Is this in any way interesting?

[Dmitry]

I know it's most likely isn't, but I would still like to understand why/why not

(pi's value to n digits - pie's value to n-1 digits)/pi's value to n-2 digits = 1/constant
the same principle applies to e

i'm just curious if there is some way to represent each next digit of pi in a simple way, so that i can represent PI as
f(PI to n digits) = PI to n+1 digits

there's no reason why I want to do this, i'm just curious

Thanks ahead of time!

[PM 2Ring]

I know it's most likely isn't, but I would still like to understand why/why not

(pi's value to n digits - pi's value to n-1 digits)/pi's value to n-2 digits = 1/constant
the same principle applies to e

Huh? I guess you're treating the strings of digits as big integers... Ok, that's not particularly interesting, since you get a similar result for a random string of digits, not just the digits of pi or e. The result is not exactly a constant, though. I haven't played with the algebra yet, I just tested it with a quick program, so I can't give you an explanation for this effect yet. But your 1/constant is approximately 90 plus an "error" term of the order of 10^(2-n). Note that 90 = 10² - 10, so this effect is probably just some simple artifact of decimal representation.

i'm just curious if there is some way to represent each next digit of pi in a simple way, so that i can represent PI as
f(PI to n digits) = PI to n+1 digits

there's no reason why I want to do this, i'm just curious

Thanks ahead of time!

No, there's no simple formula for the decimal digits of pi, but there are compact spigot algorithms that can be used to determine digits of pi in bases which are binary powers.

[scarecrovv]

(pi's value to n digits - pie's value to n-1 digits)/pi's value to n-2 digits = 1/constant
the same principle applies to e

How did you arrive at this conclusion? I only ask because it doesn't seem to be true:

(3.1415 - 3.141) / 3.14 = 0.000159235669
(3.14159 - 3.1415) / 3.141 = 0.0000286532951

[Dopefish]

Also, isn't 1/constant just a constant itself? How do you figure it's 1 over something?

[PM 2Ring]

We're looking at expressions of the form
(314159 - 31415) / 3141 ~= 90.017191977

Let A be the integer formed from the first n-2 decimal places of pi (or whatever). If the next two digits are b and c, then the integer formed from the first n-1 decimal places is 10A + b, and the integer formed from the first n decimal places is 100A + 10b + c. In the above example A = 3141, b = 5, c = 9.

So the expression becomes
((100A + 10b + c) - (10A + b)) / A
= (90A + 9b + c) / A
= 90 + (9b + c) / A
For large A, (9b + c) / A is much smaller than 0 1.

Edit: Oops! Thanks, mike-l.

[Dmitry]

((3.14-3.1)/3)^-1 = 75
((3.141-3.14)/3.1)^-1 = 3100
((3.14159265 - 3.1415926)/3.141592)^-1=62831840

that's what i meant

and e...
((2.71-2.7)/2)^-1 = 200
((2.718-2.71)/2.7)^-1 = 337.5, guess it doesn't work here O_o
((2.71828183-2.7182818)/2.718281)^-1 = also doesn't quite work.
so i guess it doesn't quite follow for e...

still curious about PI

[jaap]

Ah, what you meant isn't that you get a constant, but that you get an integer.

Let's rearrange this:

((3.14159265 - 3.1415926)/3.141592)^-1=62831840

First actually do that inversion from the start, giving this:
3.141592/(3.14159265 - 3.1415926)=62831840

Next multiply top/bottom of the fraction by the same power of ten to get rid of the decimal points:
314159200/(314159265 - 314159260)=62831840

Now rewrite it as the general case:
(Number formed by the first n-2 digits followed by 00) / (the nth digit) = an integer(?)

If the nth digit is 1, 2, 4, or 5, then it will automatically work there because any number ending in 00 is a multiple of 100, and so will also be a multiple of 1, 2, 4, or 5.

If the nth digit is 3, 6, 7, 8, or 9 then it will not always work, and it will generally just be coincidence if it does. With pi, it is just coincidence that 3141 is divisible by 9. However at that 6 you get
31415900/6 = 5235983.3333333333333333333333333
and at the next 3 you get
3141592600/3 = 1047197533.3333333333333333333333
so it doesn't always work for pi.


[Edited for typo]

[Dmitry]

Alright that explains it, thanks

[mike-l]

For large A, (9b + c) / A is much smaller than 0.

What?