Finding least upper bounds

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In summary, the conversation discusses finding the least upperbounds on N and P for a repeating decimal representation of a real number x. The initial statement is deemed unanswerable unless there are additional conditions given. However, the fact that the decimal is repeating implies that x is a fraction A/B where A and B are integers, allowing for a bound on N and P in terms of A and B. It is suggested to look at the division of A by B to determine how to find these bounds. It is also noted that while there are specific values for N and P for a given rational x, there are no overall bounds when considering all rationals.
  • #1
Ed Quanta
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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
 
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  • #2
The problem as stated has no answer. Both N and P are unbounded, unless there are other conditions which you haven't presented.
 
  • #3
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.
 
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  • #4
For any specific rational x, you have specific values for N and P. However, there are no bounds when considering all rationals.
 

What is a least upper bound?

A least upper bound is the smallest value that is greater than or equal to all the elements in a set. It is also known as a supremum and is often denoted as sup(S).

How is a least upper bound different from a maximum value?

A least upper bound is the smallest possible value that is still greater than or equal to all the elements in a set, while a maximum value is the largest value in a set. A set may have a maximum value, but not necessarily a least upper bound.

Why is finding the least upper bound important?

Finding the least upper bound is important in mathematics and science because it helps us determine the range of a set or the limit of a sequence. It allows us to understand the behavior of a set and make accurate predictions based on the data.

What methods can be used to find the least upper bound?

There are a few methods that can be used to find the least upper bound, including the completeness axiom, the supremum property, and the least upper bound property. These methods involve comparing elements in a set and narrowing down the possible values for the least upper bound.

Can a set have more than one least upper bound?

No, a set can only have one least upper bound. This is because the least upper bound is the smallest value that is greater than or equal to all the elements in a set, and there cannot be a smaller value that satisfies this condition.

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