Patterns of decimals

There are many fascinating patterns in the decimal repeat cycles of fractions. Try out some of them on the file division.xls!

For example:

1/7 = 0.1428571428571... the decimals have a repeat cycle of 6
1/13 = 0.0769230769230... the decimals also repeat every 6 digits
1/21 = 0.0476190476190... again a repeat pattern every 6 digits

What is it that 7, 13 and 21 have in common?

I started to notice some of these patterns as I was playing with division.xls. I wanted a more comprehensive table of decimal repeat patterns and I eventually realised that I didn't need to do the division itself and I didn't need any VBA either! Download the file decimalPatterns.xls below and see an explanation.


More patterns

One divided by 73, 139 and 146 all have a repeat cycle of 8 digits.
The examples above have repeat cycles of 6 and 8. What about 7?
Are there decimal repeat cycles of every length?

Some very big numbers have surprisingly short cycles. For example, one over

513,239repeats every 11 digits
265,371,653repeats every 13 digits

The longest possible repeat cycle is one less than the number itself. For example, the reciprocal of

313repeats every 312 digits
811repeats every 810 digits

Why are some patterns so short and others so long?

One divided by 384 is 0.00260416666666... It has a repeat cycle of only 1.
Why does it take a while for this pattern to settle down?

All these decimals were calculated in base 10.
Would they have the same repeat cycle in another base?

* Download decimal_patterns.zip (113kb)
* I have looked mainly at the length of the repeat cycle. Others have looked in more detail at the patterns of the repeating digits themselves. See the comments made by Terry Trotter and Bill Price.
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