Analysis - How many numbers in the interval [0,1) contain 5 consecutive 5's?

In summary, the conversation discusses finding the measure of a set E, which consists of points in the interval [0,1) that contain the block of consecutive 5's in their decimal expansion. The approach suggested is to find the measure of the complement of E, which would be the points that do not contain any 5's, or any consecutive 5's, 2 consecutive 5's, 3 consecutive 5's, or 4 consecutive 5's. The method for computing the measure of this set is not clear, but it is compared to a Cantor set.
  • #1
glacier302
35
0

Homework Statement



Let E be the set of points in [0,1) whose decimal expansion contains somewhere the block 55555. Find the measure of E.


Homework Equations





The Attempt at a Solution



I have a feeling that in order to find the measure of E, I should find the measure of the complement of E and then subtract it from 1. The complement of E would be the points in [0,1) containing no 5's, containing no consecutive 5's, containing 2 consecutive 5's, containing 3 consecutive 5's, containing 3 consecutive 5's, or containing 4 consecutive 5's. I'm not sure how to compute the measure of this set, however.

Any help would be greatly appreciated!
 
Physics news on Phys.org
  • #2
Well, start your program. What the measure of all of the points that contain no 5's at all? This is rather like a Cantor set.
 

FAQ: Analysis - How many numbers in the interval [0,1) contain 5 consecutive 5's?

What is the interval [0,1)?

The interval [0,1) refers to the set of all real numbers between 0 (inclusive) and 1 (exclusive). This means that 0 is included in the interval, but 1 is not.

How many numbers are in the interval [0,1)?

There are infinitely many numbers in the interval [0,1). This is because the interval contains all real numbers between 0 and 1, including numbers with an infinite number of decimal places.

What does it mean for a number to contain 5 consecutive 5's?

A number contains 5 consecutive 5's if there are 5 consecutive digits in the number that are all 5's. For example, the number 0.55555 contains 5 consecutive 5's, while the number 0.54555 does not.

How many numbers in the interval [0,1) contain 5 consecutive 5's?

This is a difficult question to answer definitively, as it is possible for infinitely many numbers in the interval [0,1) to contain 5 consecutive 5's. However, it is also possible that no numbers in the interval contain 5 consecutive 5's. More research and analysis would be needed to determine a precise answer.

What is the significance of finding numbers in the interval [0,1) with 5 consecutive 5's?

This question is open to interpretation and may have different answers depending on the context. However, one potential significance could be that it reveals patterns or relationships between numbers in the interval and the occurrence of consecutive digits. This could potentially have implications in various fields such as number theory, statistics, or cryptography.

Similar threads

Expected number of coin flips before N consecutive heads?
Replies
15
Views
3K
Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits
Replies
32
Views
2K
Consecutive odd natural numbers - one is composite. Prove
Replies
1
Views
4K
Is there a way to do this without brute force ?
Replies
17
Views
3K
What is the smallest closed interval that contains all limit points of (0,1)?
Replies
6
Views
2K
Prove that x is in S or x is an accumulation point of S
Replies
25
Views
3K
If A is dense in [0,1] and f(x) = 0, x in A, prove ∫fdx = 0.
Replies
2
Views
1K
Does an Open Cover of Rationals in [0,1] Necessarily Encompass All Irrationals?
Replies
10
Views
1K
Why Does the Set {1/n : n ∈ ℕ} Have an Empty Interior?
Replies
4
Views
3K
Calculus AB: Particle Speed Increase Intervals
Replies
7
Views
7K

Hot Threads

Recent Insights

Back
Top