![]() Viewing Repeating Decimals as Infinite Seriesįirst, let’s take a look at repeating decimals. In this article, I will show you a third method - a common method I call the series method - that uses the formula for infinite geometric series to create the fraction. ![]() I’ve shown you two ways to convert a bicimal to a fraction: the subtraction method and the direct method. (Read about the series method, my final article in this series.)īy Rick Regan (Copyright © 2013 Exploring Binary)Ĭonverting a Bicimal to a Fraction (Direct Method) For example, the subtraction method will show you that the denominator is of the form (2 r – 1)2 p, where r is the length of the repeating part and p is the length of the non-repeating part - in other words, r 1s followed by p 0s. You can show why this method works by using the subtraction method, as we did for decimals. Add these two fractions to get the answer: 10000/11. 01, first convert the fractional part to 1/11, and then convert 101 to 1111/11 by multiplying 101 by 11. (See the diagram in the introduction the purple digits correspond to the non-repeating part, and the red digits correspond to the repeating part.)Ĭonverting a bicimal with a whole part works the same as for decimals, except you form the improper fraction for the whole part using binary multiplication. The resulting fraction is 1110/11000, which reduces to 111/1100, which is 7/12 in decimal numerals. The numerator is the non-repeating digits and repeating digits minus the non-repeating digits: 10010 – 100 = 1110. It has a three-digit non-repeating prefix and a two-digit repeating cycle this makes the denominator of the fraction 11000. The numerator of the fraction is 1, and the denominator is 11 it converts to 1/11, which is 1/3 in decimal numerals.Ġ.100 10 is a mixed repeating bicimal. 01 is a pure repeating bicimal with a two-digit repeating cycle. I’ll demonstrate the procedure with two examples: 0. You do it the same way you convert a decimal to a fraction, except you use a denominator that contains a string of 1s instead of a string of 9s. Converting a Repeating Bicimal To a FractionĪt this point you know almost everything you need to know about how to convert a bicimal to a fraction. The mixed repeating decimal 312.13 78 is done similarly. Then, using the denominator of that fraction, write the whole part as an improper fraction, and then add the two fractions.įor example, let’s convert the pure repeating decimal 17. First, convert the fractional part to a fraction, as above. 12 or 312.13 78, is handled by treating the whole and fractional parts separately. Decimals with Whole PartsĪ decimal with a whole part, like 17. The numerator will always be the non-repeating and repeating digits minus the non-repeating digits. To generalize, if r is the length of the repeating part and p is the length of the non-repeating part, then the denominator is (10 r – 1)10 p that is, r 9s followed by p 0s. The factor 10 3 – 1 represents three leading 9s, and the factor 10 2 represents two trailing 0s. Let’s use the subtraction method to show why it works: The numerator of the fraction is 42866-42 = 42824, and the denominator is three nines followed by two 0s this makes the fraction 42824/99900, which reduces to 10706/24975. The numerator is formed by subtraction: you subtract an integer made of the non-repeating digits from an integer made of the non-repeating digits and repeating digits. ![]() ![]() The denominator is formed easily: it starts with as many 9s as there are repeating digits, and ends with as many 0s as there are non-repeating digits. Mixed Repeating DecimalsĪ mixed repeating decimal also corresponds to a fraction with a simple pattern, although you need to do a little more work to construct it. The numerator will always be the repeating digits, expressed as an integer. To generalize, if r is the length of the repeating part, then the denominator is 10 r – 1 that is, r 9s. The subtraction method shows why it works out this way: The numerator of the fraction is 142857, and the denominator is six 9s this makes the fraction 142857/999999, which reduces to 1/7. Pure Repeating DecimalsĪ pure repeating decimal corresponds to a fraction with a numerator made up of the repeating digits and a denominator made up of as many 9s as there are repeating digits. I’ll show you how to use the direct method to convert decimals, and then I’ll show you how it adapts trivially to convert bicimals. This follows directly from the subtraction method. Example of Direct Method (7/12 in Decimal)Ĭonverting a Repeating Decimal To a FractionĪ repeating decimal can be written as a fraction with a numerator derived from its digits and a denominator consisting of one or more 9s followed by zero or more 0s.
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