Discovering Pi: Unraveling the Mystery of an Irrational Number

[Miike012]

Homework Statement

The book states a proof... which says that the ratio of
circumference/Diameter which this ratio is equal to pi which is the same for all circles...

Ok that obviously make sense noting that circumference = Diameter(pi) which equals circumference/Diameter = pi

My question...
They discovered pi by a means of dividing the distance around the circle by the distance across the circle...
how did they find the distance around the circle if they havnt yet discovered the value of pi hence... they wouldn't be able to use the formula c = d(pi)

Did they just meausre with a string or something?

 

[Dick]

For surveying technology you can use a string and get an appoximation, sure. If you're good at algebra/geometry you can calculate it by approximating the circle by polygons. http://en.wikipedia.org/wiki/Pi#Antiquity

 

[LeonhardEuler]

Historically, different people did different things. In some cultures, the ratio was believed to be 3 based on crude measurements. The ancient Egyptians used 3+1/7 for a long time. Archimedes approximated pi using polygons with a large number of sides to approximate a circle, and calculating their perimeter. In modern times there are various sums used to calculate it. The easiest to derive (but not the fastest to calculate to high precision) is Liebnitz's series based on the taylor series for the arctangent function.

 

[Miike012]

That is interesting... because inorder for a polygon to approach to what appears to be a circle... I would assume the polygon would have to have an infinite number of sides.

And inorder to get it closer and closer to a cirlce... the more and more sides it would have to posses, however the polygon may look more and more like a circle with the more sides it gets... it will still never be a circle.. Hence the number pi repeats for ever and ever...
Is this correct? Or not even close? lol

 

[gb7nash]

That is interesting... because inorder for a polygon to approach to what appears to be a circle... I would assume the polygon would have to have an infinite number of sides.

Correctomundo. There's a formula for the area of an n-gon (which I can't recall). If you take the limit as n goes to infinity, you'll get lo and behold, $$\pi r^2$$.

 

[Mentallic]

Well pi doesn't "repeat" but rather the decimal expansion continues indefinitely. That's what an irrational number means and pi is an irrational number, so what you said is true.

 

[Mentallic]

Correctomundo. There's a formula for the area of an n-gon (which I can't recall). If you take the limit as n goes to infinity, you'll get lo and behold, $$\pi r^2$$.

$$A=\frac{r^2n}{2}\sin\left(\frac{2\pi}{n}\right)$$

And if you let $$\frac{2\pi}{n}=x$$ then you wil have $$A=\pi r^2\frac{\sin(x)}{x}$$ and it is well known that $$\lim_{x\to0}\frac{sin(x)}{x}=1$$ and as x approaches 0, n approaches infinite, thus you get the area of a circle.

 

[Miike012]

Sorry, repeat was the wrong word to say... I ment to say because the polygon is indefinit so is the number pie...
So I guess my question is... is the reason why pie is indefinite is because the sides of the polygon, to which we are trying to get closer and closer to a circle, has indefinite sides?
Is there a correlation between the two?
Or not really?

 

[Apphysicist]

$$A=\frac{r^2n}{2}\sin\left(\frac{2\pi}{n}\right)$$

And if you let $$\frac{2\pi}{n}=x$$ then you wil have $$A=\pi r^2\frac{\sin(x)}{x}$$ and it is well known that $$\lim_{x\to0}\frac{sin(x)}{x}=1$$ and as x approaches 0, n approaches infinite, thus you get the area of a circle.

Oooh, very nice.

 

[Mentallic]

Well no I don't believe that's the case as to why pi is irrational. It is definitely not a proof of pi's irrationality (infinite string of decimals).

Say we took the radius to be $$\frac{1}{\sqrt{\pi}}$$, well then the area of the circle will be 1. We still did the same process of taking infinitely many sides of the n-gon approximation.

Or even another way to look at it is that with a circle radius 1, for certain polygons the approximated value switches between rational and irrational values. For n=3 (an equilateral triangle) the area will be $$\frac{3\sqrt{3}}{4}$$ while for n=4 the area is 2. For n=5 the area is irrational again until n=12. After that it's back to irrational and will stay that way.