Is there a way to combine modulation with summation?

[questionposter]

I'm trying to come up with a general formula to describe irrational numbers, by using the general summation of (1/n) , like an infinite summation of (1/3) = .33333333333 = 1/3, which isn't irrational, but I suspect irrational numbers can be decribed by some sort of summation that modulates with a base 10 or possibly a base = to some arithmetic form of n as to have a different modulus every time and thus never repeat.

Though, every time I've ever thought of some cool mathematical thing, it terms out some dead scientist thought of it 100 years ago or more, so I'm just wondering if this has already been done.

Edited April 18, 2012 by questionposter

[khaled]

I'm trying to come up with a general formula to describe irrational numbers, by using the general summation of (1/n) , like an infinite summation of (1/3) = .33333333333 = 1/3, which isn't irrational, but I suspect irrational numbers can be decribed by some sort of summation that modulates with a base 10 or possibly a base = to some arithmetic form of n as to have a different modulus every time and thus never repeat.

Though, every time I've ever thought of some cool mathematical thing, it terms out some dead scientist thought of it 100 years ago or more, so I'm just wondering if this has already been done.

 

I think you're talking about this, which looks ordinary:

 

[math].333333 \approx \frac{1}{3} = .33333... = .\ddot{3}[/math]

 

[math].\ddot{k} = .k + .0k + .00k + ... = \sum_{i=-1}^{-\infty}{{k}^{i}}[/math]

Edited May 6, 2012 by khaled

[imatfaal]

[math] \sum_{i=-1}^{-\infty}{{k}^{i}} = \frac{1}{k}+\frac{1}{k^2}+\frac{1}{k^3}... [/math]

thats not what you wanted. Take k=2 for instance - the sum of hte infinite series would be 1; basically you have defined a geometric series with a = 1/k and r as 1/k

 

what would be correct would be the sum of the series

 

[math] \sum_{i=-1}^{-\infty}{{k}*10^{i}} = \frac{k}{10}+\frac{k}{100}+\frac{k}{1000}... [/math]