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Encyclopedia :
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RE :
REC :
Recurring decimal |
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Recurring decimalA recurring decimal is an expression representing a real number in the decimal numeral system, in which after some point the same sequence of digits repeats infinitely many times. The repetition may begin before, at, or after the decimal point. The repeating sequence may consist of just one digit or of any finite number of digits. If the repeating sequence is merely a repeating "0", then the decimal is said to terminate because it is not necessary to explicitly write that there is a repeating "0". Such terminating decimals represent rational numbers whose fractions in lowest terms are of the form k/(2n5m).To indicate the part of the sequence that extends infinitely, dots should be placed above the numerals to be repeated. Where this is impossible, the extension may be represented by an ellipsis (...) although this may introduce uncertainty as to exactly which digits should be repeated:
Examples of fractions of the first group are:
This is to say:
It can also be generalised to say that p − 1 digits of 1 (or 2, 3, 4, 5, 6, 7, 8, 9) is divisible by p, which is a prime number other than than 2 or 5. The following multiplications exhibit an interesting property:
Of course 142857 × 7 = 999999 Decimals of other prime fractions such as 1/17, 1/19, 1/23, 1/29, 1/47, 1/59, 1/61, 1/97, 1/109 all exhibit the same property. Fractions of the second group are:
x = 0.333333... 10x = 3.33333... (multiplying each side of the above line by 10) 9x = 3 (subtracting the 1st line from the 2nd) x = 3/9 = 1/3 (simplifying) Another example: x = 0.18181818... 100x = 18.181818... 99x = 18 x = 18/99 = 2/11 From this kind of argument, we can see that the period of the repeating decimal of a fraction n/d will be (at most) the smallest number k such that 10k − 1 is divisible by d. For example, the fraction 2/7 has d = 7, and the smallest k that makes 10k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 2/7 is therefore 6. Why rational numbers must have repeating or terminating decimal expansionsIn order to convert a rational number represented as a fraction into decimal form, one may use long division. For example, consider the rational number 5/74: 0.0675
74 ) 5.000000
4 44
560
518
420
370
500
etc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50. When we arrive at 50 as the remainder, and bring down the "0", we find ourselves dividing 500 by 74, which is the same problem we began with. Therefore the decimal repeats: 0.0675675675.... We eventually see a remainder that we have seen earlier because only finitely many different remainders -- in this case 74 possible remainders: 0, 1, 2, ..., 73 -- can occur. As soon as we only bring down zeros, the same remainder implies the same new digit in the result and the same new remainder. Therefore the whole sequence repeats itself, again and again. The case of 0.99999... The method of calculating fractions from repeated decimals, especially the case of 1 = 0.99999..., is sometimes contested by the mathematically naive: One can also think of this as the sum of a geometric progression. Where:
The above exposition using formal mathematical notation looks more impressive than the arithmetic proof but it is not persuasive as the crucial step, the division by 10n, is not actually performed. But even were the proof using limits properly completed the arithmetic proof is adequate and simpler and can be followed by those without the proper understanding of limits. Generalising this, any number with a finite decimal expression (a decimal fraction) can be written in a second way as a recurring decimal. For example 3/4 = 0.75 = 0.750000000... = 0.74999999 ... See also
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