Recreational Mathematics · Interactive Animation

Repeating Decimals and
Digit Rotation Families Generator

Created by Inder J. Taneja

Discover how the repeating digits of a fraction form cyclic patterns under multiplication. Generate the repeating block of 1/n, watch its products animate, and compare the families of rotations side by side.

Featured examples

Select from 15 fractions to explore short and long repeating blocks. Compare one-family examples such as 1/7 and 1/17, two families for 1/13, three for 1/31, or several shorter families for 1/37 and 1/41.

Repeating decimal blocks

The bar above the digits means that the block repeats indefinitely. Leading zeros remain part of the block.

Multiplication and rotation

Colored positions help you follow the cyclic shifts. Watch the automatic animation, pause it, step through the products, or select a result directly.

Two families for 1/13

Compare the six rotations of 076923 with the six rotations of 153846. Other denominators may produce one or several families.

Choose a fraction and animate its products

Featured fraction Denominator Seconds per step Animation order

Choose an example or enter a denominator from 3 to 199, excluding multiples of 2 or 5. The animation starts automatically when you select or generate a fraction. Use Pause/Resume to hold a product, or Stop to return to the first product. Family panels are arranged by rotation.

Compare the rotation families

Understanding the mathematics and the colors

Within each family, colors follow positions in its seed block. Repeated occurrences of the same digit can therefore have different colors. Every displayed multiplier applies to the original repeating block of 1/n, including the products in other families.

A rotation moves digits from one end of the block to the other without changing their cyclic order. The repeating blocks of k/n fall into families under these shifts. When k and n have a common factor, a shorter block may repeat inside the displayed block.

If B is the integer represented by the repeating block and L is its length, then n × B = 10^L − 1: a number consisting of L nines. This identity is displayed below the family panels.

The CSV download retains leading zeros through text formulas when opened in Excel. The TXT download lists the complete products and their families.