Tool to find the period of a fraction or a decimal number with repeating decimals. The period is a set of digits that is repeated at infinity in the decimals of the number (usually a rational number or a periodic fraction).

Repeating Decimals - dCode

Tag(s) : Arithmetics

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The periodic decimal expansion/development of a rational number or a fraction (numerator over denominator) is the sequence of numbers that are repeated at infinity in the decimal writing of the number.

__Example:__ 1/3 = 0.3333333333… The digit `3` is repeated to infinity

__Example:__ 1/27 = 0.037037037037037… The digits `037` are repeated to infinity

All fractions do not have a repeating decimal form, some have a terminating decimal form.

A terminating decimal indicates that no sequence of numbers repeats infinitely in the decimal writing of the number.

__Example:__ 4/25 = 0.16 the development is finished and does not continue

Any number that is written in decimal form with a finite number of digits (after the decimal dot) has is a terminating decimal number.

Multiple notations are possible.

The first uses `…` points of suspension, but does not define the part that repeats. It is practical but not rigorous and therefore not recommended.

__Example:__ $ 37/300 = 0.12333333333\dots $

Notation with a bar above the repeated part.

__Example:__ $ 37/300 = 0.12\overline{3} $

Notation with a bar below the repeated part.

__Example:__ $ 37/300 = 0.12\underline{3} $

Notation between brackets

__Example:__ $ 37/300 = 0.12[3] $

NB: For clarity, it is better to write the fraction in an irreducible form.

Divide the numerator by the denominator. Lay the Euclidean division by hand or use a calculator.

Take $ x $ a number, and $ n $ the size (the number of digits) of the periodic part of the decimal expansion. To get a fractional writing, solve $ x \times 10^n - x $.

__Example:__ $ x = 0.1\overline{6} = 0.1666666\dots $, the repeating portion is $ 6 $, a single digit so $ n=1 $. Calculate $ 10^1 \times x = 1.\overline{6} = 1.6666666\dots $ and solve $ 10x−x = 9x = 1.\overline{6}−0.1\overline{6}=1.5 \iff 9x=1.5 \iff x=1.5/9 = 15/90 = 1/6 $

The inverses of prime numbers provide long and interesting periodic decimal developments.

__Example:__ $ 1/3 = 0.333333\dots $

__Example:__ $ 1/7 = 0.142857142857\dots $

Any rational number (any fraction) has a finite developpement or a periodic decimal expansion with a finite number of digits that repeat themselves ad infinitum.

But there are real numbers that are not rational numbers (which are not fractions) which have decimals without repetition

__Example:__ $ \pi = 3.14159265\dots $ has no known repetition to date.

__Example:__ Champernowne's constant will never have any repetition, it is a universe number.

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Cite as source (bibliography):

*Repeating Decimals* on dCode.fr [online website], retrieved on 2024-11-05,

- Recurring Decimal Detection A/B
- Terminating Decimal Detection
- Fraction Finder
- What are repeating decimals in a fraction? (Definition)
- What are terminating decimals in a fraction? (Definition)
- How to write repeating decimals?
- How to find decimals from a fraction?
- How to find the fraction from decimals?
- What are the most known decimal developments?
- Is there an infinite decimal expansion with a series of digits that never repeats?

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