/sci/ - Science & Math

[Anonymous No.7139143]

If pi is the ratio between the perimeter of a circle and the diameter, how was that number calculated without knowing what the ratio was?

[Anonymous No.7139149]

It was not calculated, that's why it is called "Pi" and not "2,72"

[Anonymous No.7139154]

22/7

[Anonymous No.7139157]

Did you even go to grade eight? You get pi when you divide the circumference by the diameter. Both can be easily measured, use a string to get the circumference and then measure with a ruler.

[Anonymous No.7139165]

it was approximated.

draw a pretty good circle in the sand and grab a length of string. tie a knot in the string so that the length of the start of the string to the knot is the diameter of your circle.

now wrap your string around the circle. see how much string you need to go around. compare that (the perimeter) to your diameter (where the knot is). remark that no matter what circle you draw, you need a little more that three knot lengths of string to go around.

now get creative.

[Anonymous No.7139459]

Because there are a lot of different ways to calculate pi. One way is to measure a circle. Or you could solve <span class="math"> \{gamma} (1/2)^{2} [/spoiler]. Or a bazillion other ways.

[Anonymous No.7139505]

>>7139143
The classic, non-empirical method is to calculate the perimeters of a many sided polygon inscribed in a circle of diameter 1 and of a many sided polygon with a circle of diameter 1. This gives a range converging on pi as the number of sides goes to infinity. Archimedes did this with a 96-gon and got the range 3.1408 - 3.1428. Jamshīd al-Kāshī calculated pi to 16 digits using this method in 1424 with a polygon with 3x2^28 sides. Obviously you can see that this is quite inefficient. Today we have infinite series that converge on pi much faster and computers to do the calculating. An easy to remember example is

pi = 4/1-4/3+4/5-4/7+...

[Anonymous No.7139544]

>>7139143

I had this question myself a few years ago, so I tried to derive the value of pi myself. Ended up using the same exact method Archimedes used, which is extremely intuitive. Create a polygon with n sides and measure the size based off the distance from the center to one of the sides. The value you get will converge towards pi as n approaches infinity. If you ignore calculus you will have to do the series by hand, which is why I think Archimedes eventually stopped at a certain value of n.

[Anonymous No.7139674]

[Anonymous No.7140145]

>>7139505
I actually written a short python code for the Leibniz's formula

a=1
b=3
c=0
x=0

while c<1000000:
x+=(4/a)-(4/b)
a+=4
b+=4
c+=1
print (x)

where "a" is the numbers being added
"b" are the numbers subtracted
"c" is the counter
and "x" is the answer

this is extremely slow so I also written code for Bailey–Borwein–Plouffe formula and chudnovsky algorithm with chudnovsky being the fastest