how do i compute pi to an arbitrary precision without using a computer; just pencil and paper.?

Answer 1

take a circle with a known (exact) radius, and a known (exact) circumference and take the equation c = 2(pi)r and solve for pi.
after the algebra, it's just simple and repetitive long division.

Answer 2

Use the Taylor series for arcsin(x); for x=.5, arcsin(x) = pi/6. You can compute arcsin(.5) to any degree of precision, multiply by 6 to get pi.

The Taylor series expansion for arcsin(x) is (see http://en.wikipedia.org/wiki/Taylor_series):

sum from n=0 to inf of (2n!)/[4^n*(n!)*(2n+1)] * x^(2n+1)

for x = .5

arcsin(.5) = sum n=0 to inf (2n!)/[4^n*(n!)*(2n+1)] * (.5)^(2n+1)

Using pencil and paper, you will have to compute each term to one more place than your desired precision, and keep adding terms until the last one does not change the value of the last place. For example, if you want pi to a precision of 16 places, calculate each fraction to 17 places, and keep adding fractions until the next one is smaller than 10^-17. This would be a lot of work, but it can be done without a calculator or a computer

Note: 22/7 is an approximation, not exactly equal to pi, which cannot be expressed as a fraction. Measuring the circumference of a circle is difficult; the result will not be accurate.

Edit: my first answer used arctan(1), but the Taylor series expansion is only valid for values -1 < x < 1. Using arcsin(.5) gives a value of pi/6, and the computation deals with rational numbers only. There are other easier methods; see http://en.wikipedia.org/wiki/Pi

Answer 3

Hi:

the formulas those guy are giving you is correct but here a better one:

Sin ((360/n) /2) * n = pi n= Number of triangles {Try 1 e 06 to 1e 90 for very good results

or :

((Tan ( 360/n)/2) *n= pi n= Number of triangles {Try 1 e 06 to 1e 90 for very good results

Answer 4

Do by the same methods which computer does but by pencil and paper

Answer 5

Divide

22 by 7 - Long divisin method.

I figured out that if I did it that way, I get a repetitive decimal part repeating after every 7 digits, I think.

Funny.

Answer 6

pi/4 = 1-1/3+1/5-1/7+1/9-1/11+1/13.......

Answer 7

get cylindrical objects and measure the circumference and the diameter. since C=pi/d, then, pi=C/d. its all about proportions

Answer 8

use infinite series