How to find 'self locating digits' in irrational numbers

[prane]

Let us take the most mainstream irrational out there, (Pi).

Now write (Pi) as:

3.
14159265...

Let us number the decimals of Pi.

0 gets paired with 1
1 gets paired with 4
2 gets paired with 1
.
.
.
6 gets paired with 6

Thus 6 is a self locating digit.

My question is then how do we devise a method to find these self locating digits in a fast way.

This is how I've gone about it.

consider λ=(Pi)-3=0.14159265...

now consider digit number n, that is, the digit that is n places along:

0 gets paired with 1
1 gets paired with 4
2 gets paired with 1
.
.
.
6 gets paired with 6
.
.
.
n gets paired with x

We need an algorithm for finding out what x is without writing the whole of λ out.

Consider a new rational number, ρ.

Let ρ_n be the number which terminates at digit n.

Then ρ_n=0.1415926...x
ρ_(n-1)=0.1415926...w where w is the (n-1)th digit etc

Now consider (ρ_n)*10^(n+1) this is equal to 1415926...x. Let us call this new number β.

We can then find what x is by subtracting ρ_(n-1)*10^(n) from β.

Now if the x = n we have a self locating digit.

This method isn't terribly practical as we still have to basically know what ρ_n is.

Maybe I'll come up with an improvement after some thought but in the mean time I'd love to see what you guys come up with :)

 

[SteveL27]

My question is then how do we devise a method to find these self locating digits in a fast way.


This method isn't terribly practical as we still have to basically know what ρ_n is.

Right, the problem is that we have no formula for the n-th digit of pi that doesn't involve calculating all the preceding digits.

But it turns out that someone did find such a formula. It only works in base 16, but it's still amazing that such an algorithm exists.

http://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_formula

 

[prane]

http://upload.wikimedia.org/wikipedia/en/math/4/8/f/48f7653d58f4ad747327d271ed789415.png

how does that make sense? A forumla for the nth digit that doesn't depend on n?!

 

[mfb]

It is a formula for all digits. But if you want to calculate a specific one, you can calculate it quickly, the wikipedia page explains how.

By the way, what happens to the 11th digit in base 10, for example? A digit cannot be "11" there.

 

[blue_raver22]

Thank you for sharing your method for finding self locating digits in irrational numbers. It is an interesting approach and shows your creative thinking. However, I would like to offer a different perspective on this problem.

First, let's define what a self locating digit is. It is a digit in a number that appears in the same position as its value. For example, in the number 123456, the digit 1 is in the first position, the digit 2 is in the second position, and so on. This means that in order to find self locating digits, we need to know the position of each digit in the number.

Now, let's take a closer look at irrational numbers, specifically Pi. As you have correctly pointed out, Pi is an infinite decimal number, meaning it has an infinite number of digits after the decimal point. This means that it is impossible to write out the entire number, as you have also mentioned. So how can we find self locating digits in a number that we cannot fully write out?

One possible solution is to use an approximation method. By using an algorithm, such as the Chudnovsky algorithm, we can calculate the value of Pi to a certain number of digits. This means that we can find the position of a digit in the number Pi, and then check if that digit is in the same position as its value. However, this method is not foolproof, as it relies on the accuracy of the algorithm and the number of digits we are calculating.

Another approach could be to look for patterns in the digits of Pi. While Pi is considered a random and non-repeating number, there have been some interesting patterns found in its digits, such as the Feynman Point (a sequence of six 9s starting at the 762nd digit). By analyzing these patterns, we may be able to identify self locating digits in Pi.

In conclusion, finding self locating digits in irrational numbers is a challenging problem that requires creative thinking and mathematical analysis. While your method is a good start, it may not be practical for finding self locating digits in numbers with an infinite number of digits. I encourage you to continue exploring this problem and come up with new and innovative solutions.