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

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*!

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).

The periodic decimal development of a fraction is the sequence of numbers which is 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

It is better to write the fraction in irreducible form.

Inverses of prime numbers provide interesting periodic decimal developments.

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

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 ... $

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] $

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... $, the repeating portion is $ 6 $, a single digit so $ n=1 $. Calculate $ 10^1 \times x = 1.\overline{6} = 1.6666666... $ 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 $

dCode retains ownership of the online 'Repeating Decimals' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Repeating Decimals download for offline use on PC, tablet, iPhone or Android !

Please, check our community Discord for help requests!

period,fraction,numerator,denominator,development,repeating,decimal,writing,digit,infinite,rational,dot

Source : https://www.dcode.fr/number-repeating-decimal

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲