Arbitrary cycle of digits in rational number?

In summary, the conversation discusses the possibility of determining a fraction with a decimal expansion that consists of a cycle of n numbers. It is mentioned that if n is a primitive root of 10, the cycle of 1/n is n-1 digits long. The conversation also suggests a method to find such a number by starting with a repeating decimal and working backwards.
  • #1
dylanbyte
7
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
Ah thanks guys, I feel stupid for having to ask now :).
 

FAQ: Arbitrary cycle of digits in rational number?

1. What is an arbitrary cycle of digits in a rational number?

An arbitrary cycle of digits in a rational number refers to a repeating pattern of digits that occur infinitely in the decimal representation of a rational number. This pattern can be identified by a specific sequence of digits that repeats itself.

2. How can we identify an arbitrary cycle of digits in a rational number?

An arbitrary cycle of digits can be identified by analyzing the decimal representation of a rational number. If a certain sequence of digits repeats itself infinitely, then it can be considered as an arbitrary cycle of digits.

3. Can an irrational number have an arbitrary cycle of digits?

No, an irrational number cannot have an arbitrary cycle of digits because by definition, irrational numbers have non-terminating and non-repeating decimal representations. Therefore, they do not have a specific pattern that repeats.

4. Why is the concept of arbitrary cycle of digits important in mathematics?

The concept of arbitrary cycle of digits is important in mathematics because it helps us understand the patterns and relationships between rational numbers. It also allows us to identify and analyze the behavior of decimal representations of rational numbers.

5. How is the concept of arbitrary cycle of digits used in real-world applications?

The concept of arbitrary cycle of digits is used in real-world applications such as data compression, cryptography, and error correction. It is also used in financial calculations, such as calculating interest rates and exchange rates, where precise decimal representations of rational numbers are crucial.

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