1. ## 3.141592654...

How the constant
$\displaystyle \pi = 3.141592654...$
comes?

2. Originally Posted by SengNee
How the constant
$\displaystyle \pi = 3.141592654...$
comes?
The most elementary answer is that it comes up in trying to measure the circumfurence of a circle. Given a circle with radius $\displaystyle r$ how can we find its circumfurence? It turns out that the diameter of a circle is (directly) proportional to its circumfurence. Meaning if you double the diamter then you double the circumfurence. If you half the diameter then you half the circumfurence. Whenever you have such a proportion it tells you that $\displaystyle \tfrac{\text{circumfurence}}{\text{diameter}}$ is a constant number, meaning it never changes for any circle. If you take a circle and lay a rope around it and measure it and compute this ration you will get about $\displaystyle 3.1$. Mathematicians found ways how to better approximate this number and how to even prove it is irrational (cannot be expressed as a fraction). With this number we can easily now find the circumfurence of a circle. If $\displaystyle r$ is the radius then $\displaystyle 2r$ is the diamter. To find the circumfurence we multiply by $\displaystyle \pi$ - this constant number of ever circle and we get that $\displaystyle 2\pi r$ gives us the circumfurence. But there are many other places in math were this number $\displaystyle \pi$ appears. What I gave you just now is the most elementary explanation.

3. Originally Posted by SengNee
How the constant
$\displaystyle \pi = 3.141592654...$
comes?
If you're asking how do they calculate the value of pi, there are several methods having to do with series that converge. A simple one is based on using Taylor's rule, which says $\displaystyle f(a) = f(0) + f'(0) \cdot a + f''(0) a^2/2! + f'''(0) a^3/3! + ...$ Use f(x) = Artan(x) for a = 1, and knowing that $\displaystyle Atan(1) = \frac {\pi} 4$ yields:

$\displaystyle \pi = 4 ( 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \frac 1 9 - \frac 1 {11} ...)$

This series is easy to remember, but unfortunately converges very slowly. After 100 terms you still have only about 3 signifiant digits.

A faster converging method is to consider that $\displaystyle Atan ( \frac 1 {sqrt 3} ) = \frac { \pi } 6$ to get:

$\displaystyle \pi = 2 \sqrt 3 ( 1 - \frac 1 {3 \cdot 3 } + \frac 1 {5 \cdot 3^2} - \frac 1 { 7 \cdot 3^3 } + \frac 1 {9 \cdot 3^4 } - ... )$

Do about 20 terms of this and you'll have pi figured to 12 decimal places.

A really fast series is this:
$\displaystyle \pi = \displaystyle\sum _{n=0} ( \frac 4 {(8n+1)} - \frac 2 {(8n+4)} - \frac 1 {(8n+5)} - \frac 1 {(8n+6)} ) (1/16)^n$
After just 7 terms you have pi = 3.1415926535

4. This image shows TPH's explaination visually.

(The elementary explaination of $\displaystyle \pi$)

5. Originally Posted by ThePerfectHacker
The most elementary answer is that it comes up in trying to measure the circumfurence of a circle. Given a circle with radius $\displaystyle r$ how can we find its circumfurence? It turns out that the diameter of a circle is (directly) proportional to its circumfurence. Meaning if you double the diamter then you double the circumfurence. If you half the diameter then you half the circumfurence. Whenever you have such a proportion it tells you that $\displaystyle \tfrac{\text{circumfurence}}{\text{diameter}}$ is a constant number, meaning it never changes for any circle. If you take a circle and lay a rope around it and measure it and compute this ration you will get about $\displaystyle 3.1$. Mathematicians found ways how to better approximate this number and how to even prove it is irrational (cannot be expressed as a fraction). With this number we can easily now find the circumfurence of a circle. If $\displaystyle r$ is the radius then $\displaystyle 2r$ is the diamter. To find the circumfurence we multiply by $\displaystyle \pi$ - this constant number of ever circle and we get that $\displaystyle 2\pi r$ gives us the circumfurence. But there are many other places in math were this number $\displaystyle \pi$ appears. What I gave you just now is the most elementary explanation.
This is the same explanation I always use. It is quite simple and it explains why, how and what all in one. Note that the fraction doesn't imply rationalization because the circumference of a circle OR the radius of a circle must be irrational because $\displaystyle \pi$ is irrational.