Uncovering the Origins of Pi: A Closer Look at Archimedes' Approximation

[decibel]

what does pi mean, and where did it come from, i know its 3.141592654, but i also know there's a lot more decimal places then that, does anyone know anything about it?

 

[The_Professional]

Funny coincidence.I just finished watching that movie a couple of nights ago. I believe the number goes to infinity after the decimal point.

 

[chroot]

http://www.mathforum.org/dr.math/faq/faq.pi.html

- Warren

 

[mathman]

what does pi mean

When I was in 7th grade, I learned that pi was the ratio of the circumference to the diameter of a circle. I still think it means that today.

 

[ahrkron]

I believe the number goes to infinity after the decimal point.

I know you mean no harm, but just be careful how you phrase it. It is true that there is an infinity of digits after the decimal point, but, in math, "goes to infinity" usually means that the quantity itself grows without bound, while pi is no larger than 3.2.

 

[The_Professional]

I know you mean no harm, but just be careful how you phrase it. It is true that there is an infinity of digits after the decimal point, but, in math, "goes to infinity" usually means that the quantity itself grows without bound, while pi is no larger than 3.2.

Thanks for clearing that out :)

 

[verty]

My question is this: is PI defined as the ratio of circumference to diameter, or as the area of a circle of radius 1?

It seems easier to define it as the area of a unit circle, as we can then approximate to the value of PI.

 

[NateTG]

My question is this: is PI defined as the ratio of circumference to diameter, or as the area of a circle of radius 1?

Since the two are the same, it doesn't really matter. In practice, the ratio of diameter to circumference is much easier to deal with than the ratio of square of radius to area.

 

[mathman]

Historically, I believe the circumference to diameter ratio was the definition. Archimedes is reputed to have approximated pi by approximating the circumference by many sided polygons.

 

[arildno]

Actually, Archimedes proved the important result that the constant of proportionality between the diameter and the circumference and the constant of proportionality between the squared radius and the circle's area was the same proportionality constant (pi).
(He showed that the area of the circle had to be equal the area of the right angled triangle with base equal to circumference and height equal to radius.)

He then proceeded as mathman says.

 

[HallsofIvy]

I wouldn't use the word "reputed"! Archimedes definitely did approximate pi by using polygons with up to 96 sides. He did not (as I foolishly thought until recently) actually draw huge polygons and measure the sides! He developed an algorithm for calculating the length of a side of a polygon of 2n sides in terms of the length of side of a polygon of n sides inscribed in the same circle (96= 6*2⁴ and the side of a hexagon is the same as the radius of the circle). He also developed an algorithm for finding upper and lower bounds on square roots since his other algorithm involved square roots. He showed that pi is between 223/71 and 22/7.