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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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.

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 fraction, 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 $

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