To: [EMAIL PROTECTED]
Subject: Mersenne: n-th decimal digit in real number
Date: Tue, 09 Oct 2001 15:09:08 -0700
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"Robert Braunwart" <[EMAIL PROTECTED]> wrote
>What do you think the best is the best approach for this problem?
>
>Calculate the n th. (any number) decimal digit of the expansion of a
>regular rational 1/n (not necessarily the same n), with n a prime number,
>without calculating the preceding digits.
>
Let x be a positive real number.
If d is the n-th decimal digit of x, then
d is an integer
0 <= d <= 9
FLOOR(10^n * x) == d (mod 10)
For example, if x = sqrt(2) = 1.4142135 ..., and n = 4, then
FLOOR(10^4 * x) = FLOOR(14142.135 ...) = 14142.
Reduce this modulo 10, to find the fourth decimal digit
(to the right of the decimal point) is a 2.
>Please give a example using substitution into formula.
If r is a rational number, say r = r1/r2,
with r1, r2 positive integers, then
FLOOR(10^n * r) mod 10
= FLOOR(10*(r1*10^(n-1) mod r2)/r2)
For example, to get the fourth decimal digit of 3/7, evaluate
3 * 10^3 mod 7 = 3000 mod 7 = 4.
FLOOR(10*4/7) = FLOOR(40/7) gives 5, which agrees with 3/7 = .428571 ...
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