NationMaster - Encyclopedia: Recurring decimal

A recurring or repeating decimal is a number which when expressed as a decimal has a set of "final" digits which repeat an infinite number of times. For instance 1/3 = 0.3333333... (spoken as "0.3 recurring"). The decimal (base ten or occasionally denary) numeral system has ten as its base. ...

More fully: a recurring decimal is a rational number whose expression in the (decimal numeral system) has some point after which the same sequence of digits repeats infinitely-many times. The repetition may begin before, at, or after the decimal point and the repeating sequence may consist of just one digit or of any finite number of digits. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ...

A decimal number written with a repeating final 0 is NOT classed as a recurring decimal, and the decimal is said to terminate before the first final 0 (because it is not necessary to explicitly write that there is a repeating "0".

There is one special case where it is not necessary to express the recurrence, but sometimes useful to do so. That is where the repeating part is a single digit 9. In these cases, the final 9s may be omitted, and the previous digit increased by one. (Eg 0.999999... = 1 ; 1.77999999... = 1.78). In general, the recurring form is only used to show how the number was derived, or to express an interesting relationship (1 = 3/3 = 3 × 1/3 = 3 × 0.333333... = 0.99999...). See 0.999.... In mathematics, the recurring decimal 0. ...

The other types of decimals are terminating decimals and decimals which neither terminate nor repeat.

Terminating decimals represent rational numbers whose fractions in lowest terms are of the form k/(2ⁿ5^m). A rational number is one that can be expressed in the form a/b where a and b are integers (b must of course be non-zero, as division by zero is undefined), that is, a vulgar fraction.
A non-terminating non-recurring decimal represents an irrational number (which cannot be expressed as the ratio of two integers).

1 Notation
2 Fractions with prime denominators
3 Why all repeating or terminating decimals must be rational numbers
- 3.1 A shortcut
4 Repeating decimals as an infinite series
5 Why all rational numbers must have repeating or terminating decimal expansions
6 The case of 0.99999...
7 See also
8 External links

In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... An irreducible fraction is a fraction a/b, where the numerator a is an integer and the denominator b is a positive integer, such that there is not another fraction c/d with c smaller in absolute value than a and 0<d<b, and c and d are integers... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... In mathematics, an irrational number is any real number that is not a rational number, i. ...

Notation

One convention to indicate a recurring decimal is to put a horizontal line above the repeated numerals ( $1/3 = 0.overline{3}$ ). Another convention is to place dots above the outermost numerals of the recurring digits. Where these methods are impossible, the extension may be represented by an ellipsis (...) although this may introduce uncertainty as to exactly which digits should be repeated:

1/9 = 0.111111111111...
1/7 = 0.142857142857...
1/3 = 0.333333333333...
1/81 = 0.0123456790...
2/3 = 0.666666666666...
7/12 = 0.58333333333...

Another notation, used for example in Europe, encloses the recurring digits in brackets:

2/3 = 0.(6)
1/7 = 0.(142857)
7/12 = 0.58(3)

Fractions with prime denominators

A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a recurring decimal. The period of the recurring decimal, 1/ p, where p is prime, is either p − 1 (the first group) or a divisor of p − 1 (the second group). An irreducible fraction is a fraction a/b, where the numerator a is an integer and the denominator b is a positive integer, such that there is not another fraction c/d with c smaller in absolute value than a and 0<d<b, and c and d are integers... In mathematics, a prime number (or a prime) is a positive integer that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Coprime - Wikipedia /**/ @import /skins-1. ...

Examples of fractions of the first group are:

1/7 = 0.142857...; 6 repeating digits

1/17 = 0.0588235294117647...; 16 repeating digits

1/19 = 0.052631578947368421...; 18 repeating digits

1/23 = 0.0434782608695652173913...; 22 repeating digits

1/29 = 0.0344827586206896551724137931...; 28 repeating digits

The list can go on to include the fractions 1/47, 1/59, 1/61, 1/97, 1/109 etc.

This is to say:

10⁶ − 1 = 999,999 (6 digits of 9) is divisible by 7;

10¹⁶ − 1 = 9,999,999,999,999,999 (16 digits of 9) is divisible by 17;

10¹⁸ − 1 = 999,999,999,999,999,999 (18 digits of 9) is divisible by 19; etc.

These can be deduced from Fermat's little theorem. Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by...

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 2 or 5.

The following multiplications exhibit an interesting property:

2/7 = 2 × 0.142857... = 0.285714...

3/7 = 3 × 0.142857... = 0.428571...

4/7 = 4 × 0.142857... = 0.571428...

5/7 = 5 × 0.142857... = 0.714285...

6/7 = 6 × 0.142857... = 0.857142...

That is, these multiples can be obtained from rotating the digits of the original decimal of 1/7. The reason for the rotating behaviour of the digits is apparent from an arithmetics exercise of finding the decimal of 1/7.

Of course 142857 × 7 = 999999, and 142 + 857 = 999.

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:

1/3 = 0.333.... which has 1 repeating digit.

1/11 = 0.090909... which has 2 repeating digits.

1/13 = 0.076923... which has 6 repeating digits.

Note that the following multiples of 1/13 exhibit the discussed property of rotating digits:

1/13 = 0.076923...

3/13 = 0.230769...

4/13 = 0.307692...

9/13 = 0.692307...

10/13 = 0.769230...

12/13 = 0.923076...

And similarly these multiples:

2/13 = 0.153846...

5/13 = 0.384615...

6/13 = 0.461538...

7/13 = 0.538461...

8/13 = 0.615384...

11/13 = 0.846153...

Again, 076923 × 13 = 999999, and 076 + 923 = 999.

Why all repeating or terminating decimals must be rational numbers

Given a repeating decimal, it is possible to calculate the fraction which produced it. For example:

 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 10^k − 1 is divisible by d. Periodicity is the quality of occurring at regular intervals (e. ...

For example, the fraction 2/7 has d = 7, and the smallest k that makes 10^k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 2/7 is therefore 6.

A shortcut

If the repeating decimal is between 0.1 and 1, and the repeating block is n digits long occurring right at the decimal point, then the fraction (not necessarily reduced) will be the n-digit block over n digits of 9. For example,

0.444444... = 4/9 since the repeating block is 4 (a 1-digit block),
0.565656... = 56/99 since the repeating block is 56 (a 2-digit block),
0.789789... = 789/999 since the repeating block is 789 (a 3-digit block), etc.

If the repeating decimal is between 0 and 0.1, and the repeating n-digit block is preceded only by k digits of 0 (all of which are to the right of the decimal point), then the fraction (not necessarily reduced) will be the n-digit block over the integer consists of n digits of 9 followed by k digits of 0. For example,

0.000444... = 4/9000 since the repeating block is 4 and this block is preceded by 3 zeros,
0.005656... = 56/9900 since the repeating block is 56 and it is preceded by 2 zeros,
0.0789789... = 789/9990 since the repeating block is 789 and it is preceded by 1 zero.

For any repeating decimal not prescribed above, it can be written as a sum of a terminating decimal and a repeating decimal of one of the two above types. For example,

1.23444... = 1.23 + 0.00444... = 123/100 + 4/900 = 1107/900 + 4/900 = 1111/900
0.3789789 ... = 0.3 + 0.0789789... = 3/10 + 789/9990 = 2997/9990 + 789/9990 = 3786/9990 = 631/1665

Repeating decimals as an infinite series

Repeating decimals can also be expressed as an infinite series. That is, repeating decimals can be shown to be a sum of a sequence of numbers. To take the simplest example, In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

$sum_{n=1}^inftyfrac{1}{10^n} = {1 over 10} + {1 over 100} + {1 over 1000} + cdots = 0.overline{1}$

The above series is a geometric series with the first term as 1/10 and the common factor 1/10. Because the absolute value of the common factor is less than 1, we can say that the geometric series converges and find the exact value in the form of a fraction by using the following formula where "a" is the first term of the series and "r" is the common factor. Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ...

$frac{a}{1-r} = frac{frac{1}{10}}{1-frac{1}{10}} = frac{1}{9} = 0.overline{1}$

Why all rational numbers must have repeating or terminating decimal expansions

In 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: In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...

  0.0675 74 ) 5.00000 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...

Main article: 0.999...

The method of calculating fractions from repeated decimals, especially the case of 1 = 0.99999..., is sometimes contested by the mathematically naive: In mathematics, the recurring decimal 0. ...

 x = 0.99999... 10x = 9.9999... 10x − x = 9.9999... − 0.99999... 9x = 9 x = 1

Some argue that, in the second step of the equation given above, 10x is 9.999...0 and not 9.999... but this is not the case: the right-hand side does not terminate (it is recurring) and so there is no end to which a zero can be appended. In mathematics, LHS is informal shorthand for the left-hand side of an equation. ...

One can also think of this as the sum of a geometric progression. Where: Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ...

$S_a = sum_{n=0}^{a} frac{0.9}{10^n}$

$S_a = 0.9 sum_{n=0}^{a} frac{1}{10^n}$

By standard result:

$S_a = 0.9 frac{10^{-a-1} - 1}{10^{-1}-1}$

From definition:

$lim_{a rightarrow infty} S_a = 0.99999 ldots$

So applying this on the sum of the geometric series:

$lim_{a rightarrow infty} 0.9 frac{10^{-a-1} - 1}{10^{-1}-1} = 0.9 frac{-1}{-0.9}$

$0.9 frac{-1}{-0.9} = 1$

Therefore:

$.99999 ldots = 1$

For a less persuasive but more formal-looking proof, consider the formula:

$x = {10^n-1 over 10^n}$

$n = 1: x = {9 over 10} = 0.9$

$n = 2: x = {99over 100} = 0.99$

It follows that

$lim_{n to infty}{10^n-1 over 10^n} = 0.9999dots$

On the other hand we can evaluate this limit easily as 1, also, by dividing top and bottom by 10ⁿ.

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 nonzero number with a finite decimal expression (a decimal fraction) can be written in a second way as a recurring decimal. Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus...

For example 3/4 = 0.75 = 0.750000000... = 0.74999999 ...

External links

Mathworld

Categories: Numeration | Elementary arithmetic

Results from FactBites:

Decimal - Encyclopedia, History, Geography and Biography (1484 words)
	Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called digits) for ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent numbers.
	Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator.
	That a rational must produce a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only (q-1) possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q-1.

Decimal - ExampleProblems.com (1138 words)
	Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) to represent numbers.
	Decimal arithmetic is used in computers so that fractional results can be computed exactly, which is not possible using a binary fractional representation.
	Decimal fractions can be expressed without a denominator, the decimal point being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator.

More results at FactBites »

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