So, you want to find a rational number approximation for a given positive real number?
well kind of...It's more to find an irrational number. or numbers that just don't exist yet. Not saying that it can be done, I'm just trying it out.
Can you post an image of the graph to help explain?
http://c3.ac-images.myspacecdn.com/images02/43/l\_f987e0e9b6e84375ad7f5c299df57efa.jpg The dotted line in the picture is the plotting of a and b, were a is the x-axis and b is the y-axis.
I came up with a way to approximate the random number faster then adding 1 to either a or b, by multiplying both a and b by 10 if the product of a/b has a closer approximation then it had before. The plot doesn't multiply by 10 but instead adds 10 to x and y so that they don't go over there bit limit. the same angle is achieved.
The random number I used is: 0.811319243339041120212239080712172745200117254351360228364246460925262842220
and after about five minutes I ended up with:
Which is accurate to 35 decimals: 0.81131924333904112021223908071217274
I'm not sure what you mean by saying that "the angle goes off course". Clearly the n vs. d curve should converge to a slope n/d equal to the number being approximated.
When trying to use the angle to predict the rest of the random number though it will be accurate up to a and b, and cannot be used to even get a couple extra decimals, which i could then use to make a and b more accurate, recalculate the angle to a higher precision and repeat the steps to have a way to predict the rest of the random sequence. i have a few idea's why the angle cannot achieve even one extra decimal but nothing that is well defined.