how do you extract the value of pi?

Question

show some math equation

Answer 1

There are several methods for computing π, explained in more detail on wikipedia. The easiest methods to understand are those based on the arctangent function. The arctangent function can be given by a power series, namely:

arctan (x) = [k=0, ∞]∑((-1)^k x^(2k + 1)/(2k + 1)) = x - x³/3 + x⁵/5 - x⁷/7 + x⁹/9...

This series converges whenever |x|≤1, so the arctangent of a number may be computed to arbitrary precision using this series. Now, since tan (π/4) = 1, it follows immediately that arctan (1) = π/4, which leads to the following well-known formula:

π/4 = [k=0, ∞]∑(-1)^k/(2k+1) = 1 - 1/3 + 1/5 - 1/7 + 1/9...

Of course, while this formula is completely accurate and allows computation of π to arbitrary precision, it has the disadvantage of being extremely slow (which isn't surprising, considering that 1 is literally on the edge of the radius of convergence). For this reason, trigonometric identities are employed to represent π in terms of arctangents of smaller values, so as to obtain faster convergence. This leads to the Machin-like formulae. Some examples are:

π/4 = 4 arctan (1/5) - arctan (1/239) (the original)
π/4 = 2 arctan (1/2) - arctan (1/7)
π/4 = 12 arctan (1/49) + 32 arctan (1/57) - 5 arctan (1/239) + 12 arctan (1/110443)

And so on. There are more methods of computing π, but this one is the easiest to understand.

Also note that contrary to the assertion of one of the previous posters, 22/7 is NOT the "best approximation for π". In fact, it's not even a good approximation. Consider:

π ≈ 3.141592 653589 793238 462643 383279 5
22/7 ≈ 3.142857 142857 142857 142857 142857

As you can see, they differ as soon as the thousandths place. 355/113 is a better approximation:

355/113 = 3.141592 920353 982300 884955 752212 4

And of course, far more accurate approximations may be obtained by using the machin-like formulas provided above.

Answer 2

pi is 4 times the inverse tangent of 1. That is, pi = 4 arctan(1)
This is because tan(pi/4) = 1, so then take the inverse function.

I don't know if this is the kind of answer you want... but this is how to do it in a programming language that doesn't have an exact constant for pi as part of the language.

Answer 3

Pi is 3.14 and is a never ending number. Some guy I think remembered Pi out to the 1 millionth decimal which was insane.

Answer 4

i'm doing a similar situation on masteringchemistry. and that i said via with Roger's reasoning, and it extremely works. except i imagine there is an mathematics situation consisting of his artwork. because after dividing via by technique of each and everything and searching after the sig figs, the answer is 36.7kg. And that is ideal, because the internet website usual it. =]

Answer 5

OK

Well it is not as easy as you think. The best approximation would be 22/7.

But if you want a REAL detailed explanation try
http://en.wikipedia.org/wiki/Pi or
http://mathforum.org/isaac/problems/pi1.html

I think it will have more than enough information for you!

Hope this helps.

Answer 6

pi = 3.14... im not sure what exactly u are asking tho ?