Search for a tool
Repeating Decimals

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).

Results

Repeating Decimals -

Tag(s) : Arithmetics

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Feedback and suggestions are welcome so that dCode offers the best 'Repeating Decimals' tool for free! Thank you!

Repeating Decimals

Recurring Decimal Detection A/B




Terminating Decimal Detection




Fraction Finder



Answers to Questions (FAQ)

What are repeating decimals in a fraction? (Definition)

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.

What are terminating decimals in a fraction? (Definition)

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.

How to write repeating decimals?

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.

How to find decimals from a fraction?

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

How to find the fraction from decimals?

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 $

What are the most known decimal developments?

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

Example: $ 1/3 = 0.333333\dots $

Example: $ 1/7 = 0.142857142857\dots $

Is there an infinite decimal expansion with a series of digits that never repeats?

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.

Source code

dCode retains ownership of the "Repeating Decimals" source code. Except explicit open source licence (indicated Creative Commons / free), the "Repeating Decimals" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Repeating Decimals" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Repeating Decimals" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Repeating Decimals" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Repeating Decimals on dCode.fr [online website], retrieved on 2024-11-05, https://www.dcode.fr/number-repeating-decimal

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Repeating Decimals' tool for free! Thank you!


https://www.dcode.fr/number-repeating-decimal
© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.
 
Feedback