How Do You Determine the Least Upper Bounds of N and P in Repeating Decimals?

[Ed Quanta]

I have to find the least upperbounds on N and P where x is an element of the reals and represented by the repeating decimal

x=m.d1d2...dNdN+1...dN+P instead of underlining I meant for this to be an overline representing the repeating sequence of digits in the decimal

 

[mathman]

The problem as stated has no answer. Both N and P are unbounded, unless there are other conditions which you haven't presented.

 

[mathwonk]

I beg to differ with mathman.

the fact that the decimal is repeating implies that the real number x is a fraction A/B where A and B are integers. then it makes sense to give a bound on N as well as on P in terms of A and B.

I.e. N is the number of terms until the decimal starts to repeat and P is the length of a cycle after it starts repeating in cycles of the same length. It seems to me that if you just look at what happens when you divide A by B, you will see how to do this.

 

[mathman]

For any specific rational x, you have specific values for N and P. However, there are no bounds when considering all rationals.