Repeating decimal

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Repeating decimal refers to a decimal representation of a number where, after a certain point, a sequence of one or more digits repeats infinitely. It is also known as a recurring decimal. Every repeating decimal represents a rational number, that is, a number that can be expressed as the division of two integers (a fraction).

Definition

A repeating decimal is written with a bar (¯) over the digits that repeat. For example, 0.333... is written as 0.\overline{3}, indicating that the digit 3 repeats indefinitely. If a sequence of more than one digit repeats, the bar covers the entire sequence, as in 0.142857\overline{142857} or more succinctly as 0.\overline{142857}.

Conversion to Fraction

To convert a repeating decimal to a fraction, one can use algebraic methods. For a repeating decimal of the form 0.\overline{a}, where \(a\) is the repeating digit, the fraction is given by \(\frac{a}{9}\). For a repeating sequence of two digits, \(ab\), the fraction is \(\frac{ab}{99}\), and so on. This method leverages the properties of geometric series.

Examples

- 0.\overline{3} = \(\frac{1}{3}\) - 0.\overline{12} = \(\frac{12}{99} = \frac{4}{33}\) - 0.1\overline{6} = \(\frac{1}{6}\), where the non-repeating part is handled separately.

Characteristics

- Every repeating decimal is a rational number. - Not all rational numbers have a terminating decimal representation, but all can be expressed as repeating decimals. - Non-repeating decimals represent irrational numbers, which cannot be expressed as fractions.

Mathematical Significance

Repeating decimals are significant in mathematics for illustrating the concept of limits and the density of rational numbers within the real numbers. They serve as a bridge between the discrete world of integers and the continuous world of real numbers.

Related Concepts

- Terminating decimal: A decimal that has a finite number of digits after the decimal point. - Irrational number: A number that cannot be expressed as a fraction of two integers, and has a non-repeating, non-terminating decimal representation. - Fraction: A mathematical expression representing the division of two integers. - Geometric series: A series with a constant ratio between successive terms, used in the conversion of repeating decimals to fractions.