Exact number to pi

1. Nov 10, 2005

Anything

Pi

does anyone know the exact number to pi (yes the one over 1000 characters long:P) i need it a.s.a.p

2. Nov 10, 2005

amcavoy

That is only an approximation. You can find certain amounts of digits on Google, but they aren't exact.

3. Nov 10, 2005

VietDao29

I dunno what's that for... But anyway, here it is:
Pi to 1,000,000 places.

4. Nov 10, 2005

Zurtex

You can't write it fully in decimal form as it is irrational.

5. Nov 10, 2005

hypermorphism

You can calculate pi's decimal expansion using the algorithms here.

6. Nov 10, 2005

finchie_88

Why do you need pi to that accuracy, and why have people got computers working out pi to that number of decimal places? What is the use, I don't get it...

7. Nov 10, 2005

shaner-baner

pi approximations

There is a mathematician named Borwien who has come up with some neat algorithms. The best one in terms of speed of convergence vs. complexity
converge quartically, i.e. the number of correct digit roughly quadruples every iteration, furthermore the algorithms only involve square roots and fourth roots, nothing as complicated as Ramujan's stuff. Using this algorithm a mere 15 iterations gives something like a billion digits. His derivation of it is an interesting mix of elliptic intergrals and the arithmetic-geometric mean of a number. What takes computers so long these days is just the mechanics of doing arithmatic on numbers with a billion digits. Look up Borwein if you are interested,

8. Nov 11, 2005

matt grime

Conjecture: pi is normal. Normal roughly means that the digits of pi behave randomly in some sense. We can't prove this, but knowing pi to many places has provided strong supporting evidence for it to be true.

Further, having a known (transcendetal) constant is useful for benchmarking alogrithms (ie how fast they converge)

9. Nov 11, 2005

GregA

I remember trying to figure out a formula for calculating pi a few months ago and came up with: 2x(sin(90/x)) where a very big value of x gets a close approximation for pi.
My calculator could only evaluate it to 10 digits though...the other millions of digits where given in standard form

10. Nov 11, 2005

Manchot

Taylor expand sin(90°/x) = sin($\frac{\pi}{2x}$) around x=0, and see what you get. :)

Last edited: Nov 11, 2005
11. Nov 12, 2005

Can anybody tell me how to plot pi on anumber line?

12. Nov 12, 2005

GregA

Gonna be a while before I can do that

13. Nov 12, 2005

uart

Sure. Just place a dot at about 3.14 and then label it $$\pi$$.

14. Nov 12, 2005

HallsofIvy

Staff Emeritus
What do you mean by "plot pi"?

If you mean just mark a point on a line, pi is approximately 3.14. Marking that point should be sufficient. If you mean "construct a line segment, using compass and straight edge, having length pi, given a line segment of length 1", it is impossible. pi is transcendental and so not constructible.

15. Nov 12, 2005

bomba923

16. Nov 12, 2005

Manchot

Ok, well basically, using calculus, you can show that

$$\sin{x} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$$

where x is in radians. (You can easily verify this for yourself on a graphing calculator, if you have one.) This infinite series converges for all x, but it is centered around zero, so it converges for low numbers "before" it converges for high ones. In fact, for numbers that are infinitely close to zero, you can just drop the higher order terms and say that $\sin(x) = x$. You were taking x to be infinitely high, so $\frac{\pi}{2x}$ would be very low, and you can say that $2x\sin(\frac{\pi}{2x}) = 2x(\frac{\pi}{2x}) = 2(\frac{\pi}{2}) = \pi$. Nice, huh?

(Incidentally, I arrived at a formula very similar to yours when I was a sophomore in high school, and knew geometry but not calculus. Would I be correct in guessing that you arrived at it by calculating the circumference of a regular polygon with an infinite number of sides?)

17. Nov 13, 2005

Robokapp

There is no exact number. It is Irrational. It's sum{Pi^2/2}=1/n^2 If I'm not mistaking...and imagine how many natural numbers there are...

18. Nov 13, 2005

Curious3141

I think you're driving at zeta (2).
$$\zeta(2) = \Sigma_{i=1}^{\infty}\frac{1}{i^2} = \frac{\pi^2}{6}$$
The proof of the irrationality (or indeed transcendence) of pi does not depend on that equality, though.

19. Nov 14, 2005

Robokapp

I wonder...why do you need it? are you trying to crack it? If you succeed let me know!

20. Nov 18, 2005