is not correct (as is often the case when you try to draw a conclusion from observation....) (1) It is ALMOST always the case that the decimal representation is repeating if the denominator is a prime. The exceptions are 2 and 5. (2) It IS always true that the decimal representation is repeating if the denominator is a multiple of 3. (3) There are many repeating decimals in which the denominator is neither prime nor a multiple of 3 - such as 14 and 35. To characterize the denominators of common fractions that will have repeating decimal representations, it is easier to characterize the denominators that will produce terminating decimal fractions. Then any denominator that doesn't produce a terminating decimal will produce a repeating decimal. The key to identifying the denominators that produce terminating decimals lies in the names for the decimal numbers. Here are some fractions that have terminating decimal representations: .5 = "five tenths" = 5/10 ( = 1/2 in reduced form) .35 = "thirty-five hundredths" = 35/100 ( = 7/20) .123 = "one hundred twenty-three thousandths" = 123/1000 And here are a few that have repeating decimal representations: .33333... = "point 3 repeating" (?) ( = 1/3) .027027... = "point 027 repeating" (?) ( = 1/37, believe it or not...) .416666... = ??? ( = 5/12; I don't know how you would say it in words) Look at the denominators in these examples and see what distinguishes the denominators in the first group from those in the second group. Write back if you want more help with this. You can also undoubtedly find more information on this subject in the Dr. Math archives. Click on the "Search the Archives" link on the main Dr. Math page and try "terminating decimal" or "repeating decimal" as the phrase to search for. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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