repeating decimal
recurring decimal

Repeating decimals can also be expressed as an infinite series.

Thus, it is often useful to convert repeating decimals into fractions.

Knowing just that, a general repeating decimal can be expressed as a fraction without having to solve an equation.

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10.

The preferred way to indicate a repeating decimal is to place a bar over the digits that repeat, for example 0.

Given a repeating decimal, it is possible to calculate the fraction that produced it.

The characterization of such numbers can be done using repeating decimals (and thus the related fractions), or directly.

Sometimes an infinite number of repeating decimals is required to convey the same kind of precision.

That is, repeating decimals can be shown to be a sum of a sequence of numbers.

Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.

Nikky nodded back, oblivious to the irony, "I am not fond of repeating decimals myself.

Those reciprocals of primes can be associated with several sequences of repeating decimals.

This illustrates that cyclic permutations are somehow related to repeating decimals and the corresponding fractions.

So this particular repeating decimal corresponds to the fraction 1/(10 1), where the denominator is the number written as n digits 9.

Ross, Kenneth A. Repeating decimals: a period piece.

Conversely, suppose we are faced with a repeating decimal, we can prove that it is a fraction of two integers.

In a repeating decimal, the vinculum is used to indicate the group of repeating digits:

This type of repeating decimal can be obtained by long division if one uses a modified form of the usual division algorithm.

One convention to indicate a repeating decimal is to put a horizontal line (known as a vinculum) above the repeated numerals ().

A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal.

For any integer coprime to 10, its reciprocal is a repeating decimal without any non-recurring digits.

If you divide a number by the amount of 9s corresponding to its number of digits, the number is turned into a repeating decimal.

And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works.

It is possible to get a general formula expressing a repeating decimal with an n digit period, beginning right after the decimal point, as a fraction:

The underlying mathematical issue is that while rational numbers can always be expressed as a repeating decimal, the length of the repeating string may be very long.