Arbitrary cycle of digits in rational number?

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Discussion Overview

The discussion revolves around the determination of rational numbers that have a decimal expansion consisting of a cycle of n digits. Participants explore methods to find such fractions, particularly focusing on examples with long repeating sequences.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant inquires about finding a fraction with a repeating decimal of a specific length, expressing a desire to understand the process independently.
  • Another participant mentions that if n is a primitive root of 10, then the decimal expansion of 1/n will have a cycle of n-1 digits, providing the example of 1/7 which has a repeating sequence of 6 digits.
  • A different approach is suggested, where participants are encouraged to manipulate a known repeating decimal by multiplying it by a power of ten and subtracting the original number to explore the properties of repeating decimals.
  • A participant expresses gratitude for the responses but also conveys a sense of embarrassment for needing assistance.

Areas of Agreement / Disagreement

Participants present various methods and insights, but no consensus is reached on a singular approach or solution to the original inquiry.

Contextual Notes

The discussion does not clarify the specific mathematical conditions or definitions that might affect the determination of repeating decimals, nor does it resolve the steps involved in the suggested methods.

Who May Find This Useful

This discussion may be of interest to individuals exploring properties of rational numbers, repeating decimals, or those preparing for studies in mathematics.

dylanbyte
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Hello all, I have recently been wondering whether there is a way to determine a fraction for which the decimal expansion is a cycle of n numbers?

I would like to be able to work this out myself, but I can't wait until I start my mathematics degree. So any help would be greatly appreciated.

For example, if I wanted a rational number with a repeated 500 digit sequence, is there a way to efficiently work out an example of such a number?

Penny for your thoughts guys.

Cheers.
 
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If n is a primitive root of 10 then the cycle of 1/n is n-1 digits long.

For example 1/7=.142857142857... has 6 repeating digits.
 
Start with a repeating decimal like, for example, 0.123412341234... Try multiplying it by an appropriate power of ten, and then subtracting the original number.

In this example you started with a repeating decimal; try to run the exercise backwards, to see if you can end up obtaining a repeating decimal.
 
Ah thanks guys, I feel stupid for having to ask now :).
 

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